diff --git a/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf b/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3d0c0a63dc3b6f4469a880e0f641b8e779b1d9fb Binary files /dev/null and b/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf differ diff --git a/-NAyT4oBgHgl3EQf3flQ/content/tmp_files/2301.00769v1.pdf.txt b/-NAyT4oBgHgl3EQf3flQ/content/tmp_files/2301.00769v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..78b48d51ddb288e182fb7eaa2871376c1aa40057 --- /dev/null +++ b/-NAyT4oBgHgl3EQf3flQ/content/tmp_files/2301.00769v1.pdf.txt @@ -0,0 +1,197 @@ +arXiv:2301.00769v1 [math.AP] 2 Jan 2023 +SHARP NORM ESTIMATES FOR THE CLASSICAL HEAT EQUATION +ERIK TALVILA +Abstract. Sharp estimates of solutions of the classical heat equation are proved in Lp +norms on the real line. +1. Introduction +In this paper we give sharp estimates of solutions of the classical heat equation on the +real line with initial value data that is in an Lp space (1 ≤ p ≤ ∞). +For u:R × (0, ∞) → R write ut(x) = u(x, t). +The classical problem of the heat equation on the real line is, given a function f ∈ Lp +for some 1 ≤ p ≤ ∞, find a function u : R × (0, ∞) → R such that ut ∈ C2(R) for each +t > 0, u(x, ·) ∈ C1((0, ∞)) for each x ∈ R and +∂2u(x, t) +∂x2 +− ∂u(x, t) +∂t += 0 for each (x, t) ∈ R × (0, ∞) +(1.1) +lim +t→0+∥ut − f∥p = 0. +(1.2) +If p = ∞ then f is also assumed to be continuous. +A solution is given by the convolution ut(x) = F ∗Θt(x) = +� ∞ +−∞ F(x−y)Θt(y) dy where +the Gauss–Weierstrass heat kernel is Θt(x) = exp(−x2/(4t))/(2 +√ +πt). For example, see +[4]. Under suitable growth conditions on u the solution is unique. See [5] and [9]. Refer- +ences [3] and [9] contain many results on the classical heat equation, including extensive +bibliographies. +The heat kernel has the following properties. +Let t > 0 and let s ̸= 0 such that +1/s + 1/t > 0. Then +Θt ∗ Θs = Θt+s +(1.3) +∥Θt∥q = +αq +t(1−1/q)/2 where αq = + + + +1, +q = 1 +1 +(2√π)1−1/q q1/(2q) , +1 < q < ∞ +1 +2√π, +q = ∞. +(1.4) +The last of these follows from the probability integral +� ∞ +−∞ e−x2 dx = √π. +Theorem 1.1. Let 1 ≤ p ≤ ∞ and f ∈ Lp. +(a) If p ≤ s ≤ ∞ then f ∗ Θt ∈ Ls. +(b) Let q, r ∈ [1, ∞] such that 1/p + 1/q = 1 + 1/r. There is a constant Kp,q such that +∥f ∗ Θt∥r ≤ Kp,q∥f∥p t−(1−1/q)/2 for all t > 0. The estimate is sharp in the sense that if +ψ : (0, ∞) → (0, ∞) such that ψ(t) = o(t−(1−1/q)/2) as t → 0+ or t → ∞ then there is +Date: Preprint January 2, 2023. +2020 Mathematics Subject Classification. Primary 35K05, 46E30; Secondary 26A42. +Key words and phrases. Heat equation, Lebesgue space. +1 + +2 +ERIK TALVILA +G ∈ Lp such that ∥G ∗ Θt∥r/ψ(t) is not bounded as t → 0+ or t → ∞. The constant +Kp,q = (cpcq/cr)1/2αq, where cp = p1/p/(p′)1/p′ with p, p′ being conjugate exponents. It +cannot be replaced with any smaller number. +(c) If 1 ≤ s < p then f ∗ Θt need not be in Ls. +When r = p and q = 1 the inequality in part (b) reads ∥f ∗ Θt∥p ≤ ∥f∥p. When r = ∞ +then p and q are conjugates and the inequality in part (b) reads ∥f ∗Θt∥∞ ≤ ∥f∥pt−1/(2p). +The condition for sharpness in Young’s inequality is that both functions be Gaussians. +This fact is exploited in the proof of part (b). See [7, p. 99], [2] and [8]. Our proof also +uses ideas from [5, Theorem 9.2, p. 195] and [1, pp. 115-120]. +The estimates are known, for example [6, Proposition 3.1], but we have not been able +to find a proof in the literature that they are sharp. +Proof. (a), (b) Young’s inequality gives +(1.5) +∥f ∗ Θt∥r ≤ Cp,q∥f∥p∥Θt∥q = Cp,q∥f∥pαq +t(1−1/q)/2 , +where αq is given in (1.4). The sharp constant, given in [7, p. 99], is Cp,q = (cpcq/cr)1/2 +where cp = p1/p/(p′)1/p′ with p, p′ being conjugate exponents. Note that c1 = c∞ = 1. +Also, 0 < Cp,q ≤ 1. We then take Kp,q = Cp,qαq. +To show the estimate ∥ut∥r = O(t−(1−1/q)/2) is sharp as t → 0+ and t → ∞, let ψ be as +in the statement of the theorem. Fix p ≤ r ≤ ∞. Define the family of linear operators St: +Lp → Lr by St[f](x) = f ∗ Θt(x)/ψ(t). The estimate ∥St[f]∥r ≤ Kp,q∥f∥pt−(1−1/q)/2/ψ(t) +shows that, for each t > 0, St is a bounded linear operator. Let ft = Θt. Then, from (1.3) +and (1.4), +∥St[ft]∥r +∥ft∥p += ∥Θt ∗ Θt∥r +ψ(t)∥Θt∥p += +∥Θ2t∥r +ψ(t)∥Θt∥p += +αr +αp2(1−1/r)/2ψ(t)t(1−1/q)/2 . +This is not bounded in the limit t → 0+. Hence, St is not uniformly bounded. By the +Uniform Bounded Principle it is not pointwise bounded. Therefore, there is a function +f ∈ Lp such that ∥f ∗ Θt∥r ̸= O(ψ(t)) as t → 0+. And, the growth estimate ∥f ∗ Θt∥r = +O(t−(1−1/q)/2)) as t → 0+ is sharp. Similarly for sharpness as t → ∞. +Now show the constant Kp,q cannot be reduced. A calculation shows we have equality +in (1.5) when f = Θβ +t and β is given by the equation +(1.6) +β1−1/q +(β + 1)1−1/r = cpcq +cr +�αpαq +αr +�2 += +� +1 − 1 +p +�1−1/p � +1 − 1 +q +�1−1/q � +1 − 1 +r +�−(1−1/r) +. +First consider the case p ̸= 1 and q ̸= 1. Notice that 1 − 1/r = (1 − 1/q) + (1 − 1/p) > +1 − 1/q. Let g(x) = xA(x + 1)−B with B > A > 0. Then g is strictly increasing on +(0, A/(B − A)) and strictly decreasing for x > A/(B − A) so there is a unique maximum +for g at A/(B − A). Put A = 1 − 1/q and B = 1 − 1/r. Then +g +� +A +B − A +� += +β1−1/q +(β + 1)1−1/r = +� +1 − 1 +p +�1−1/p � +1 − 1 +q +�1−1/q � +1 − 1 +r +�−(1−1/r) +. +Hence, (1.6) has a unique positive solution for β given by β = (1 − 1/q)/(1 − 1/p). + +HEAT EQUATION +3 +If p = 1 then q = r. In this case, (1.6) reduces to (1 + 1/β)1−1/q = 1 and the solution +is given in the limit β → ∞. Sharpness of (1.5) is then given in this limit. It can also be +seen that taking f to be the Dirac distribution gives equality. +If q = 1 then p = r. Now, (1.6) reduces to (β +1)1−1/p = 1 and β = 0. There is equality +in (1.5) when f = 1. This must be done in the limit β → 0+. +If p = q = r = 1 then there is equality in (1.5) for each β > 0. +Hence, the constant in (1.5) is sharp. +(c) Suppose f ≥ 0 and f is decreasing on [c, ∞) for some c ∈ R. Let x > c. Then +f ∗ Θt(x) +≥ +� x +c +f(y)Θt(x − y) dy ≥ f(x) +� x +c +Θt(x − y) dy += +f(x) +√π +� (x−c)/(2 +√ +t) +0 +e−y2 dy ∼ f(x)/2 +as x → ∞. +Now put f(x) = 1/[x1/p log2(x)] for x ≥ e and f(x) = 0, otherwise. For p = ∞ replace +x1/p by 1. +□ +References +[1] S. Axler, P. Bourdon and W. Ramey, Harmonic function theory, New York, Springer-Verlag, 2001. +[2] W. Beckner, Inequalities in Fourier analysis, Ann. of Math. (2) 102(1975), 159–182. +[3] J.R. Cannon, The one-dimensional heat equation, Menlo Park, Addison–Wesley, 1984. +[4] G.B. Folland, Introduction to partial differential equations, Princeton, Princeton University Press, +1995. +[5] I.I. Hirschman and D.V. Widder, The convolution transform, Princeton, Princeton University Press, +1955. +[6] T. Iwabuchi, T. Matsuyama and K. Taniguchi, Boundedness of spectral multipliers for Schr¨odinger +operators on open sets, Rev. Mat. Iberoam. 34(2018), 1277–1322. +[7] E.H. Lieb and M. Loss, Analysis, Providence, American Mathematical Society, 2001. +[8] G. Toscani, Heat equation and the sharp Young’s inequality, arXiv:1204.2086 (2012). +[9] D.V. Widder, The heat equation, New York, Academic Press, 1975. +Department of Mathematics & Statistics, University of the Fraser Valley, Abbots- +ford, BC Canada V2S 7M8 +Email address: Erik.Talvila@ufv.ca + diff --git a/-NAyT4oBgHgl3EQf3flQ/content/tmp_files/load_file.txt b/-NAyT4oBgHgl3EQf3flQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bfa7c556ccbbbe28f5d4a0fb4ebd8f6bb5cc373c --- /dev/null +++ b/-NAyT4oBgHgl3EQf3flQ/content/tmp_files/load_file.txt @@ -0,0 +1,146 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf,len=145 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='00769v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='AP] 2 Jan 2023 SHARP NORM ESTIMATES FOR THE CLASSICAL HEAT EQUATION ERIK TALVILA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Sharp estimates of solutions of the classical heat equation are proved in Lp norms on the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Introduction In this paper we give sharp estimates of solutions of the classical heat equation on the real line with initial value data that is in an Lp space (1 ≤ p ≤ ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' For u:R × (0, ∞) → R write ut(x) = u(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' The classical problem of the heat equation on the real line is, given a function f ∈ Lp for some 1 ≤ p ≤ ∞, find a function u : R × (0, ∞) → R such that ut ∈ C2(R) for each t > 0, u(x, ·) ∈ C1((0, ∞)) for each x ∈ R and ∂2u(x, t) ∂x2 − ∂u(x, t) ∂t = 0 for each (x, t) ∈ R × (0, ∞) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='1) lim t→0+∥ut − f∥p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='2) If p = ∞ then f is also assumed to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' A solution is given by the convolution ut(x) = F ∗Θt(x) = � ∞ −∞ F(x−y)Θt(y) dy where the Gauss–Weierstrass heat kernel is Θt(x) = exp(−x2/(4t))/(2 √ πt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' For example, see [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Under suitable growth conditions on u the solution is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' See [5] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Refer- ences [3] and [9] contain many results on the classical heat equation, including extensive bibliographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' The heat kernel has the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Let t > 0 and let s ̸= 0 such that 1/s + 1/t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Then Θt ∗ Θs = Θt+s (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='3) ∥Θt∥q = αq t(1−1/q)/2 where αq = \uf8f1 \uf8f2 \uf8f3 1, q = 1 1 (2√π)1−1/q q1/(2q) , 1 < q < ∞ 1 2√π, q = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='4) The last of these follows from the probability integral � ∞ −∞ e−x2 dx = √π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Let 1 ≤ p ≤ ∞ and f ∈ Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' (a) If p ≤ s ≤ ∞ then f ∗ Θt ∈ Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' (b) Let q, r ∈ [1, ∞] such that 1/p + 1/q = 1 + 1/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' There is a constant Kp,q such that ∥f ∗ Θt∥r ≤ Kp,q∥f∥p t−(1−1/q)/2 for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' The estimate is sharp in the sense that if ψ : (0, ∞) → (0, ∞) such that ψ(t) = o(t−(1−1/q)/2) as t → 0+ or t → ∞ then there is Date: Preprint January 2, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Primary 35K05, 46E30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Secondary 26A42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Heat equation, Lebesgue space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' 1 2 ERIK TALVILA G ∈ Lp such that ∥G ∗ Θt∥r/ψ(t) is not bounded as t → 0+ or t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' The constant Kp,q = (cpcq/cr)1/2αq, where cp = p1/p/(p′)1/p′ with p, p′ being conjugate exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' It cannot be replaced with any smaller number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' (c) If 1 ≤ s < p then f ∗ Θt need not be in Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' When r = p and q = 1 the inequality in part (b) reads ∥f ∗ Θt∥p ≤ ∥f∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' When r = ∞ then p and q are conjugates and the inequality in part (b) reads ∥f ∗Θt∥∞ ≤ ∥f∥pt−1/(2p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' The condition for sharpness in Young’s inequality is that both functions be Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' This fact is exploited in the proof of part (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' See [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' 99], [2] and [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Our proof also uses ideas from [5, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' 195] and [1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' 115-120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' The estimates are known, for example [6, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='1], but we have not been able to find a proof in the literature that they are sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' (a), (b) Young’s inequality gives (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='5) ∥f ∗ Θt∥r ≤ Cp,q∥f∥p∥Θt∥q = Cp,q∥f∥pαq t(1−1/q)/2 , where αq is given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' The sharp constant, given in [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' 99], is Cp,q = (cpcq/cr)1/2 where cp = p1/p/(p′)1/p′ with p, p′ being conjugate exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Note that c1 = c∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Also, 0 < Cp,q ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' We then take Kp,q = Cp,qαq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' To show the estimate ∥ut∥r = O(t−(1−1/q)/2) is sharp as t → 0+ and t → ∞, let ψ be as in the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Fix p ≤ r ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Define the family of linear operators St: Lp → Lr by St[f](x) = f ∗ Θt(x)/ψ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' The estimate ∥St[f]∥r ≤ Kp,q∥f∥pt−(1−1/q)/2/ψ(t) shows that, for each t > 0, St is a bounded linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Let ft = Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Then, from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='4), ∥St[ft]∥r ∥ft∥p = ∥Θt ∗ Θt∥r ψ(t)∥Θt∥p = ∥Θ2t∥r ψ(t)∥Θt∥p = αr αp2(1−1/r)/2ψ(t)t(1−1/q)/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' This is not bounded in the limit t → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Hence, St is not uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' By the Uniform Bounded Principle it is not pointwise bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Therefore, there is a function f ∈ Lp such that ∥f ∗ Θt∥r ̸= O(ψ(t)) as t → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' And, the growth estimate ∥f ∗ Θt∥r = O(t−(1−1/q)/2)) as t → 0+ is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Similarly for sharpness as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Now show the constant Kp,q cannot be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' A calculation shows we have equality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='5) when f = Θβ t and β is given by the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='6) β1−1/q (β + 1)1−1/r = cpcq cr �αpαq αr �2 = � 1 − 1 p �1−1/p � 1 − 1 q �1−1/q � 1 − 1 r �−(1−1/r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' First consider the case p ̸= 1 and q ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Notice that 1 − 1/r = (1 − 1/q) + (1 − 1/p) > 1 − 1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Let g(x) = xA(x + 1)−B with B > A > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Then g is strictly increasing on (0, A/(B − A)) and strictly decreasing for x > A/(B − A) so there is a unique maximum for g at A/(B − A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Put A = 1 − 1/q and B = 1 − 1/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Then g � A B − A � = β1−1/q (β + 1)1−1/r = � 1 − 1 p �1−1/p � 1 − 1 q �1−1/q � 1 − 1 r �−(1−1/r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Hence, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='6) has a unique positive solution for β given by β = (1 − 1/q)/(1 − 1/p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' HEAT EQUATION 3 If p = 1 then q = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' In this case, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='6) reduces to (1 + 1/β)1−1/q = 1 and the solution is given in the limit β → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Sharpness of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='5) is then given in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' It can also be seen that taking f to be the Dirac distribution gives equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' If q = 1 then p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Now, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='6) reduces to (β +1)1−1/p = 1 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' There is equality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='5) when f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' This must be done in the limit β → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' If p = q = r = 1 then there is equality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='5) for each β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Hence, the constant in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='5) is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' (c) Suppose f ≥ 0 and f is decreasing on [c, ∞) for some c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Let x > c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Then f ∗ Θt(x) ≥ � x c f(y)Θt(x − y) dy ≥ f(x) � x c Θt(x − y) dy = f(x) √π � (x−c)/(2 √ t) 0 e−y2 dy ∼ f(x)/2 as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Now put f(x) = 1/[x1/p log2(x)] for x ≥ e and f(x) = 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' For p = ∞ replace x1/p by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' □ References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Axler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Bourdon and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Ramey, Harmonic function theory, New York, Springer-Verlag, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Beckner, Inequalities in Fourier analysis, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' (2) 102(1975), 159–182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Cannon, The one-dimensional heat equation, Menlo Park, Addison–Wesley, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Folland, Introduction to partial differential equations, Princeton, Princeton University Press, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' [5] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Hirschman and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Widder, The convolution transform, Princeton, Princeton University Press, 1955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' [6] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Iwabuchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Matsuyama and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Taniguchi, Boundedness of spectral multipliers for Schr¨odinger operators on open sets, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Iberoam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' 34(2018), 1277–1322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Lieb and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Loss, Analysis, Providence, American Mathematical Society, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Toscani, Heat equation and the sharp Young’s inequality, arXiv:1204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='2086 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Widder, The heat equation, New York, Academic Press, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content=' Department of Mathematics & Statistics, University of the Fraser Valley, Abbots- ford, BC Canada V2S 7M8 Email address: Erik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='Talvila@ufv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} +page_content='ca' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQf3flQ/content/2301.00769v1.pdf'} diff --git a/.gitattributes b/.gitattributes index 63a114bcd03de2356859379819dfd4e7f14662bd..538e2e881f9933dac1d204e731e0dcc9c2e0b6e4 100644 --- a/.gitattributes +++ b/.gitattributes @@ -1947,3 +1947,57 @@ ctFAT4oBgHgl3EQf6B4X/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex WdAzT4oBgHgl3EQfmP2O/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text 29A0T4oBgHgl3EQfNP8E/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text 6dA0T4oBgHgl3EQfN__p/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text 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P. McCandless∗ and V. Protasenko +School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA +D. Rowe and N. Pieczulewski +Department of Material Science and Engineering, Cornell University, Ithaca, New York 14853, USA +M. Alonso-Orts, M. S. Williams, and M. Eickhoff +Institute of Solid-State Physics, University Bremen, Otto-Hahn-Allee 1, 28359 Bremen, Germany +H. G. Xing +School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA +Department of Material Science and Engineering, Cornell University, Ithaca, New York 14853, USA and +Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, New York 14853, USA +D. A. Muller +Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, New York 14853, USA and +School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA +D. Jena +Department of Material Science and Engineering, Cornell University, Ithaca, New York 14853, USA +School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA and +Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, New York 14853, USA +P. Vogt +Department of Material Science and Engineering, Cornell University, Ithaca, New York 14853, USA and +Institute of Solid-State Physics, University Bremen, Otto-Hahn-Allee 1, 28359 Bremen, Germany +We report the growth of α-Ga2O3 on 𝑚-plane α-Al2O3 by conventional plasma-assisted molecular-beam epitaxy +(MBE) and In-mediated metal-oxide-catalyzed epitaxy (MOCATAXY). We report a growth-rate-diagram for +α-Ga2O3(10¯10), and observe (i) a growth rate increase, (ii) an expanded growth window, and (iii) reduced out-of- +lane mosaic spread when MOCATAXY is employed for the growth of α-Ga2O3. Through the use of In-mediated +catalysis, growth rates over 0.2 μm hr−1 and rocking curves with full width at half maxima of ∆𝜔 ≈ 0.45◦ are +achieved. Faceting is observed along the α-Ga2O3film surface and is explored through scanning transmission +electron microscopy. +INTRODUCTION +Over the past decade, Ga2O3 has gained much attention as +a wide-band gap semiconductor. Monoclinic β-Ga2O3 pos- +sesses an ultra-wide bang gap of ∼ 4.7 eV [1], and it has +been the most studied phase owing to its thermal stability +and the availability of large-area, native, semi-insulating and +conductive substrates [2, 3]. +To further increase its band +gap β-Ga2O3 can be alloyed with Al to form β-(Al,Ga)2O3, +but achieving high Al content has remained challenging due +to the tendency to have phase segregation [4]. In contrast, +α-(Al,Ga)2O3 becomes more stable as the Al is increased be- +cause the crystal is isostructural with the α-Al2O3 substrate, +and the lattice mismatch is reduced as the Al concentration +is increased [5]. This has enabled the entire compositional +range of α-(Al,Ga)2O3 to be readily achieved [5, 6], and it has +enabled band gaps exceeding those of AlN, BN, or diamond +[7, 8]. +With the recent advances enabling α-Ga2O3 to remain stable +during high-temperature anneals [9], the next challenge is +to achieve electrical conductivity. To date, conductivity in +α-Ga2O3 has been achieved by chemical vapor deposition +(CVD) [10, 11], but has remained elusive for films grown by +molecular-beam epitaxy (MBE). Additionally, conductive β- +Ga2O3 films grown by MBE on α-Al2O3 have yet to be +achieved [7]. While the exact reasons these films remain +insulating are unknown, the thermodynamics during MBE +growth and the low formation energy of defects may cause +these Ga2O3 films on Al2O3 to be insulating. +Multiple compensating point defects (e.g., cation vacancies, +oxygen vacancies, donor impurities [12, 13]) and extended +crystallographic defects (e.g., rotational domains and thread- +ing dislocations[14]) occur within the Ga2O3 films grown +on sapphire. +Reasons for the emergence of these defects +include the lattice mismatch between the film and the sub- +strate [14], and the growth regime in which the films are +grown [12, 13, 15]. For example, Ga vacancies (𝑉Ga) may +act as compensating acceptors for introduced 𝑛-type dopants +arXiv:2301.13053v1 [cond-mat.mtrl-sci] 30 Jan 2023 + +2 +in grown Ga2O3 thin films [15]. +O-rich growth environ- +ments are likely to generate a significant amount of 𝑉Ga (due +to their low formation energy) whereas Ga-rich growth en- +vironments are found to significantly suppress the formation +of 𝑉Ga (due to their high formation energy) [13]. Thus, the +growth of Ga2O3 in the highly Ga-rich regime—accessed by +new epitaxial growth concepts [16]—may improve the trans- +port properties of Ga2O3 thin films since the Ga-rich growth +regimes lead to higher 𝑉Ga formation energies, resulting in +lower 𝑉Ga densities within the Ga2O3 layers. +One approach to address these issues is through the use of +metal-oxide-catalyzed epitaxy (MOCATAXY) [17]. This is +a growth process where a catalytic element (e.g. In) is in- +troduced to the growth system and results in metal-exchange +catalysis [18]. This growth mode has been observed for β- +(Al,Ga)2O3 on different substrates and surface orientations, +as well as for different epitaxial growth techniques [19–23]. +Many benefits arise from using MOCATAXY during the +growth of Ga2O3. For example: (i) It can improve the surface +morphologies of β-Ga2O3-based films [20]. (ii) The synthe- +sis of Ga2O3 can occur in previously inaccessible kinetic and +thermodynamic growth regimes (e.g. in highly metal-rich +regimes) which can be beneficial for the suppression of un- +desired point (such as𝑉Ga) defects in Ga2O3 [12, 13, 18]. (iii) +The formation of thermodynamically unstable Ga2O3 phases +becomes energetically favorable [16, 18, 23], e.g., the for- +mation of the ϵ/κ-phase of Ga2O3, which has enabled novel +ϵ/κ-Ga2O3-based heterostructures [22]. (iv) The growth rate +(𝛤), possible growth temperatures (𝑇G), and crystalline qual- +ity of β-(Al,Ga)2O3-based thin films can be vastly enhanced +[17]. +In this work, we introduce the growth of α-Ga2O3 by MO- +CATAXY, resulting in an expansion of the α-Ga2O3 growth +window combined with an increased 𝛤 and an improvement +in its out-of-plane mosaic spread. It is the first demonstration +of a catalytic growth process during the growth of α-Ga2O3. +EXPERIMENTAL +Samples were grown in a Veeco GEN930 plasma MBE sys- +tem with standard Ga and In effusion cells. For all samples, +the substrates were cleaned in acetone and isopropanol for 10 +minutes and the α-Ga2O3 samples were grown for 60 minutes. +The growth temperature (𝑇G) was measured by a thermocou- +ple located within the substrate heater. The Ga flux (𝜙Ga) +and In flux (𝜙In) were monitored by beam equivalent pres- +sure (BEP) chamber readings. For conventional MBE and +MOCATAXY, the O2 flux (𝜙O) was measured in standard +cubic centimeters per minute (SCCM) and a radio-frequency +plasma power of 250 W was employed during all growths. To +convert the measured values of 𝜙Ga (BEP), 𝜙In (BEP), and 𝜙O +(SCCM) into units of nm−2 s−1 conversion factors are taken +from Ref. [24]. Note, when using In-mediated catalysis, the +available 𝜙O for Ga to Ga2O3 oxidation is about 2.8 times +larger than for Ga oxidation in the absence of In [16, 18]. +TABLE I. Collected growth parameters used in this work, val- +ues of 𝜙Ga, 𝜙In, 𝜙O, and 𝑇G, for samples grown by conven- +tional MBE and MOCATAX are listed. +The conversion for +𝜙Ga and 𝜙In from BEP to nm min−1 to nm−2 s−1 are 𝜙Ga = +2.5 × 10−8 Torr ˆ= 1.1 nm min−1 ˆ= 1 nm−2 s−1 and 𝜙In = 1.1 × +10−7 Torr ˆ= 2.6 nm−2 s−1, respectively. +Growth parameters +Conventional MBE +MOCATAXY +𝜙Ga (nm−2 s−1) +0.8 – 2.0 +1.1 – 5.5 +𝜙In (nm−2 s−1) +0 +2.6 – 2.8 +𝜙O (SCCM) +1.4 +0.7 – 1.0 +𝜙O (nm−2 s−1) +2.2 +3.2 – 4.6 +𝑇G (◦C) +640 – 800 +680 +For samples grown by conventional MBE and MOCATAXY, +the impact of 𝜙Ga and 𝑇G is studied. In the case of MO- +CATAXY growth, the impact of 𝜙In is also investigated. All +growth parameters used in this work are collected in Table I. +For scanning transmission electron microscopy (STEM), +samples were prepared using Thermo Fisher Helios G4 UX +Focused Ion Beam with a final milling step of 5 keV to +minimize damage. Carbon and Au-Pd layers were sputtered +to reduce charging during sample preparation. Carbon and +platinum protective layers were also deposited to minimize +ion-beam damage. STEM measurements were taken with +an aberration-corrected Thermo Fisher Spectra 300 CFEG +operated at 300 keV. +RESULTS AND DISCUSSION +Figure 1 shows the growth-rate-diagram of α-Ga2O3(10¯10) +grown on α-Al2O3(10¯10) by conventional MBE (the gray +shaded area) and MOCATAXY (the purple shaded area). For +conventionally grown samples two distinct growth regimes +are observed: (i) the O-rich regime where O adsorbates are in +excess over Ga adsorbates (i.e., the Ga flux limited regime), +and (ii) the 𝛤-plateau regime (i.e., the Ga2O desorption lim- +ited regime). The O-rich regime is characterized by an in- +creasing 𝛤 with increasing 𝜙Ga, whereas the plateau regime +is characterized by a constant 𝛤, being independent of 𝜙Ga. +Within this regime, however, 𝛤 may decrease with increasing +𝑇G (see inset in Fig. 1) as the desorption of the volatile sub- +oxide Ga2O becomes thermally more active [27]. The data in +the inset of Fig. 1 plot 𝛤 as a function of 𝑇G: (i) for α-Ga2O3 +grown the O-rich regime and (ii) for α-Ga2O3 grown in the +𝛤-plateau regime. +To expand the accessible growth window of α-Ga2O3 to +higher 𝜙Ga and higher 𝑇G, combined with increased 𝛤 and +improved crystalline quality, In-mediated catalysis was em- +ployed to the formation of α-Ga2O3 [18]. The red stars in +Fig. 1 show the resulting 𝛤 as a function of 𝜙Ga at con- +stant 𝑇G. +The gray shaded and purple shaded areas in +Fig. 1 depict model-based descriptions of 𝛤 for α-Ga2O3 +grown by conventional MBE and MOCATAXY, respec- + +3 +1.6 +1.4 +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +Growth Rate, Γ (nm +-2s +-1) +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +Growth Rate, Γ (nm/min) +10 +8 +6 +4 +2 +0 +Ga Flux, φGa (nm +-2s +-1) +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +Ga Flux, φGa (10 +7× torr) +1.6 +1.2 +0.8 +0.4 +0.0 +Growth Rate (nm/min) +800 +750 +700 +650 +Growth Temp(ºC) +MOCATAXY +T = 680°C +Conventional +T = 680°C +1.2 +0.8 +0.4 +0.0 +Growth Rate (nm/min) +800 +750 +700 +650 +Growth Temp(ºC) +1.6 +1.2 +0.8 +0.4 +0.0 +Growth Rate (nm/min) +800 +750 +700 +650 +Growth Temp(ºC) +Fig 1: Shows the growth rate (Γ) of α-Ga O (10-10) grown on α-Al O (10-10) as a function of the +1.6 +1.2 +0.8 +0.4 +0.0 +Growth Rate (nm/min) +800 +750 +700 +650 +Growth Temp(ºC) +φGa = 0. 9 +φGa = 1.6 +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +Growth Rate (nm/min) +10 +8 +6 +4 +2 +0 +Ga Flux (atoms/nm +2s) +1.4 +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +Growth Rate (atoms/nm +2s) +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +Ga BEP (torr)×10 +-7 +MOCATAXY +T = 680°C +Conventional +T = 680°C +(i) φGa = 0. 9 +(ii) φGa = 1.6 +(i) +(ii) +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +Growth Rate (nm/min) +10 +8 +6 +4 +2 +0 +Ga Flux (atoms/nm +2s) +1.4 +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +Growth Rate (atoms/nm +2s) +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +Ga BEP (torr)×10 +-7 +1.6 +1.2 +0.8 +0.4 +0.0 +Γ (nm/min) +800 +750 +700 +650 +TG (ºC) +MOCATAXY +Conventional +Ga Flux, φGa (nm-2 s-1) +FIG. 1. +Growth-rate-diagram of α-Ga2O3(10¯10) grown on α- +Al2O3(10¯10). The growth rate 𝛤 as a function of 𝜙Ga at 𝑇G = +680 ◦C is plotted for the growth by conventional MBE (blue trian- +gles) and MOCATAXY (red stars). The 𝛤-data is fit by a 𝛤-model +taken from Ref. [25]. The gray shaded region shows the parameter +space under which the formation of α-Ga2O3by conventional MBE +may occur. The purple shaded area depicts the growth regime of +α-Ga2O3assisted by MOCATAXY. Both fitted data sets were ob- +tained at constant 𝑇G and 𝜙O (values given in Table I). Inset: 𝛤 as +a function of 𝑇G at two different fluxes of (i) 𝜙Ga = 0.9 nm−2 s−1 +(the O-rich regime, solid squares) and (ii) 𝜙Ga = 1.6 nm−2 s−1 (the +𝛤-plateau regime, solid discs). A growth-rate-diagram of α-Ga2O3 +as a function of 𝜙O is given in Ref. [26]. +tively. The maximum 𝛤 obtained for each growth technique +is 𝛤 ≈ 1.5 nm min−1 and 𝛤 ≈ 3.3 nm min−1, respectively. +Using MOCATAXY, a more than 2-times increase in 𝛤 for +α-Ga2O3 at given growth conditions, as well as a shift far +into the adsorption-controlled regime (i.e, far into the Ga rich +flux regime) is observed. This direct comparison between +the two growth types clearly shows the expanded growth +window made possible with MOCATAXY, for example, en- +abling 𝛤 ≈ 1.8 nm min−1 for α-Ga2O3 at 𝜙Ga = 5.5 nm−2 s−1. +In contrast, at these growth conditions, no growth of α-Ga2O3 +is obtained by conventional MBE. The catalytic effect on 𝛤 +of α-Ga2O3 is modeled as a function of 𝜙O within the sup- +plemental section [26]. We note that the depicted models +use arbitrary kinetic parameters, based on kinetic parameters +extracted for the growth of β-Ga2O3 [25]. +To describe the growth of α-Ga2O3 by MOCATAXY, 𝜙O +is scaled by a factor of 2.8 compared with the growth of +α-Ga2O3 by conventional MBE. This additional O comes +from the catalytic nature of In forming a catalytic adlayer +(𝐴) with O adsorbates, e.g., 𝐴 = In–O, which provides more +active O for the Ga to α-Ga2O3 oxidation. In other words, +𝐴 increases the reaction probability of Ga with O on the +respective growth surface, facilitating the formation of the +final Ga2O3 compound at much higher 𝜙Ga and 𝑇G, which +-1 +0 +1 +ω(º) +10 +-1 +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +7 +10 +8 +Intensity (a.u.) +68 +67 +66 +65 +64 +2θ-ω (°) +jpm432@cornell.edu +3 +ɑ-Ga2O3 with MOCATAXY +Conventional +MOCATAXY +Fig. 2 Longitudinal XRD scans recorded for optimized Ga2O3 films grown on α-Al2O3(10 +by conventional MBE and MOCATAXY. The reflections of the films coincide with the +Ga2O3(10-10) phase grown by conventional MBE (the black trace) and MOCATAXY (the blue +trace). The Ga2O3 films were grown at !Ga … and TG = …, respectively, where an O flux +… was provided. The growth rates and surface morphologies of both films are shown in Figs +1(?) and 2(?) and depicted as … and … +MOCATAXY AFM +(a) +(d) +(e) +Conventional +Rq = 0.64 nm +MOCATAXY +Rq = 0.96nm +Best individual AFM and Best individual XRD RC +Conventional AFM +X-ray Intensity (arb. unit) +30$30 +ɑ-Ga2O3 +30$30 +ɑ-Al2O3 +1.92 in = 10um +2μm +2μm +Δω = 0.55° +(b) +(c) Δω = 0.45° +X-ray Intensity (arb. unit) +FIG. 2. (a) Longitudinal XRD scans of optimized α-Ga2O3 films. +The reflections of the films coincide with the α-Ga2O3(10¯10) +phase grown by conventional MBE (the blue trace) and MO- +CATAXY (the red trace). The used growth parameters were 𝜙Ga += 2.9 nm−2 s−1, 𝜙O = 1.4 SCCM ˆ= 2.2 nm−2 s−1, and 𝑇G = 750 ◦C +(conventional MBE), and 𝜙Ga = 2.9 nm−2 s−1, 𝜙In = 2.8 nm−2 s−1, +𝜙O = 0.7 SCCM ˆ= 3.2 nm−2 s−1, and 𝑇G = 680 ◦C (MOCATAXY). +(b) and (c) Transverse XRD scans across the 30¯30 peak with +their FWHM of ∆𝜔 = 0.55◦ (conventionally MBE-grown) and +∆𝜔 = 0.45◦ (MOCATAXY-grown). These obtained ∆𝜔 are de- +picted in Fig. 3 at given 𝜙Ga and 𝑇G. (d) and (e) Surface morpholo- +gies obtained by 10 × 10 μm AFM scans for α-Ga2O3(10¯10) sur- +faces grown by conventional MBE and MOCATAXY, respectively. +Growth conditions for the samples plotted in (d) and (e) are the same +as for the ones plotted in panels (a)–(c), except a slightly lower 𝑇G= +730 ◦C used for the conventionally grown sample and a slightly +higher supplied 𝜙O = 1.0 SCCM for the MOCATAXY grown film. +This resulted in ∆𝜔 = 0.61◦ and 𝛤 ≈ 1.2 nm min−1 for the conven- +tionally grown sample, and ∆𝜔 = 0.48◦ and 𝛤 > 3.0 nm min−1 for +the MOCATAXY grown sample. +enables excellent crystal quality [16, 18]. We further note +that the same factor of 2.8 was needed for modeling the +MOCATAXY growth of β-Ga2O3 on different substrates and +different surface orientations [16, 18]. We note, however, that +for a quantitative extraction of all kinetic growth parameters +more 𝛤-studies of α-Ga2O3 are needed and are beyond the +scope of this work. Nevertheless, the models help validate +the 𝛤-data and provide insight into the growth regimes and +growth mechanisms of α-Ga2O3. +For example, once 𝜙Ga +exceeds the active O flux, i.e., for 𝜙Ga > 𝜙O, the growth +will enter the Ga-rich regime and 𝛤 will start to decrease, as +shown by the gray shaded area in Fig. 1. Thus, this is the first +direct indication that the growth of α-Ga2O3 is limited by the +formation and subsequent desorption of Ga2O, like what is +observed for β-Ga2O3 grown by conventional MBE [25]. + +oμm +2 +4 +6 +8 +10.0nm +oum +8.0 +N +6.0 +4.0 +2.0 +0.04 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +Rocking Curve FWHM (º) +760 +720 +680 +640 +Growth Temperature (ºC) +6 +5 +4 +3 +2 +1 +0 +Ga Flux (atms/nm +2s) +6 +5 +4 +3 +2 +1 +0 +Ga Flux (atms/nm +2s) +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +Rocking Curve FWHM (º) +760 +720 +680 +640 +Growth Temperature (ºC) +Fig. 3: (a) Full width at half maxima (FWHM) as a function of the growth temperature (TG), obtained by +transverse XRD scans across the 30-30 peaks of Ga2O3 [e.g., see Fig. 3]. (b) The root means square (rms) +roughnesses as a function of TG; measured by AFM [see Fig. ? or supplement ... ]. Three distinct growth regimes +are indicated in panels (a) and (b): (i) the O-rich rich regime (depicted as squares), (ii) the adsorption-controlled +regime (depicted as stars), and the MOCATAXY regime (depicted as triangles). Panels (c) and (d) show the +impact of the Ga flux on the FWHM of the 30-30 peak and rms roughnesses, respectively, of the grown α- +Ga2O3(10-10) thin films. +(a) +%-Plateau Regime +MOCATAXY +3500 +3000 +2500 +2000 +1500 +1000 +500 +0 +Rocking Curve FWHM (arcsec.) +760 +720 +680 +640 +Growth Temperature (ºC) +O-rich Regime +Tsub = 680°C +φGa = 0.95 (O-rich) +φGa = 1.24 (%-Plateau) +φGa = 5.5 (MOCATAXY) +Ga Flux, φGa (nm-2 s-1) +TG (°C) +(b) +FIG. 3. (a) and (b) FWHM (i.e., ∆𝜔 values) are plotted as a function +of 𝑇G and 𝜙Ga are plotted, respectively. Values are obtained by +transverse XRD scans of the 30¯30 peaks of α-Ga2O3 grown films +(XRD data not shown). Three distinct growth regimes are studied +in panels (a) and (b): (i) the O-rich rich regime (blue squares), (ii) +the 𝛤-plateau regime (green circles), and (iii) the MOCATAXY +regime (red stars). The lowest value of ∆𝜔 is indicated by a dashed +line. Note that for the samples grown by MOCATAXY, 𝜙In = (2.6 – +2.8) nm−2 s−1 was supplied and might explain the slight variations +observed in ∆𝜔 for α-Ga2O3 grown at 𝜙Ga = 2.9 nm−2 s−1 in panel +(b)]. +Figure 2 directly compares the impact of both MBE growth +techniques on the structural quality of the epitaxially grown +films. In Fig. 2 (a), 2𝜃-𝜔 XRD scans of two selected α- +Ga2O3 films are shown, one grown by conventional MBE +(depicted as the blue trace) and one grown by MOCATAXY +(depicted as the red trace). The reflections of the films co- +incide with the α-Ga2O3 30¯30 peak. This, along with the +absence of other diffraction peaks, indicates phase-pure α- +Ga2O3(10¯10) with In incorporation of < 1% in the grown +α-Ga2O3 layers, similar to what is observed for β-(Al,Ga)2O3 +grown by MOCATAXY [17]. Fig. 2(b) and 2(c) plot trans- +verse scans (rocking curves) for the conventional MBE and +MOCATXY grown α-Ga2O3 samples as plotted in Fig. 2(a). +The rocking curves are measured across the symmetric 30¯30 +peak. The full width at half maxima (FWHM) of 𝜔 quan- +tifies the out-of-plane mosaic spread of the α-Ga2O3 film. +For conventionally grown films the out-of-plane crystal dis- +tribution is ∆𝜔 ≈ 0.55◦ and for MOCATAXY grown films +it is ∆𝜔 ≈ 0.45◦. The film thicknesses 𝑑 of the conven- +tionally and MOCATAXY grown films are 𝑑 = 73 nm and +𝑑 = 127 nm, respectively. Jinno et al., reported that α-Ga2O3 +films are fully relaxed for 𝑑 > 60 nm [5]. Since lattice mis- +match and relaxation are not impacted by MOCATAXY, it is +noteworthy that despite the MOCATAXY film being thicker, +∆𝜔 is substantially smaller compared to what is obtained by +conventional growth. The same MOCATAXY grown sample +shown here is studied by STEM and shown in Fig. 4. +Surface morphologies and root mean square roughnesses +(𝑅q) are measured by AFM and depicted in Figs. 2(d) and +2(e). The best surface roughness for conventionally grown α- +Ga2O3with 𝑑 = 66 nm is 𝑅q = 0.64 nm, while the smoothest +one for MOCATAXY grown samples with 𝑑 ∼ 270 nm has +an 𝑅q = 0.94 nm. The larger surface roughness for the MO- +CATAXY grown sample is likely due to facetting on the top +surface of the α-Ga2O3 thin film [see Fig. 4(a)]. We specu- +late that In does not only act as a catalyst but also acts as a +surface active agent (surfactant) for the growth α-Ga2O3 thin +films. It is widely understood that In can act as a surfactant +for the epitaxial growth of GaN-based films [28], and has +also been observed during the growth of β-Ga2O3 [20] and +β-(Al, Ga)2O3 [29]. +Depending on the growth conditions +and growth surface, which can affect the surface diffusion ki- +netics, surface chemical potentials, and the assessed growth +mode, the suppression of facetting may be accomplished +through the use of optimized conditions using In as a surfac- +tant, enabling a modification in the surface free energies of +the growing α-Ga2O3 thin film and a change in its growth +mode [17, 20, 30, 31]. However, surfactant-induced mor- +phological phase-transitions from 2-dimensional (2D) layer +growth to 3-dimensional (3D) island growth have also been +observed during MBE growth [32]. We believe that a simi- +lar effect occurs for the α-Ga2O3 surfaces studied here when +In may act as an (anti)surfactant during the growth of these +films. Note, we have not fully explored all growth regimes +made accessible through MOCATAXY in this study. Further +studies may lead to additional improvements in the crystalline +quality and surface morphologies of the α-Ga2O3 thin films. +In Figs. 3(a) and 3(b), the impact of 𝜙Ga and 𝑇G, respectively, +on ∆𝜔 for samples grown by conventional MBE in the O- +rich regime (blue squares) and in the 𝛤-plateau regime (green +circles), as well as for samples grown by MOCATAXY (red +stars), are shown. XRD data and ∆𝜔 are obtained by the +same methods as described above for Fig. 2. Within the O- +rich regime at 𝑇G = 640 ◦C, a large ∆𝜔 is observed, Fig. 3(a). +At higher growth temperatures (i.e. 𝑇G ≥ 660 ◦C), ∆𝜔 are +similar (or slightly improving) with increasing temperature, +regardless of growth regime. We speculate that the reason +the crystal quality improves with 𝑇G, is that there is an in- +crease in the kinetic energy and a subsequent increase in the +diffusion length of the adsorbates, allowing the Ga and O +to reach the proper lattice site. However, if 𝑇G is increased +too much, a decrease in the surface lifetime of Ga adsorbates +may occur, resulting in a reduction in the crystalline quality +of the growing thin films. Using MOCATAXY in the Ga-rich +regime and fixed 𝑇G, excess Ga may now reduce the needed +surface diffusion length, improving the crystalline quality of +the obtained α-Ga2O3 layers. More studies to separate the +effects of 𝜙Ga and 𝑇G on ∆𝜔 need to be performed, but to the +best of our knowledge, the obtained ∆𝜔 values are the lowest +reported in the literature for α-Ga2O3 grown on α-Al2O3. +Finally, to directly quantify and identify how MOCATAXY +affects the crystal structure of α-Ga2O3 thin films, high-angle +annular dark-field STEM (HAADF-STEM) was performed +along the < 0¯110 > zone axis, and is plotted in Fig. 4. The +sample shown here is the same as the one shown in Fig. 2(c). +In Fig. 4(a), a clear contrast differentiates the sapphire sub- +strate, the epitaxial film (α-Ga2O3), and the protective Au-Pd + +5 +Fig 4. HAADF-STEM images showing an overview of Alpha-Ga2O3 film grown on m-plane sapphire. A) The film shows thickness of ___nm with +faceting on film surface. Line defects are running from the interface to surface on average every ___nm. Increased brightness at the interface as a result +of scattering shows high density of defects. B) Enlarged image of interface show presence of defect due to strain relaxation. +25 nm +200 nm +5 nm +(a) +(b) +!→ +!→ +(c) +2 0 2 4 +1 0 1 10 +(1011) +[0001] +[0110] +Ga2O3 +Al2O3 +Al2O3 +Ga2O3 +FIG. 4. HAADF-STEM images show an overview of an α-Ga2O3(10¯10) film grown on α-Al2O3(10¯10). (a) The epitaxial film shows +increased contrast due to misfit dislocations at the film/substrate interface. Threading dislocation propagate through the film and terminating +at the intersection of its surface periodic faceting. (b) Enlarged image of the film-substrate interface (i.e., the α-Al2O3-α-Ga2O3 interface) +is shown. Burger circuits are drawn around the edge dislocations. (c) Fast Fourier transform (FFT) of the interface region is shown. +Diffraction peak separation at (20¯2¯4) and (10¯110) indicate strain relaxation of the α-Ga2O3(10¯10) on α-Al2O3(10¯10). +sputtered coating. The bright contrast observed at the sub- +strate/film interface (see Fig. 4(b) and Ref. [26]) is due to +additional scattering of the electron beam and indicates the +presence of misfit dislocations. These dislocations arise due +to the film relaxation caused by strain. A subset of the ob- +served misfit dislocations propagate and lead to threading +dislocations. From the contrast variation observed within +the film [see Fig. 4(a)], an average frequency of one thread- +ing dislocation every 30 nm laterally along the film/substrate +interface is observed. While more investigation is needed to +determine the cause of the faceting and verify the above hy- +pothesis (e.g., due to the changed growth mode when using +In-mediated catalysis), it is observed that the threading dislo- +cations can merge and then continue to propagate toward the +film surface. These dislocations terminate at the bottom of +intersecting surface planes, where faceting along the (10¯11) +plane is observed. The complimentary facet is unidentified +since the facet is not perpendicular to the beam and tilts out +of plane. This tilting is detected in Fig. 4(a) by the fading of +contrast along the surface, in contrast to the sharp change in +contrast on the (10¯11) plane. +Figure 4(b) shows an enlarged image of the film/substrate +interface. +A pair of edge dislocations is observed and is +highlighted with their Burgers circuits. This edge dislocation +pair is observed along the film/substrate interface, and its +dislocation density is estimated to be 5 × 105 cm−1 (or ∼ +1011 cm−2), i.e., occurring every 20 nm. This is similar to +what is reported by conventional MBE [5]. To quantify Al/Ga +inter-diffusion at the interface, a line scan (see S-Fig. 2 [26]) +was performed to quantify the contrast change. An interface +width of 𝜎 ≈ 0.9 nm was measured from an error function +fitted to the Al intensity line scan profile (see S-Fig. 2 [26]). +A fast Fourier transform (FFT), of the interface region shown +in Fig. 4(b), is displayed in Fig. 4(c). A thin film completely +strained to the substrate will show a singular diffraction peak. +However, when the film relaxes its interplanar spacing 𝑑ℎ𝑘𝑙 +changes, resulting in an additional peak, shifted from the sub- +strate peak. However, shifted peaks in the in-plane direction +are not visible because the α-Ga2O3 (000¯6) reflection peak is +approximately 10x weaker than in α-Al2O3. The strain relax- +ation is observed in the 20¯2¯4 and 10¯110 diffraction peaks of +α-Ga2O3. The strain relaxation is accomplished by the for- +mation of edge dislocations at the interface, where the 20¯2¯4 +peak is correlated to the yellow Burgers circuit and the 10¯110 +peak to the cyan Burgers circuit. In addition, no phase separa- +tion or secondary phases were observed by STEM within the +α-Ga2O3 film grown by MOCATAXY. However, a bi-layer +structure from overlapping α-Ga2O3 grains when viewed in +projection is observed with a slip along the [10¯2¯2] direction +(see S-Fig. 3 [26]). The presence of this bi-layer structure +indicates that the film is not single-crystalline. The bi-layer +structure was confirmed using an ab initio TEM (abTEM) +simulation [33] which produced a matching HAADF image +from the crystallographic information framework. +This TEM investigation of MOCATAXY grown α-Ga2O3 +shows comparable crystal quality to what is measured for +conventional MBE [5] with regards to edge dislocation den- +sity and phase purity. We note that the difference in pro- +jection direction may have prevented imaging of the bi-layer +structure in this previous report. No faceting of α-Ga2O3 +was observed by conventional MBE when grown on 𝑚-plane +α-Al2O3 [5, 9]. +CONCLUSION +Phase-pure α-Ga2O3(10¯10) on α-Al2O3(10¯10) was grown +using conventional MBE and MOCATAXY with thickness +up to 𝑑 = 262 nm. We mapped out the 𝛤-dependence on 𝜙Ga +and 𝑇G and its impact on the crystalline quality and surface +morphologies. We identified and explored previously inac- +cessible growth regimes by MOCATAXY, and showed that + +6 +it vastly extends the growth regime and improves the out- +of-plane mosaic spread of the grown α-Ga2O3 films. Using +In-mediated catalysis, we observe facetting on top of the α- +Ga2O3(10¯10) layers. This study confirms that this new MBE +growth mode can be applied to the growth of α-Ga2O3– and +is not limited to the growth of the β-Ga2O3 and β-(Al,Ga)2O3 +polymorphs. We emphasize more studies are needed to de- +termine the kinetic parameters that form α-Ga2O3 during +conventional MBE and MOCATAXY growth, as well as to +further improve the quality of the grown α-Ga2O3/α-Al2O3 +heterostructures, and to understand the mechanisms leading +to the surface faceting of α-Ga2O3. +ACKNOWLEDGEMENTS +This research is supported by the Air Force Research +Laboratory-Cornell Center for Epitaxial Solutions (AC- +CESS), monitored by Dr. Ali Sayir (FA9550-18-1-0529). +JPM acknowledges the support of a National Science Foun- +dation Graduate Research Fellowship under Grant No. +DGE–2139899. +M. A-O acknowledges financial support +from the Central Research Development Fund (CRDF) of the +University of Bremen. 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Speck, Metal +oxide catalyzed epitaxy (MOCATAXY) of β-Ga2O3 films in +various orientations grown by plasma-assisted molecular beam +epitaxy, APL Materials 8, 021104 (2020). +[21] P. Mazzolini, A. Falkenstein, C. Wouters, R. Schewski, +T. Markurt, Z. Galazka, M. Martin, M. Albrecht, and O. Bier- + +7 +wagen, Substrate-orientation dependence of β-Ga2O3 (100), +(010), (001), and (¯201) homoepitaxy by indium-mediated +metal-exchange catalyzed molecular beam epitaxy (MEXCAT- +MBE), APL Materials 8, 011107 (2020). +[22] Y. Kuang, X. Chen, T. Ma, Q. Du, Y. Zhang, J. Hao, F. F. +Ren, B. Liu, S. Zhu, S. Gu, R. Zhang, Y. Zheng, and J. Ye, +Band Alignment and Enhanced Interfacial Conductivity Ma- +nipulated by Polarization in a Surfactant-Mediated Grown +κ-Ga2O3/In2O3 Heterostructure, ACS Applied Electronic Ma- +terials 3, 795 (2021). +[23] M. Kracht, A. Karg, J. Schörmann, M. Weinhold, D. Zink, +F. Michel, M. Rohnke, M. Schowalter, B. Gerken, A. Rose- +nauer, P. J. Klar, J. Janek, and M. Eickhoff, Tin-Assisted Syn- +thesis of ϵ-Ga2O3 by Molecular Beam Epitaxy, Physical Re- +view Applied 8, 054002 (2017). +[24] P. Vogt, Growth Kinetics , Thermodynamics , and Phase For- +mation of group-III and IV oxides during Molecular Beam +Epitaxy, Ph.D. thesis (2017). +[25] P. Vogt and O. Bierwagen, Quantitative subcompound- +mediated reaction model for the molecular beam epitaxy of +III-VI and IV-VI thin films: Applied to Ga2O3 ,In2O3, and +SnO2, Physical Review Materials 2, 1 (2018). +[26] See Supplemental Material at [URL will be inserted by pub- +lisher] for a model of 𝛤 as a function of 𝜙O at 𝑇G = 680 ◦C +(S-Fig. 1) as well as images obtained by STEM (S-Fig. 2 and +S-Fig. 3) and an included crystallographic model. +[27] P. Vogt and O. Bierwagen, Reaction kinetics and growth win- +dow for plasma-assisted molecular beam epitaxy of 𝐺𝑎2𝑂3: +Incorporation of Ga vs. 𝐺𝑎2𝑂 desorption, Applied Physics +Letters 108, 072101 (2016). +[28] J. Neugebauer, T. K. Zywietz, M. Scheffler, J. E. Northrup, +H. Chen, and R. M. Feenstra, Adatom Kinetics On and Be- +low the Surface: The Existence of a New Diffusion Channel, +Physical Review Letters 90, 1 (2003). +[29] P. Vogt, A. Mauze, F. Wu, B. Bonef, and J. S. Speck, +Metal-oxide-catalyzed epitaxy (MOCATAXY): the exam- +ple of O plasma-assisted molecular beam epitaxy of +𝛽-(𝐴𝑙𝑥𝐺𝑎1−𝑥)2𝑂3/𝛽 − 𝐺𝑎2𝑂3 heterostructures, Applied +Physics Express 11, 115503 (2018). +[30] M. Copel, M. Reuter, E. Kaxiras, and M. Tromp, Surfactants +in Epitaxial Growth, Physical Review Letters 63, 632 (1989). +[31] J. Neugebauer, Surfactants and antisurfactants on group-III- +nitride surfaces, Physica Status Solidi C: Conferences 0, 1651 +(2003). +[32] R. B. Lewis, P. Corfdir, H. Li, J. Herranz, C. Pfuller, O. Brandt, +and L. Geelhaar, Quantum Dot Self-Assembly Driven by a +Surfactant-Induced Morphological Instability, Physical Re- +view Letters 119, 1 (2017). +[33] J. Madsen and T. Susi, The abtem code: transmission electron +microscopy from first principles., Open Research Europe 1, +13015 (2021). + diff --git a/0dFPT4oBgHgl3EQfTjRk/content/tmp_files/load_file.txt b/0dFPT4oBgHgl3EQfTjRk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fc9c49429b9f62c5ddbf0636e5805a925df8daee --- /dev/null +++ b/0dFPT4oBgHgl3EQfTjRk/content/tmp_files/load_file.txt @@ -0,0 +1,754 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf,len=753 +page_content='Growth of α-Ga2O3 on α-Al2O3 by conventional molecular-beam epitaxy and metal-oxide-catalyzed epitaxy J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' McCandless∗ and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Protasenko School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Rowe and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Pieczulewski Department of Material Science and Engineering, Cornell University, Ithaca, New York 14853, USA M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Alonso-Orts, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Williams, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Eickhoff Institute of Solid-State Physics, University Bremen, Otto-Hahn-Allee 1, 28359 Bremen, Germany H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Xing School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA Department of Material Science and Engineering, Cornell University, Ithaca, New York 14853, USA and Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, New York 14853, USA D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Muller Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, New York 14853, USA and School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Jena Department of Material Science and Engineering, Cornell University, Ithaca, New York 14853, USA School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14853, USA and Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, New York 14853, USA P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Vogt Department of Material Science and Engineering, Cornell University, Ithaca, New York 14853, USA and Institute of Solid-State Physics, University Bremen, Otto-Hahn-Allee 1, 28359 Bremen, Germany We report the growth of α-Ga2O3 on 𝑚-plane α-Al2O3 by conventional plasma-assisted molecular-beam epitaxy (MBE) and In-mediated metal-oxide-catalyzed epitaxy (MOCATAXY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' We report a growth-rate-diagram for α-Ga2O3(10¯10), and observe (i) a growth rate increase, (ii) an expanded growth window, and (iii) reduced out-of- lane mosaic spread when MOCATAXY is employed for the growth of α-Ga2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Through the use of In-mediated catalysis, growth rates over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 μm hr−1 and rocking curves with full width at half maxima of ∆𝜔 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='45◦ are achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Faceting is observed along the α-Ga2O3film surface and is explored through scanning transmission electron microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' INTRODUCTION Over the past decade, Ga2O3 has gained much attention as a wide-band gap semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Monoclinic β-Ga2O3 pos- sesses an ultra-wide bang gap of ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='7 eV [1], and it has been the most studied phase owing to its thermal stability and the availability of large-area, native, semi-insulating and conductive substrates [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' To further increase its band gap β-Ga2O3 can be alloyed with Al to form β-(Al,Ga)2O3, but achieving high Al content has remained challenging due to the tendency to have phase segregation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' In contrast, α-(Al,Ga)2O3 becomes more stable as the Al is increased be- cause the crystal is isostructural with the α-Al2O3 substrate, and the lattice mismatch is reduced as the Al concentration is increased [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This has enabled the entire compositional range of α-(Al,Ga)2O3 to be readily achieved [5, 6], and it has enabled band gaps exceeding those of AlN, BN, or diamond [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' With the recent advances enabling α-Ga2O3 to remain stable during high-temperature anneals [9], the next challenge is to achieve electrical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' To date, conductivity in α-Ga2O3 has been achieved by chemical vapor deposition (CVD) [10, 11], but has remained elusive for films grown by molecular-beam epitaxy (MBE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Additionally, conductive β- Ga2O3 films grown by MBE on α-Al2O3 have yet to be achieved [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' While the exact reasons these films remain insulating are unknown, the thermodynamics during MBE growth and the low formation energy of defects may cause these Ga2O3 films on Al2O3 to be insulating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Multiple compensating point defects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=', cation vacancies, oxygen vacancies, donor impurities [12, 13]) and extended crystallographic defects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=', rotational domains and thread- ing dislocations[14]) occur within the Ga2O3 films grown on sapphire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Reasons for the emergence of these defects include the lattice mismatch between the film and the sub- strate [14], and the growth regime in which the films are grown [12, 13, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' For example, Ga vacancies (𝑉Ga) may act as compensating acceptors for introduced 𝑛-type dopants arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='13053v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='mtrl-sci] 30 Jan 2023 2 in grown Ga2O3 thin films [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' O-rich growth environ- ments are likely to generate a significant amount of 𝑉Ga (due to their low formation energy) whereas Ga-rich growth en- vironments are found to significantly suppress the formation of 𝑉Ga (due to their high formation energy) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Thus, the growth of Ga2O3 in the highly Ga-rich regime—accessed by new epitaxial growth concepts [16]—may improve the trans- port properties of Ga2O3 thin films since the Ga-rich growth regimes lead to higher 𝑉Ga formation energies, resulting in lower 𝑉Ga densities within the Ga2O3 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' One approach to address these issues is through the use of metal-oxide-catalyzed epitaxy (MOCATAXY) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This is a growth process where a catalytic element (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' In) is in- troduced to the growth system and results in metal-exchange catalysis [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This growth mode has been observed for β- (Al,Ga)2O3 on different substrates and surface orientations, as well as for different epitaxial growth techniques [19–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Many benefits arise from using MOCATAXY during the growth of Ga2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' For example: (i) It can improve the surface morphologies of β-Ga2O3-based films [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' (ii) The synthe- sis of Ga2O3 can occur in previously inaccessible kinetic and thermodynamic growth regimes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' in highly metal-rich regimes) which can be beneficial for the suppression of un- desired point (such as𝑉Ga) defects in Ga2O3 [12, 13, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' (iii) The formation of thermodynamically unstable Ga2O3 phases becomes energetically favorable [16, 18, 23], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=', the for- mation of the ϵ/κ-phase of Ga2O3, which has enabled novel ϵ/κ-Ga2O3-based heterostructures [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' (iv) The growth rate (𝛤), possible growth temperatures (𝑇G), and crystalline qual- ity of β-(Al,Ga)2O3-based thin films can be vastly enhanced [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' In this work, we introduce the growth of α-Ga2O3 by MO- CATAXY, resulting in an expansion of the α-Ga2O3 growth window combined with an increased 𝛤 and an improvement in its out-of-plane mosaic spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' It is the first demonstration of a catalytic growth process during the growth of α-Ga2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' EXPERIMENTAL Samples were grown in a Veeco GEN930 plasma MBE sys- tem with standard Ga and In effusion cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' For all samples, the substrates were cleaned in acetone and isopropanol for 10 minutes and the α-Ga2O3 samples were grown for 60 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The growth temperature (𝑇G) was measured by a thermocou- ple located within the substrate heater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The Ga flux (𝜙Ga) and In flux (𝜙In) were monitored by beam equivalent pres- sure (BEP) chamber readings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' For conventional MBE and MOCATAXY, the O2 flux (𝜙O) was measured in standard cubic centimeters per minute (SCCM) and a radio-frequency plasma power of 250 W was employed during all growths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' To convert the measured values of 𝜙Ga (BEP), 𝜙In (BEP), and 𝜙O (SCCM) into units of nm−2 s−1 conversion factors are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Note, when using In-mediated catalysis, the available 𝜙O for Ga to Ga2O3 oxidation is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 times larger than for Ga oxidation in the absence of In [16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Collected growth parameters used in this work, val- ues of 𝜙Ga, 𝜙In, 𝜙O, and 𝑇G, for samples grown by conven- tional MBE and MOCATAX are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The conversion for 𝜙Ga and 𝜙In from BEP to nm min−1 to nm−2 s−1 are 𝜙Ga = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 × 10−8 Torr ˆ= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='1 nm min−1 ˆ= 1 nm−2 s−1 and 𝜙In = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='1 × 10−7 Torr ˆ= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 nm−2 s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Growth parameters Conventional MBE MOCATAXY 𝜙Ga (nm−2 s−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='1 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 𝜙In (nm−2 s−1) 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 𝜙O (SCCM) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='7 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 𝜙O (nm−2 s−1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 𝑇G (◦C) 640 – 800 680 For samples grown by conventional MBE and MOCATAXY, the impact of 𝜙Ga and 𝑇G is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' In the case of MO- CATAXY growth, the impact of 𝜙In is also investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' All growth parameters used in this work are collected in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' For scanning transmission electron microscopy (STEM), samples were prepared using Thermo Fisher Helios G4 UX Focused Ion Beam with a final milling step of 5 keV to minimize damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Carbon and Au-Pd layers were sputtered to reduce charging during sample preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Carbon and platinum protective layers were also deposited to minimize ion-beam damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' STEM measurements were taken with an aberration-corrected Thermo Fisher Spectra 300 CFEG operated at 300 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' RESULTS AND DISCUSSION Figure 1 shows the growth-rate-diagram of α-Ga2O3(10¯10) grown on α-Al2O3(10¯10) by conventional MBE (the gray shaded area) and MOCATAXY (the purple shaded area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' For conventionally grown samples two distinct growth regimes are observed: (i) the O-rich regime where O adsorbates are in excess over Ga adsorbates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=', the Ga flux limited regime), and (ii) the 𝛤-plateau regime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=', the Ga2O desorption lim- ited regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The O-rich regime is characterized by an in- creasing 𝛤 with increasing 𝜙Ga, whereas the plateau regime is characterized by a constant 𝛤, being independent of 𝜙Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Within this regime, however, 𝛤 may decrease with increasing 𝑇G (see inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 1) as the desorption of the volatile sub- oxide Ga2O becomes thermally more active [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The data in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 1 plot 𝛤 as a function of 𝑇G: (i) for α-Ga2O3 grown the O-rich regime and (ii) for α-Ga2O3 grown in the 𝛤-plateau regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' To expand the accessible growth window of α-Ga2O3 to higher 𝜙Ga and higher 𝑇G, combined with increased 𝛤 and improved crystalline quality, In-mediated catalysis was em- ployed to the formation of α-Ga2O3 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The red stars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 1 show the resulting 𝛤 as a function of 𝜙Ga at con- stant 𝑇G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The gray shaded and purple shaded areas in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 1 depict model-based descriptions of 𝛤 for α-Ga2O3 grown by conventional MBE and MOCATAXY, respec- 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='4 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Growth Rate, Γ (nm/min) 10 8 6 4 2 0 Ga Flux, φGa (nm 2s 1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Ga Flux, φGa (10 7× torr) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Growth Rate (nm/min) 800 750 700 650 Growth Temp(ºC) MOCATAXY T = 680°C Conventional T = 680°C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Growth Rate (nm/min) 800 750 700 650 Growth Temp(ºC) Fig 1: Shows the growth rate (Γ) of α-Ga O (10-10) grown on α-Al O (10-10) as a function of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Growth Rate (nm/min) 800 750 700 650 Growth Temp(ºC) φGa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 9 φGa = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Growth Rate (nm/min) 10 8 6 4 2 0 Ga Flux (atoms/nm 2s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Growth Rate (atoms/nm 2s) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Ga BEP (torr)×10 7 MOCATAXY T = 680°C Conventional T = 680°C (i) φGa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 9 (ii) φGa = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 (i) (ii) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Growth Rate (nm/min) 10 8 6 4 2 0 Ga Flux (atoms/nm 2s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Growth Rate (atoms/nm 2s) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Ga BEP (torr)×10 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Γ (nm/min) 800 750 700 650 TG (ºC) MOCATAXY Conventional Ga Flux, φGa (nm-2 s-1) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Growth-rate-diagram of α-Ga2O3(10¯10) grown on α- Al2O3(10¯10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The growth rate 𝛤 as a function of 𝜙Ga at 𝑇G = 680 ◦C is plotted for the growth by conventional MBE (blue trian- gles) and MOCATAXY (red stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The 𝛤-data is fit by a 𝛤-model taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The gray shaded region shows the parameter space under which the formation of α-Ga2O3by conventional MBE may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The purple shaded area depicts the growth regime of α-Ga2O3assisted by MOCATAXY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Both fitted data sets were ob- tained at constant 𝑇G and 𝜙O (values given in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Inset: 𝛤 as a function of 𝑇G at two different fluxes of (i) 𝜙Ga = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='9 nm−2 s−1 (the O-rich regime, solid squares) and (ii) 𝜙Ga = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 nm−2 s−1 (the 𝛤-plateau regime, solid discs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' A growth-rate-diagram of α-Ga2O3 as a function of 𝜙O is given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The maximum 𝛤 obtained for each growth technique is 𝛤 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 nm min−1 and 𝛤 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='3 nm min−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Using MOCATAXY, a more than 2-times increase in 𝛤 for α-Ga2O3 at given growth conditions, as well as a shift far into the adsorption-controlled regime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='e, far into the Ga rich flux regime) is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This direct comparison between the two growth types clearly shows the expanded growth window made possible with MOCATAXY, for example, en- abling 𝛤 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 nm min−1 for α-Ga2O3 at 𝜙Ga = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 nm−2 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' In contrast, at these growth conditions, no growth of α-Ga2O3 is obtained by conventional MBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The catalytic effect on 𝛤 of α-Ga2O3 is modeled as a function of 𝜙O within the sup- plemental section [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' We note that the depicted models use arbitrary kinetic parameters, based on kinetic parameters extracted for the growth of β-Ga2O3 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' To describe the growth of α-Ga2O3 by MOCATAXY, 𝜙O is scaled by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 compared with the growth of α-Ga2O3 by conventional MBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This additional O comes from the catalytic nature of In forming a catalytic adlayer (𝐴) with O adsorbates, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=', 𝐴 = In–O, which provides more active O for the Ga to α-Ga2O3 oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' In other words, 𝐴 increases the reaction probability of Ga with O on the respective growth surface, facilitating the formation of the final Ga2O3 compound at much higher 𝜙Ga and 𝑇G, which 1 0 1 ω(º) 10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=') 68 67 66 65 64 2θ-ω (°) jpm432@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='edu 3 ɑ-Ga2O3 with MOCATAXY Conventional MOCATAXY Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 2 Longitudinal XRD scans recorded for optimized Ga2O3 films grown on α-Al2O3(10 by conventional MBE and MOCATAXY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The reflections of the films coincide with the Ga2O3(10-10) phase grown by conventional MBE (the black trace) and MOCATAXY (the blue trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The Ga2O3 films were grown at !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='Ga … and TG = …, respectively, where an O flux … was provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The growth rates and surface morphologies of both films are shown in Figs 1(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=') and 2(?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=') and depicted as … and … MOCATAXY AFM (a) (d) (e) Conventional Rq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='64 nm MOCATAXY Rq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='96nm Best individual AFM and Best individual XRD RC Conventional AFM X-ray Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' unit) 30$30 ɑ-Ga2O3 30$30 ɑ-Al2O3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='92 in = 10um 2μm 2μm Δω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='55° (b) (c) Δω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='45° X-ray Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' unit) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' (a) Longitudinal XRD scans of optimized α-Ga2O3 films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The reflections of the films coincide with the α-Ga2O3(10¯10) phase grown by conventional MBE (the blue trace) and MO- CATAXY (the red trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The used growth parameters were 𝜙Ga = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='9 nm−2 s−1, 𝜙O = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='4 SCCM ˆ= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 nm−2 s−1, and 𝑇G = 750 ◦C (conventional MBE), and 𝜙Ga = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='9 nm−2 s−1, 𝜙In = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 nm−2 s−1, 𝜙O = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='7 SCCM ˆ= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 nm−2 s−1, and 𝑇G = 680 ◦C (MOCATAXY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' (b) and (c) Transverse XRD scans across the 30¯30 peak with their FWHM of ∆𝜔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='55◦ (conventionally MBE-grown) and ∆𝜔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='45◦ (MOCATAXY-grown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' These obtained ∆𝜔 are de- picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 3 at given 𝜙Ga and 𝑇G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' (d) and (e) Surface morpholo- gies obtained by 10 × 10 μm AFM scans for α-Ga2O3(10¯10) sur- faces grown by conventional MBE and MOCATAXY, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Growth conditions for the samples plotted in (d) and (e) are the same as for the ones plotted in panels (a)–(c), except a slightly lower 𝑇G= 730 ◦C used for the conventionally grown sample and a slightly higher supplied 𝜙O = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 SCCM for the MOCATAXY grown film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This resulted in ∆𝜔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='61◦ and 𝛤 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 nm min−1 for the conven- tionally grown sample, and ∆𝜔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='48◦ and 𝛤 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 nm min−1 for the MOCATAXY grown sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' enables excellent crystal quality [16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' We further note that the same factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 was needed for modeling the MOCATAXY growth of β-Ga2O3 on different substrates and different surface orientations [16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' We note, however, that for a quantitative extraction of all kinetic growth parameters more 𝛤-studies of α-Ga2O3 are needed and are beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Nevertheless, the models help validate the 𝛤-data and provide insight into the growth regimes and growth mechanisms of α-Ga2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' For example, once 𝜙Ga exceeds the active O flux, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=', for 𝜙Ga > 𝜙O, the growth will enter the Ga-rich regime and 𝛤 will start to decrease, as shown by the gray shaded area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Thus, this is the first direct indication that the growth of α-Ga2O3 is limited by the formation and subsequent desorption of Ga2O, like what is observed for β-Ga2O3 grown by conventional MBE [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' oμm 2 4 6 8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0nm oum 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Rocking Curve FWHM (º) 760 720 680 640 Growth Temperature (ºC) 6 5 4 3 2 1 0 Ga Flux (atms/nm 2s) 6 5 4 3 2 1 0 Ga Flux (atms/nm 2s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='0 Rocking Curve FWHM (º) 760 720 680 640 Growth Temperature (ºC) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 3: (a) Full width at half maxima (FWHM) as a function of the growth temperature (TG), obtained by transverse XRD scans across the 30-30 peaks of Ga2O3 [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=', see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' (b) The root means square (rms) roughnesses as a function of TG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' measured by AFM [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' or supplement .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Three distinct growth regimes are indicated in panels (a) and (b): (i) the O-rich rich regime (depicted as squares), (ii) the adsorption-controlled regime (depicted as stars), and the MOCATAXY regime (depicted as triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Panels (c) and (d) show the impact of the Ga flux on the FWHM of the 30-30 peak and rms roughnesses, respectively, of the grown α- Ga2O3(10-10) thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' (a) %-Plateau Regime MOCATAXY 3500 3000 2500 2000 1500 1000 500 0 Rocking Curve FWHM (arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=') 760 720 680 640 Growth Temperature (ºC) O-rich Regime Tsub = 680°C φGa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='95 (O-rich) φGa = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='24 (%-Plateau) φGa = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='5 (MOCATAXY) Ga Flux, φGa (nm-2 s-1) TG (°C) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' (a) and (b) FWHM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=', ∆𝜔 values) are plotted as a function of 𝑇G and 𝜙Ga are plotted, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Values are obtained by transverse XRD scans of the 30¯30 peaks of α-Ga2O3 grown films (XRD data not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Three distinct growth regimes are studied in panels (a) and (b): (i) the O-rich rich regime (blue squares), (ii) the 𝛤-plateau regime (green circles), and (iii) the MOCATAXY regime (red stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The lowest value of ∆𝜔 is indicated by a dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Note that for the samples grown by MOCATAXY, 𝜙In = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='6 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='8) nm−2 s−1 was supplied and might explain the slight variations observed in ∆𝜔 for α-Ga2O3 grown at 𝜙Ga = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='9 nm−2 s−1 in panel (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Figure 2 directly compares the impact of both MBE growth techniques on the structural quality of the epitaxially grown films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 2 (a), 2𝜃-𝜔 XRD scans of two selected α- Ga2O3 films are shown, one grown by conventional MBE (depicted as the blue trace) and one grown by MOCATAXY (depicted as the red trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The reflections of the films co- incide with the α-Ga2O3 30¯30 peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This, along with the absence of other diffraction peaks, indicates phase-pure α- Ga2O3(10¯10) with In incorporation of < 1% in the grown α-Ga2O3 layers, similar to what is observed for β-(Al,Ga)2O3 grown by MOCATAXY [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 2(b) and 2(c) plot trans- verse scans (rocking curves) for the conventional MBE and MOCATXY grown α-Ga2O3 samples as plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The rocking curves are measured across the symmetric 30¯30 peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The full width at half maxima (FWHM) of 𝜔 quan- tifies the out-of-plane mosaic spread of the α-Ga2O3 film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' For conventionally grown films the out-of-plane crystal dis- tribution is ∆𝜔 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='55◦ and for MOCATAXY grown films it is ∆𝜔 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The film thicknesses 𝑑 of the conven- tionally and MOCATAXY grown films are 𝑑 = 73 nm and 𝑑 = 127 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Jinno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=', reported that α-Ga2O3 films are fully relaxed for 𝑑 > 60 nm [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Since lattice mis- match and relaxation are not impacted by MOCATAXY, it is noteworthy that despite the MOCATAXY film being thicker, ∆𝜔 is substantially smaller compared to what is obtained by conventional growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The same MOCATAXY grown sample shown here is studied by STEM and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Surface morphologies and root mean square roughnesses (𝑅q) are measured by AFM and depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 2(d) and 2(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The best surface roughness for conventionally grown α- Ga2O3with 𝑑 = 66 nm is 𝑅q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='64 nm, while the smoothest one for MOCATAXY grown samples with 𝑑 ∼ 270 nm has an 𝑅q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='94 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The larger surface roughness for the MO- CATAXY grown sample is likely due to facetting on the top surface of the α-Ga2O3 thin film [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 4(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' We specu- late that In does not only act as a catalyst but also acts as a surface active agent (surfactant) for the growth α-Ga2O3 thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' It is widely understood that In can act as a surfactant for the epitaxial growth of GaN-based films [28], and has also been observed during the growth of β-Ga2O3 [20] and β-(Al, Ga)2O3 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Depending on the growth conditions and growth surface, which can affect the surface diffusion ki- netics, surface chemical potentials, and the assessed growth mode, the suppression of facetting may be accomplished through the use of optimized conditions using In as a surfac- tant, enabling a modification in the surface free energies of the growing α-Ga2O3 thin film and a change in its growth mode [17, 20, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' However, surfactant-induced mor- phological phase-transitions from 2-dimensional (2D) layer growth to 3-dimensional (3D) island growth have also been observed during MBE growth [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' We believe that a simi- lar effect occurs for the α-Ga2O3 surfaces studied here when In may act as an (anti)surfactant during the growth of these films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Note, we have not fully explored all growth regimes made accessible through MOCATAXY in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Further studies may lead to additional improvements in the crystalline quality and surface morphologies of the α-Ga2O3 thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 3(a) and 3(b), the impact of 𝜙Ga and 𝑇G, respectively, on ∆𝜔 for samples grown by conventional MBE in the O- rich regime (blue squares) and in the 𝛤-plateau regime (green circles), as well as for samples grown by MOCATAXY (red stars), are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' XRD data and ∆𝜔 are obtained by the same methods as described above for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Within the O- rich regime at 𝑇G = 640 ◦C, a large ∆𝜔 is observed, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' At higher growth temperatures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 𝑇G ≥ 660 ◦C), ∆𝜔 are similar (or slightly improving) with increasing temperature, regardless of growth regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' We speculate that the reason the crystal quality improves with 𝑇G, is that there is an in- crease in the kinetic energy and a subsequent increase in the diffusion length of the adsorbates, allowing the Ga and O to reach the proper lattice site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' However, if 𝑇G is increased too much, a decrease in the surface lifetime of Ga adsorbates may occur, resulting in a reduction in the crystalline quality of the growing thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Using MOCATAXY in the Ga-rich regime and fixed 𝑇G, excess Ga may now reduce the needed surface diffusion length, improving the crystalline quality of the obtained α-Ga2O3 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' More studies to separate the effects of 𝜙Ga and 𝑇G on ∆𝜔 need to be performed, but to the best of our knowledge, the obtained ∆𝜔 values are the lowest reported in the literature for α-Ga2O3 grown on α-Al2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Finally, to directly quantify and identify how MOCATAXY affects the crystal structure of α-Ga2O3 thin films, high-angle annular dark-field STEM (HAADF-STEM) was performed along the < 0¯110 > zone axis, and is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The sample shown here is the same as the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 4(a), a clear contrast differentiates the sapphire sub- strate, the epitaxial film (α-Ga2O3), and the protective Au-Pd 5 Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' HAADF-STEM images showing an overview of Alpha-Ga2O3 film grown on m-plane sapphire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' A) The film shows thickness of ___nm with faceting on film surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Line defects are running from the interface to surface on average every ___nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Increased brightness at the interface as a result of scattering shows high density of defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' B) Enlarged image of interface show presence of defect due to strain relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 25 nm 200 nm 5 nm (a) (b) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='→ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='→ (c) 2 0 2 4 1 0 1 10 (1011) [0001] [0110] Ga2O3 Al2O3 Al2O3 Ga2O3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' HAADF-STEM images show an overview of an α-Ga2O3(10¯10) film grown on α-Al2O3(10¯10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' (a) The epitaxial film shows increased contrast due to misfit dislocations at the film/substrate interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Threading dislocation propagate through the film and terminating at the intersection of its surface periodic faceting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' (b) Enlarged image of the film-substrate interface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=', the α-Al2O3-α-Ga2O3 interface) is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Burger circuits are drawn around the edge dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' (c) Fast Fourier transform (FFT) of the interface region is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Diffraction peak separation at (20¯2¯4) and (10¯110) indicate strain relaxation of the α-Ga2O3(10¯10) on α-Al2O3(10¯10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' sputtered coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The bright contrast observed at the sub- strate/film interface (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 4(b) and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' [26]) is due to additional scattering of the electron beam and indicates the presence of misfit dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' These dislocations arise due to the film relaxation caused by strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' A subset of the ob- served misfit dislocations propagate and lead to threading dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' From the contrast variation observed within the film [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 4(a)], an average frequency of one thread- ing dislocation every 30 nm laterally along the film/substrate interface is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' While more investigation is needed to determine the cause of the faceting and verify the above hy- pothesis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=', due to the changed growth mode when using In-mediated catalysis), it is observed that the threading dislo- cations can merge and then continue to propagate toward the film surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' These dislocations terminate at the bottom of intersecting surface planes, where faceting along the (10¯11) plane is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The complimentary facet is unidentified since the facet is not perpendicular to the beam and tilts out of plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This tilting is detected in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 4(a) by the fading of contrast along the surface, in contrast to the sharp change in contrast on the (10¯11) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Figure 4(b) shows an enlarged image of the film/substrate interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' A pair of edge dislocations is observed and is highlighted with their Burgers circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This edge dislocation pair is observed along the film/substrate interface, and its dislocation density is estimated to be 5 × 105 cm−1 (or ∼ 1011 cm−2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=', occurring every 20 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This is similar to what is reported by conventional MBE [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' To quantify Al/Ga inter-diffusion at the interface, a line scan (see S-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 2 [26]) was performed to quantify the contrast change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' An interface width of 𝜎 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='9 nm was measured from an error function fitted to the Al intensity line scan profile (see S-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 2 [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' A fast Fourier transform (FFT), of the interface region shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 4(b), is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' A thin film completely strained to the substrate will show a singular diffraction peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' However, when the film relaxes its interplanar spacing 𝑑ℎ𝑘𝑙 changes, resulting in an additional peak, shifted from the sub- strate peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' However, shifted peaks in the in-plane direction are not visible because the α-Ga2O3 (000¯6) reflection peak is approximately 10x weaker than in α-Al2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The strain relax- ation is observed in the 20¯2¯4 and 10¯110 diffraction peaks of α-Ga2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The strain relaxation is accomplished by the for- mation of edge dislocations at the interface, where the 20¯2¯4 peak is correlated to the yellow Burgers circuit and the 10¯110 peak to the cyan Burgers circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' In addition, no phase separa- tion or secondary phases were observed by STEM within the α-Ga2O3 film grown by MOCATAXY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' However, a bi-layer structure from overlapping α-Ga2O3 grains when viewed in projection is observed with a slip along the [10¯2¯2] direction (see S-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' 3 [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The presence of this bi-layer structure indicates that the film is not single-crystalline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' The bi-layer structure was confirmed using an ab initio TEM (abTEM) simulation [33] which produced a matching HAADF image from the crystallographic information framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This TEM investigation of MOCATAXY grown α-Ga2O3 shows comparable crystal quality to what is measured for conventional MBE [5] with regards to edge dislocation den- sity and phase purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' We note that the difference in pro- jection direction may have prevented imaging of the bi-layer structure in this previous report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' No faceting of α-Ga2O3 was observed by conventional MBE when grown on 𝑚-plane α-Al2O3 [5, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' CONCLUSION Phase-pure α-Ga2O3(10¯10) on α-Al2O3(10¯10) was grown using conventional MBE and MOCATAXY with thickness up to 𝑑 = 262 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' We mapped out the 𝛤-dependence on 𝜙Ga and 𝑇G and its impact on the crystalline quality and surface morphologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' We identified and explored previously inac- cessible growth regimes by MOCATAXY, and showed that 6 it vastly extends the growth regime and improves the out- of-plane mosaic spread of the grown α-Ga2O3 films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Using In-mediated catalysis, we observe facetting on top of the α- Ga2O3(10¯10) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This study confirms that this new MBE growth mode can be applied to the growth of α-Ga2O3– and is not limited to the growth of the β-Ga2O3 and β-(Al,Ga)2O3 polymorphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' We emphasize more studies are needed to de- termine the kinetic parameters that form α-Ga2O3 during conventional MBE and MOCATAXY growth, as well as to further improve the quality of the grown α-Ga2O3/α-Al2O3 heterostructures, and to understand the mechanisms leading to the surface faceting of α-Ga2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This research is supported by the Air Force Research Laboratory-Cornell Center for Epitaxial Solutions (AC- CESS), monitored by Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' Ali Sayir (FA9550-18-1-0529).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' JPM acknowledges the support of a National Science Foun- dation Graduate Research Fellowship under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' DGE–2139899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' A-O acknowledges financial support from the Central Research Development Fund (CRDF) of the University of Bremen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This work makes use of PARADIM under Cooperative Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' DMR-2039380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' This work uses the CCMR and CESI Shared Facilities partly sponsored by the NSF MRSEC program (DMR-1719875) and MRI DMR-1338010, and the Kavli Institute at Cornell (KIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content=' ∗ Electronic mail: jpm432@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFPT4oBgHgl3EQfTjRk/content/2301.13053v1.pdf'} +page_content='edu [1] H.' metadata={'source': 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Finn, Son Doan, Paul Wang, Elly W. Yang, Daniel S. Zisook + +Abstract +Objective The 2022 n2c2 NLP Challenge posed identification of social determinants of health +(SDOH) in clinical narratives. We present three systems that we developed for the challenge +and discuss the distinctive task formulation used in each of the three systems. +Materials and Methods The first system identifies target pieces of information independently +using machine learning classifiers. The second system uses a large language model (LLM) to +extract complete structured outputs per document. The third system extracts candidate +phrases using machine learning and identifies target relations with hand-crafted rules. +Results The three systems achieved F1 scores of 0.884, 0.831, and 0.663 in the Subtask A of the +Challenge, which are ranked third, seventh, and eighth among the 15 participating teams. The +review of the extraction results from our systems reveals characteristics of each approach and +those of the SODH extraction task. +Discussion Phrases and relations annotated in the task is unique and diverse, not conforming to +the conventional event extraction task. These annotations are difficult to model with limited +training data. The system that extracts information independently, ignoring the annotated +relations, achieves the highest F1 score. Meanwhile, LLM with its versatile capability achieves +the high F1 score, while respecting the annotated relations. The rule-based system tackling +relation extraction obtains the low F1 score, while it is the most explainable approach. +Conclusion The F1 scores of the three systems vary in this challenge setting, but each approach +has advantages and disadvantages in a practical application. The selection of the approach +depends not only on the F1 score but also on the requirements in the application. + + + + +Background and Significance +Clinical notes are a rich source of information, containing, among others, patient-reported +information and clinicians’ assessments that are not coded in structured records. Automated +extraction and coding of information has been widely studied 1. Among different types of +information sought in clinical notes, social determinants of health (SDOH) have gained attention +in the last several years, due to their significance on one’s health as well as to their unique +availability in clinical notes 2,3. In the Track-2 of the 2022 n2c2 NLP Challenge 4,5, extraction of +SDOH from clinical notes was posed as a shared task, and a corpus annotated with SDOH was +prepared by the challenge organizer. The availability of the annotated corpus would further +increase interests in this information extraction task and advance the technology toward real- +world applications. +This paper focuses on three information extraction systems that we developed for our +submissions of the Track-2 in the 2022 n2c2 NLP Challenge, while we defer the background of +this Challenge task and the review of related studies to the publications by the Challenge +organizers3–6. In the corpus prepared for the Challenge, types of annotated phrases are unique +and diverse. Relations to be identified among them are difficult to characterize, making the task +very different from the conventional event extraction. Each of the three systems we developed +employs a different task formulation to tackle this challenge. + +Objective +Natural language processing (NLP) technology has undergone many changes over the years, +especially in the last several years 7. New methods as well as long-standing methods have been +evaluated for different clinical NLP tasks in shared-task challenges 8,9. Besides the performance +evaluation results, the task formulation considered for each shared-task challenge has +contributed to the clinical NLP field, providing the baselines in designing an information +extraction system for the same or related task. Given these backgrounds, two objectives of this +paper are as follows: + +1. We describe three systems that we developed for the submissions for the 2022 n2c2 +NLP Challenge Track-2, which employ both recent and long-standing methods and were +ranked high among the participating systems. +2. We present three different formulations of the task that we used in our three systems +and discuss the motivation, results, and advantages and disadvantages of each +approach. + +Materials and Methods +The Subtask-A of the Track-2, in which we participated, used the Social History Annotation +Corpus (SHAC) 3. The data consisted of 1,316, 188, and 373 clinical narrative texts from MIMIC +III 10, which were released respectively as the training, development, and test set. During the +challenge period, the training and development sets were made available for the participants to +develop systems, and the test set was released for the final evaluation. All these data sets were +provided as brat annotation files, consisting of narrative text files (.txt) and corresponding +annotation files (.ann). Further information of the brat annotation tool can be found in the brat +tool paper 11 and on the brat web page 12. +In the SHAC corpus, texts are annotated with trigger phrases for five types of SDOH +(Alcohol, Tobacco, Drug, Employment, and Living Status) along with their associated argument +phrases. A subset of the argument phrases, named labeled arguments, are normalized to +predefined labels (e.g., Status is a labeled argument for Alcohol, normalized to one of the three +status values: none, current, or past). The rest of the argument phrases, named span-only +arguments, do not have labels to normalized to and are “spans only” (e.g., Duration is a span- +only argument for Alcohol, annotated for the duration of alcohol use, such as “for eight years”). +Further information of the corpus, including annotation examples, can be found in the SHAC +corpus paper and in the evaluation guideline document 3,6. The evaluation script used in the +challenge is provided by the organizer on GitHub 13. + +During our participation in the challenge, we considered three formulations of the task +and implemented three systems as described next. We did not explore the use of additional +texts or annotations or the augmentation of the provided data. + +System 1: Sentence classification and sequence labeling +There are many triggers and arguments in the current task. We observed difficult topics in NLP +are involved for their detection (e.g., phrase boundary ambiguity; nested phrase annotations; +trigger-argument across sentences; one or more annotated phrases per argument type). +However, a good fraction of triggers and arguments look easy to identify (e.g., repeatedly +annotated unambiguous phrases). Also, the evaluation metric used in the challenge is forgiving +(e.g., phrase spans are not required for the labeled argument). Considering these factors, we +convert the given task into a set of simpler tasks that can be tackled by common methods. +In this approach, an input narrative is first split into sentences using a regular expression +pattern, and then, two common methods are applied to each sentence, independently: +1. Text classification to identify sentences containing triggers and labeled arguments +2. Sequence labeling to extract triggers and span-only arguments in each sentence. +There are two key observations behind this approach. First, most of the trigger-argument +relations are within a single sentence, and there is usually at most one trigger of the same kind +within each sentence, which is also noted in the SHAC corpus paper 3. Second, phrase spans are, +in effect, not required in the evaluation of triggers and labeled arguments. That is, labeled +arguments are evaluated by the inferred label values only. Triggers are evaluated by the span, +but any overlap between the predicted span and the annotated gold span is counted. +Therefore, the trigger span is not required in effect if a long enough span is proposed. +The two observations lead to a task formulation that, for triggers and labeled +arguments, we only need to classify each sentence whether it implies a particular trigger type +or a particular labeled argument, e.g., “Does this sentence report a patient’s alcohol abuse?” or +“Does this sentence report a patient’s current alcohol abuse?” As for the span-only argument, + +the task needs to be tackled as sequence labeling, specifically the common BIO labeling of +tokens 14. A separate model is prepared for each span-only label type and for each trigger type +because phrases annotated for span-only arguments and triggers sometimes overlap each +other. After triggers and arguments are detected independently, the predictions are merged +per sentence. When an argument is predicted by any of the models, the corresponding trigger +must be present for it to be reported, and the trigger is additionally predicted, if it is not +predicted by the trigger detection methods. +For the text classification, a multi-label text classification model was trained using the +Hugging Face Transformers library 15, which is used to make binary classification for 28 targets: +5 triggers and 23 labeled arguments. The implementation follows a publicly available Jupyter +notebook example, “Fine-tune BERT for Multi-label Classification” 16. For the BERT model, +Bio_Discharge_Summary_BERT was selected 17, because it seemed to yield slightly +better performance than the other model we tested, BioBERT 18, during the development. +For sequence labeling (2), 33 models were trained also using the Hugging Face Transformers +library, each of which extracts phrases for a specific trigger and span-only label: 5 triggers and +28 span-only labels. The implementation follows the tutorial “Token classification” in the +Hugging Face Course 19. For the BERT model, between the two models tested, BioBERT was +used. + + + +Figure 1. Few-shotting GPT-J with alcohol narratives +To few-shot GPT-J for the social history extraction task, we convert the .ann format of the annotated +text into a structured table prompt that maintains the essential content but is more compact and +amenable for data capture. The word “unknown” is used in all cases where the .ann file does not +have an annotation for the given element. +Of note, all information regarding spans is eliminated in the conversion in Figure A. We create a +column labeled “Inference” to store categorical annotations. Each E line in the .ann file is translated +into a single row in the table prompt, allowing for the possibility of multiple triggers of the same +type. +Sample GPT-J generator parameters are shown in Figure B. We use “###” to hint to the model when +language generation should cease. +Figure C shows perfect matching of the model output and the gold .ann representation in prompt +format. + +A +.ann Representation +GPT-J prompt +E +A +prompt = "" +Alcohol:"drinking" +StatusTimeVal:"current" +Make a table about alcohol use in the following story. Use exact words or phrases from the story. +Status:"reports" +She reports drinking 2 alcoholic drinks per month. +Frequency:"per month" +I Alcohol I Amount I Duration I Frequency I History I Type I Status I Inference I +Amount: "2 alcoholic drinks +I drinking I 2 alcoholic drinks I unknown I per month I unknown I unknown I reports I current I +### +E +A +Make a table about alcohol use in the following story. Use exact words or phrases from the story. +Alcohol:"ETOH" +StatusTimeVal:"none" +Denies ETOH +Status:"Denies" +I Alcohol I Amount I Duration I Frequency I History I Type I Status I Inference | +I ETOH I unknown I unknown I unknown I unknown I unknown I Denies I none I +### +E: +A +Make a table about alcohol use in the following story. Use exact words or phrases from the story. +Alcohol: "alcoholic drinks" +StatusTimeVal: "current" +Four to five alcoholic drinks per night. +Amount: "Four to five alcoholic drinks" +I Alcohol I Amount I Duration I Frequency I History I Type I Status I Inference I +Status: "drinks" +I alcoholic drinks I Four to five alcoholic drinks I unknown I per night I unknown I unknown I drinks I current I +Frequency: “per night" +### +E: +Make a table about alcohol use in the following story. Use exact words or phrases from the story. +Alcohol: "Alcohol" +A +Status:"no longer drinking" +StatusTimeVal:“past" +I Alcohol I Amount I Duration I Frequency I History I Type Status I Inference I +History: "in 22 months" +I Alcohol I unknown I unknown I unknown I in 22 months I unknown I no longer drinking I past I +### +? +A +Make a table about alcohol use in the following story. Use exact words or phrases from the story. +? +h/o EtOH abuse but last drink in 2001. +I Alcohol I Amount I Duration I Frequency I History I Type I Status I Inference | +B +end_sequence="###" +generator_kwargs = ( +"max_new_tokens':100, +‘T:,d-do, +"temperature':.01, +'clean_up_tokenization_spaces':True, +'do_sample': True, +"early_stopping': True, +"return_full_text': False, +"pad_token_id':tokenizer.eos_token_id, +"eos_token_id: int(tokenizer.convert_tokens_to_ids(end_sequence) +res = generator(prompt, **generator_kwargs) +print(res) +c +:ndno +:plog +I EtOH I unknown I unknown I unknown I in 2001 I unknown I h/o I past |System 2: Fine-tuned GPT-J model +With the general availability of medium and large size language models we were curious to +explore how much of the social history information extraction task could be performed by +leveraging the knowledge encoded in an LLM as opposed to layering additional knowledge on +top. To this end, we attempted to create a system that performed minimal re-representation of +the input data in terms of supplementation with linguistic, structural, or clinical context. A toy +example of our thought process is shown in Figure 1. Here we illustrate how few-shotting GPT-J +with four brief alcohol related narratives allows the model to correctly generate annotations for +an unknown example. Figure 1A demonstrates how we sandwich each narrative with a natural +language prompt above and a desired table format below. For the few-shot examples we +include data from the E lines (“Event” annotation in brat, i.e., a tuple of phrases) and A lines +(“Attribute” annotation in brat, i.e., a label value assigned to a phrase) in the brat .ann files 11 +but re-formatted to fit the table structure. The rows with gold annotations are placed beneath +the header row. We chose the table representation as we suspected it would be more “in +distribution” (vs. out of distribution) for GPT-J than other possibilities, including the raw .ann +format. In addition, the table format offers flexibility to encompass all E and A information for a +given trigger on a single line. For the cases where there are multiple triggers of the same type, +we simply add additional rows to the table. +The few-shot example shown in Figure 1A was run with parameters indicated in Figure +1B. The generated text and gold annotation can be seen in Figure 1C. As formulated, the few- +shot task is essentially asking the LLM to function as both a question/answer language model +(for span extraction) and a few-shot classification model (for categorical assignments). GPT-J +performs both tasks admirably in this toy example. +While few-shotting demonstrates the power of models like GPT-J to “learn” from a +minimal number of examples, the setup is fragile and does not yield high performance across +significant numbers of new inferences. The context window for GPT-J does not permit enough +few-shot samples to represent the range of annotations for a given social history element. + +Thus, for the actual extraction task we fine-tuned GPT-J using the entirety of the gold .ann files +provided. +Fine-tuning was performed on a machine with the following specs: 2 V100 32 Gb +graphics cards, Intel Xeon 20 core processor, 11Tb of storage and 512 Gb of RAM. The fine- +tuning python script was written in-house but calls Hugging Face's highly abstracted +API. DeepSpeed 20 was used to accomplish offloading as follows: stage 1 shards optimizer states +across GPUs, stage 2 adds sharding of gradients, stage 3 adds sharding of model parameters +and allows offloading of parameters, weights, and optimizer state. Of note, stage 3 allows +offloading to NVMe and CPU + memory. While offloading incurs significant I/O burden, it allows +for training arbitrarily large models at the expense of memory, CPU compute, and speed. +Just as in the few-shot examples in Figure 1, input to the fine-tuning procedure was +provided as single “sandwiches” of natural language prompt, social history narrative, table +format, and table rows generated from the annotations in .ann files. Specifically, we +incorporated unedited narratives stripped of new lines and span annotations and injected only +the “knowledge” that the categorical text is an inference of some sort, and the type of social +history data we are looking to generate (in the form of our natural language prompt). In almost +every other respect our fine-tuning data is equivalent to using the original files themselves. +While we performed a few experiments with different natural language prompts, we do +not have data on the effectiveness of our chosen verbiage “Make a table about **** in the +following story. Use exact words or phrases from the story.” Anecdotally the choice of prompt +did not seem to impact performance significantly and in this use case, and possibly others, the +prompt may have been superfluous. The LLM demonstrated considerable ability to memorize +the training data, achieving 93-94% F1 score when applied to the training data. +We did not generate exact spans from GPT-J, hypothesizing that would be challenging. +Due to time constraints, we created some simple heuristics to map the model evaluation text +back to the narrative string. We did attempt some experiments where we included an +additional word on either side of the gold annotation to try and increase specificity in the +eventual map back to the narrative. We do not have results on performance from these + +experiments but anecdotally it seemed to decrease. The loss of information about spans likely +resulted in a decreased recall for our effort. + +System 3: NLP pipeline reuse +In this approach, we regard SDOH information as an event just as the information is annotated +and tackle trigger detection, argument detection, and trigger-argument relation extraction. This +general framework has been widely used for event extraction 21,22, and subtasks are commonly +organized in a pipeline manner, unless they are solved jointly, e.g., System 2. In System 3, we +reuse an in-house NLP pipeline built on the UIMA framework 23 to accommodate these subtasks. +The pipeline also provides necessary preprocessing, including tokenization, sentence splitting, +part-of-speech tagging, and syntactic parsing. The pipeline components integrate different +methods and software libraries. For example, for part-of-speech tagging and syntactic parsing, +CoreNLP library 24 is used to derive constituent parse trees and dependency graphs. +After preprocessing, an existing pipeline component for phrase detection is applied for +the extraction of triggers and argument candidates, where trigger phrases are also assigned with +the SDOH type. To this end, a sequence labeling model is trained on trigger and argument phrases +annotated in the training corpus using Conditional Random Field (CRF) 25. Next, a custom +component developed for the current task is applied, which links each detected trigger with +argument candidates within the same sentence. For linking, hand-crafted rules are implemented, +which are based on the constituent span, the dependency link, or any selected text pattern. Rules +were developed following the corpus annotation guidelines and provided examples 3,6 and tested +on annotations collected from a few notes in the training set. +The existing NLP pipeline, which can provide the system framework and reusable +preprocessing components, allowed us to put together this layered system quickly. During our +participation in the challenge, however, we could not allocate sufficient time to write rules for +many relations and to test them beyond the few examples used the initial development. As +reported in the next section, the precision and recall were rather low for this reason. The +performance metric reported on this system, therefore, should be interpreted accordingly. + + +Results +The three systems were used in our submission of the Subtask-A in the Track-2, where the +training, development and test data set were from MIMIC III 10. Table 2 shows the counts of +true positives and predicted positives per target type, obtained on the test data using the +evaluation script provided by the challenge organizer. As the table rows show, the evaluation +script counts triggers and arguments separately, rather than as trigger-argument pairs or +trigger-arguments tuples. Then, it computes the final performance metric from the total counts, +which are shown in the first row “OVERALL” in Table 2. The span-only arguments are relatively +rare, and the performance metric is mostly based on triggers and labeled arguments. The +evaluation results in the two tables show the characteristics of each system as well as that of +the evaluation metric. +System 1 tends to predict more triggers and arguments than the other two systems. +That would be attributed to multiple models in the system that independently predict targets +without considering trigger-argument relations. The current scoring metric favors independent +prediction because, as stated above, triggers and arguments are counted separately toward the +scoring. For example, if a trigger is predicted correctly, it is counted as one true positive +independent of its arguments; if an argument is predicted correctly, the argument and the +associated trigger are both counted. + +System 2 achieved a good performance metric, and it may be improved further with a +larger model and/or larger data. It is notable that this system generates a complete table, +where many relations must be considered together, e.g., trigger-argument, argument- +argument, and trigger-trigger. The complete relations among triggers and arguments, though +restrictive, must help identify consistent answers. For instance, History is a span-only argument +type used for a phrase concerning a patient’s last use of substance, e.g., “7 years ago.” +Therefore, it is always related to the Status argument, Status=past, in addition to the trigger, +and they should be considered together to report coherent outputs. Among the three systems, +this system achieved good or the best results for the three History arguments as in Table 2. + + +The overall performance of System 3 is not as high as the other two systems in Table 1, +but the trigger extraction performance is close to the other two in Table 2. In fact, the baseline +performance of trigger extraction is high in this task because there is a relatively small number +of recurrent and unambiguous trigger phrases, such as “ETOH” (Alcohol), “IVDU” (Drug), +“Tobacco” (Tobacco), “works” (Employment), and “lives” (Living Status). If a system can +memorize those terms, the trigger extraction looks reasonably good. This suggests that the +main interest and challenge in this task is argument detection. A major hurdle for System 3 is +that there are so many arguments, and it takes time to manually review relations and develop +good rules. When a good rule is created, the precision of the extraction can be high, e.g., Drug +Status=none or Employment Status=retired. Yet, many rules are needed to boost the recall. + +Table 1. The evaluation results of our three systems and the first rank system on the Subtask A +test set. There are 3,471 annotated instances (positives). +Subtask + +True Positives +Predicted Positives +Precision +Recall +F1 +A +System 1 +3,070 +3,472 +0.8842 +0.8845 +0.8843 +System 2 +2,776 +3,210 +0.8648 +0.7998 +0.8310 +System 3 +2,157 +3,032 +0.7114 +0.6214 +0.6634 +Rank 1 system +N/A +N/A +0.9093 +0.9078 +0.9008 +B +System 1 +18,376 +23,261 +0.7900 +0.7477 +0.7683 +Rank 1 system +N/A +N/A +0.8109 +0.7703 +0.7739 + + + + +Table 2. The detailed evaluation results of the three systems on the Subtask A test set. In the +gold annotation, triggers and labeled arguments are mandatory per “event” and are highlighted +in the table. + + + + +True Positives +Predicted Positives +SDOH type +argument +subtype +Positives +Sys. 1 +Sys. 2 +Sys. 3 +Sys. 1 +Sys. 2 +Sys. 3 +OVERALL +- +- +3471 +3070 +2776 +2157 +3472 +3210 +3032 +Alcohol +Trigger +N/A +308 +302 +288 +273 +310 +307 +290 + +Status +current +110 +102 +87 +90 +118 +108 +224 + + +none +151 +144 +136 +53 +148 +139 +64 + + +past +47 +37 +37 +0 +44 +60 +2 + +Amount +N/A +47 +32 +27 +15 +45 +38 +35 + +Duration +N/A +6 +3 +3 +0 +6 +5 +7 + +Frequency +N/A +51 +36 +29 +22 +48 +49 +31 + +History +N/A +32 +14 +16 +9 +28 +26 +19 + +Type +N/A +26 +21 +16 +6 +29 +23 +21 +Drug +Trigger +N/A +189 +182 +165 +166 +190 +179 +178 + +Status +current +18 +12 +11 +13 +19 +21 +130 + + +none +153 +148 +135 +47 +152 +142 +48 + + +past +18 +11 +10 +0 +15 +16 +0 + +Amount +N/A +2 +0 +0 +0 +0 +4 +2 + +Duration +N/A +0 +0 +0 +0 +1 +1 +3 + +Frequency +N/A +6 +1 +2 +0 +4 +4 +3 + +History +N/A +10 +6 +5 +1 +15 +12 +7 + +Method +N/A +35 +20 +23 +5 +23 +26 +6 + +Type +N/A +112 +90 +89 +17 +115 +110 +26 +Tobacco +Trigger +N/A +321 +306 +283 +280 +323 +302 +306 + +Status +current +69 +61 +44 +49 +77 +61 +201 + + +none +137 +129 +123 +57 +135 +131 +85 + + +past +115 +93 +95 +10 +104 +109 +20 + +Amount +N/A +105 +76 +65 +47 +99 +93 +71 + +Duration +N/A +51 +41 +34 +32 +48 +45 +40 + +Frequency +N/A +36 +31 +25 +20 +34 +34 +28 + +History +N/A +87 +57 +67 +42 +83 +81 +55 + +Method +N/A +1 +0 +0 +0 +0 +0 +2 + +Type +N/A +20 +17 +13 +4 +22 +22 +14 + +Employment +Trigger +N/A +168 +161 +113 +135 +175 +122 +157 + +Status +employed +64 +57 +43 +47 +67 +43 +108 + + +homemaker +1 +0 +0 +0 +0 +0 +0 + + +on_disability +10 +10 +2 +0 +15 +5 +2 + + +retired +38 +34 +32 +33 +35 +32 +35 + + +student +4 +1 +0 +3 +1 +0 +3 + + +unemployed +51 +47 +34 +8 +54 +42 +9 + +Duration +N/A +4 +1 +0 +0 +3 +1 +5 + +History +N/A +6 +3 +0 +2 +9 +3 +6 + +Type +N/A +130 +91 +56 +48 +129 +89 +75 +LivingStatus +Trigger +N/A +242 +236 +227 +228 +252 +242 +243 + +Status +current +234 +227 +220 +220 +242 +236 +241 + + +past +8 +4 +5 +2 +5 +5 +2 + +Type +alone +60 +59 +57 +44 +62 +66 +46 + + +homeless +4 +4 +3 +0 +5 +3 +0 + + +with_family +139 +136 +131 +129 +143 +137 +177 + + +with_others +39 +25 +25 +0 +34 +35 +0 + +Duration +N/A +4 +1 +0 +0 +5 +0 +1 + +History +N/A +2 +1 +0 +0 +1 +0 +2 + +Discussion +SDOH information in the SHAC corpus is regarded as an “event,” and it is annotated as a trigger +with associated arguments. This annotation framework is widely used in event extraction tasks +22. Meanwhile, it is reported that “[v]arious versions of the event extraction task exist, +depending on the goal” 26 and “[t]he definition of an event varies in granularity depending on +the desired application of event extraction” 27. SDOH annotations in the SHAC corpus are +particularly unique, in that they are reports on patients’ conditions, rather than event +occurrences 2. Additionally, the annotations include both direct reports (e.g., “past smoker” or +“unemployed”) and indirect reports, from which patients’ conditions are inferred (e.g., “He quit +smoking” → Smoking Status:past or “former nurse” → Employment Status:unemployed). All +these factors make the current task different from the conventional event extraction task. + +In the conventional event extraction task, usually, a trigger is a verb, or its +nominalization denoting an event occurrence, and arguments are terms syntactically related to +the trigger. However, triggers in the SHAC corpus are a mixture of clues indicative of SDOH +reports, including section headers (e.g., “Tobacco history: …”), verbs or derivative nouns used +to state habits or status (e.g., “smokes” or “smoker”), and any keywords suggestive of SDOH +reports (e.g., “cigarettes” or “ppd” (packs per day)). Then, there are four to seven different +kinds of arguments for each of the five SDOH targets. Given many kinds of triggers and +arguments, relations between them are diverse and complex. Compared to the conventional +event extraction task, it is particularly difficult to characterize relations between triggers and +arguments. + +To mitigate the challenge, System 1 avoided modeling relations and considered +independent information extraction tasks. The advantage of this approach is the ease of the +complexity in the relation extraction. There are many methods and techniques applicable to the +simplified tasks. The disadvantage is that this approach does not extract phrases and relations +as in the corpus annotation guidelines. The system is inherently limited, and it cannot extract +two triggers in one sentence or trigger-argument across sentences. + +System 2 does not simplify the task and generate complete structured outputs. The +advantage of this approach is the complete outputs as well as the single end-to-end model +dealing with all the relations simultaneously. The disadvantage is that the model behavior +cannot be easily understood or modified because it is a single end-to-end model. Also, a large +computing resource is needed for LLM, while that can help improve the performance further +and can be considered an advantage. + +System 3 is based on trigger-argument relation extraction, conforming to the corpus +annotation guidelines. The advantage of this approach is the transparency of the extraction +procedure and the interpretability of outputs owing to the pipeline architecture and human- +readable rules. The disadvantage is that, given many relations in the task, it is time-consuming +to analyze them and write good rules. Also, the management of many rules and many pipeline +components can be difficult in practice. + + +As discussed above, the three task formulations have different advantages and +disadvantages. Notably, though these systems were evaluated in the same challenge, they are +not comparable for building an application. For example, if the goal is to automatically populate +a structured database with extracted phrases, System 1, which does not extract trigger phrases +and labeled argument phrases, is not applicable. Systems 2 and 3 are applicable to such an +application, but the user experience as well as the system maintenance effort is vastly different. +If users expect explanation for outputs, a rule-based system like System 2 may be necessary 28. +It must be crucial to understand users’ needs and expectation in the application 29. + +Conclusion +In this paper, we describe three information extraction systems that we developed for our +participation in the Task-A of the Track-2 in the 2022 n2c2 NLP Challenge, extraction of SDOH +from clinical narratives. While the SDOH information is annotated using the event-based +annotation framework in the challenge corpus, the meaning of the “trigger” and “argument” is +different from the conventional event extraction task. A commonly used approach to event +extraction is difficult to apply, due to the diverse and complex relations annotated in this +corpus. To overcome this challenge, two alternative task formulations are explored. These +approaches have different advantages and disadvantages. The practical utility of the +approaches depends on the requirements and expectation in the application. + +This paper focuses on SDOH extraction, but the analysis and discussion are applicable to +other information extraction tasks in the clinical NLP domain, where target information is often +not an “event,” but patients’ conditions, clinicians’ observations and assessments, or various +other properties, e.g., severity of a symptom, laterality of an anatomy, a measurement +reported for a lab test or a radiographic study, or their combinations. It is desirable if +information extraction framework suitable for such targets are investigated further in the +clinical NLP domain. + + +Acknowledgments +We thank the organizers and the corpus annotators of the 2022 n2c2 NLP Challenge and the +MIMIC project for the data used in the study. + +References +1. Wang Y, Wang L, Rastegar-Mojarad M, et al. Clinical information extraction applications: A +literature review. J Biomed Inform. 2018;77:34-49. doi:10.1016/j.jbi.2017.11.011 +2. Conway M, Keyhani S, Christensen L, et al. Moonstone: a novel natural language processing +system for inferring social risk from clinical narratives. J Biomed Semant. 2019;10(1):6. +doi:10.1186/s13326-019-0198-0 +3. Lybarger K, Ostendorf M, Yetisgen M. Annotating social determinants of health using active +learning, and characterizing determinants using neural event extraction. J Biomed Inform. +2021;113:103631. doi:10.1016/j.jbi.2020.103631 +4. Track 2 Extracting Social Determinants of Health. National NLP Clinical Challenges (n2c2). +Accessed November 26, 2022. https://n2c2.dbmi.hms.harvard.edu/2022-track-2 +5. Lybarger K, Yetisgen M, Uzuner Ö. The 2022 n2c2/UW Shared Task on Extracting Social +Determinants of Health. Published online January 13, 2023. Accessed January 25, 2023. +http://arxiv.org/abs/2301.05571 +6. Kevin Lybarger. Social Determinants of Health Extraction Challenge - Evaluation Criteria. +Published online February 17, 2022. +https://github.com/Lybarger/brat_scoring/blob/main/docs/sdoh_scoring.pdf +7. Manning CD. Human Language Understanding & Reasoning. Daedalus. 2022;151(2):127- +138. doi:10.1162/daed_a_01905 +8. Chapman WW, Nadkarni PM, Hirschman L, D’Avolio LW, Savova GK, Uzuner O. Overcoming +barriers to NLP for clinical text: the role of shared tasks and the need for additional creative +solutions. J Am Med Inform Assoc JAMIA. 2011;18(5):540-543. doi:10.1136/amiajnl-2011- +000465 +9. Gao Y, Dligach D, Christensen L, et al. 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J Am Med Inform +Assoc JAMIA. 2022;29(10):1810-1817. doi:10.1093/jamia/ocac121 + + diff --git a/2NFIT4oBgHgl3EQf4ivI/content/tmp_files/load_file.txt b/2NFIT4oBgHgl3EQf4ivI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c0760faf28486181f87f4744d4dcea4e505f1629 --- /dev/null +++ b/2NFIT4oBgHgl3EQf4ivI/content/tmp_files/load_file.txt @@ -0,0 +1,921 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf,len=920 +page_content='Task formulation for Extracting Social Determinants of Health from Clinical Narratives Manabu Torii, Ian M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Finn, Son Doan, Paul Wang, Elly W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Yang, Daniel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Zisook Abstract Objective The 2022 n2c2 NLP Challenge posed identification of social determinants of health (SDOH) in clinical narratives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' We present three systems that we developed for the challenge and discuss the distinctive task formulation used in each of the three systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Materials and Methods The first system identifies target pieces of information independently using machine learning classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The second system uses a large language model (LLM) to extract complete structured outputs per document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The third system extracts candidate phrases using machine learning and identifies target relations with hand-crafted rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Results The three systems achieved F1 scores of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='884, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='831, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='663 in the Subtask A of the Challenge, which are ranked third, seventh, and eighth among the 15 participating teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The review of the extraction results from our systems reveals characteristics of each approach and those of the SODH extraction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Discussion Phrases and relations annotated in the task is unique and diverse, not conforming to the conventional event extraction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' These annotations are difficult to model with limited training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The system that extracts information independently, ignoring the annotated relations, achieves the highest F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Meanwhile, LLM with its versatile capability achieves the high F1 score, while respecting the annotated relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The rule-based system tackling relation extraction obtains the low F1 score, while it is the most explainable approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Conclusion The F1 scores of the three systems vary in this challenge setting, but each approach has advantages and disadvantages in a practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The selection of the approach depends not only on the F1 score but also on the requirements in the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Background and Significance Clinical notes are a rich source of information, containing, among others, patient-reported information and clinicians’ assessments that are not coded in structured records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Automated extraction and coding of information has been widely studied 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Among different types of information sought in clinical notes, social determinants of health (SDOH) have gained attention in the last several years, due to their significance on one’s health as well as to their unique availability in clinical notes 2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' In the Track-2 of the 2022 n2c2 NLP Challenge 4,5, extraction of SDOH from clinical notes was posed as a shared task, and a corpus annotated with SDOH was prepared by the challenge organizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The availability of the annotated corpus would further increase interests in this information extraction task and advance the technology toward real- world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' This paper focuses on three information extraction systems that we developed for our submissions of the Track-2 in the 2022 n2c2 NLP Challenge, while we defer the background of this Challenge task and the review of related studies to the publications by the Challenge organizers3–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' In the corpus prepared for the Challenge, types of annotated phrases are unique and diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Relations to be identified among them are difficult to characterize, making the task very different from the conventional event extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Each of the three systems we developed employs a different task formulation to tackle this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Objective Natural language processing (NLP) technology has undergone many changes over the years, especially in the last several years 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' New methods as well as long-standing methods have been evaluated for different clinical NLP tasks in shared-task challenges 8,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Besides the performance evaluation results, the task formulation considered for each shared-task challenge has contributed to the clinical NLP field, providing the baselines in designing an information extraction system for the same or related task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Given these backgrounds, two objectives of this paper are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' We describe three systems that we developed for the submissions for the 2022 n2c2 NLP Challenge Track-2, which employ both recent and long-standing methods and were ranked high among the participating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' We present three different formulations of the task that we used in our three systems and discuss the motivation, results, and advantages and disadvantages of each approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Materials and Methods The Subtask-A of the Track-2, in which we participated, used the Social History Annotation Corpus (SHAC) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The data consisted of 1,316, 188, and 373 clinical narrative texts from MIMIC III 10, which were released respectively as the training, development, and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' During the challenge period, the training and development sets were made available for the participants to develop systems, and the test set was released for the final evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' All these data sets were provided as brat annotation files, consisting of narrative text files (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='txt) and corresponding annotation files (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='ann).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Further information of the brat annotation tool can be found in the brat tool paper 11 and on the brat web page 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' In the SHAC corpus, texts are annotated with trigger phrases for five types of SDOH (Alcohol, Tobacco, Drug, Employment, and Living Status) along with their associated argument phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' A subset of the argument phrases, named labeled arguments, are normalized to predefined labels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', Status is a labeled argument for Alcohol, normalized to one of the three status values: none, current, or past).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The rest of the argument phrases, named span-only arguments, do not have labels to normalized to and are “spans only” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', Duration is a span- only argument for Alcohol, annotated for the duration of alcohol use, such as “for eight years”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Further information of the corpus, including annotation examples, can be found in the SHAC corpus paper and in the evaluation guideline document 3,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The evaluation script used in the challenge is provided by the organizer on GitHub 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' During our participation in the challenge, we considered three formulations of the task and implemented three systems as described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' We did not explore the use of additional texts or annotations or the augmentation of the provided data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' System 1: Sentence classification and sequence labeling There are many triggers and arguments in the current task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' We observed difficult topics in NLP are involved for their detection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', phrase boundary ambiguity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' nested phrase annotations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' trigger-argument across sentences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' one or more annotated phrases per argument type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' However, a good fraction of triggers and arguments look easy to identify (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', repeatedly annotated unambiguous phrases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Also, the evaluation metric used in the challenge is forgiving (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', phrase spans are not required for the labeled argument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Considering these factors, we convert the given task into a set of simpler tasks that can be tackled by common methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' In this approach, an input narrative is first split into sentences using a regular expression pattern, and then, two common methods are applied to each sentence, independently: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Text classification to identify sentences containing triggers and labeled arguments 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Sequence labeling to extract triggers and span-only arguments in each sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' There are two key observations behind this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' First, most of the trigger-argument relations are within a single sentence, and there is usually at most one trigger of the same kind within each sentence, which is also noted in the SHAC corpus paper 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Second, phrase spans are, in effect, not required in the evaluation of triggers and labeled arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' That is, labeled arguments are evaluated by the inferred label values only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Triggers are evaluated by the span, but any overlap between the predicted span and the annotated gold span is counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Therefore, the trigger span is not required in effect if a long enough span is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The two observations lead to a task formulation that, for triggers and labeled arguments, we only need to classify each sentence whether it implies a particular trigger type or a particular labeled argument, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', “Does this sentence report a patient’s alcohol abuse?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' or “Does this sentence report a patient’s current alcohol abuse?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' As for the span-only argument, the task needs to be tackled as sequence labeling, specifically the common BIO labeling of tokens 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' A separate model is prepared for each span-only label type and for each trigger type because phrases annotated for span-only arguments and triggers sometimes overlap each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' After triggers and arguments are detected independently, the predictions are merged per sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' When an argument is predicted by any of the models, the corresponding trigger must be present for it to be reported, and the trigger is additionally predicted, if it is not predicted by the trigger detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' For the text classification, a multi-label text classification model was trained using the Hugging Face Transformers library 15, which is used to make binary classification for 28 targets: 5 triggers and 23 labeled arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The implementation follows a publicly available Jupyter notebook example, “Fine-tune BERT for Multi-label Classification” 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' For the BERT model, Bio_Discharge_Summary_BERT was selected 17, because it seemed to yield slightly better performance than the other model we tested, BioBERT 18, during the development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' For sequence labeling (2), 33 models were trained also using the Hugging Face Transformers library, each of which extracts phrases for a specific trigger and span-only label: 5 triggers and 28 span-only labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The implementation follows the tutorial “Token classification” in the Hugging Face Course 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' For the BERT model, between the two models tested, BioBERT was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Few-shotting GPT-J with alcohol narratives To few-shot GPT-J for the social history extraction task, we convert the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='ann format of the annotated text into a structured table prompt that maintains the essential content but is more compact and amenable for data capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The word “unknown” is used in all cases where the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='ann file does not have an annotation for the given element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Of note, all information regarding spans is eliminated in the conversion in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' We create a column labeled “Inference” to store categorical annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Each E line in the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='ann file is translated into a single row in the table prompt, allowing for the possibility of multiple triggers of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Sample GPT-J generator parameters are shown in Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' We use “###” to hint to the model when language generation should cease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Figure C shows perfect matching of the model output and the gold .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='ann representation in prompt format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='ann Representation GPT-J prompt E A prompt = "" Alcohol:"drinking" StatusTimeVal:"current" Make a table about alcohol use in the following story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Use exact words or phrases from the story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Status:"reports" She reports drinking 2 alcoholic drinks per month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Frequency:"per month" I Alcohol I Amount I Duration I Frequency I History I Type I Status I Inference I Amount: "2 alcoholic drinks I drinking I 2 alcoholic drinks I unknown I per month I unknown I unknown I reports I current I ### E A Make a table about alcohol use in the following story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Use exact words or phrases from the story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Alcohol:"ETOH" StatusTimeVal:"none" Denies ETOH Status:"Denies" I Alcohol I Amount I Duration I Frequency I History I Type I Status I Inference | I ETOH I unknown I unknown I unknown I unknown I unknown I Denies I none I ### E: A Make a table about alcohol use in the following story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Use exact words or phrases from the story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Alcohol: "alcoholic drinks" StatusTimeVal: "current" Four to five alcoholic drinks per night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Amount: "Four to five alcoholic drinks" I Alcohol I Amount I Duration I Frequency I History I Type I Status I Inference I Status: "drinks" I alcoholic drinks I Four to five alcoholic drinks I unknown I per night I unknown I unknown I drinks I current I Frequency: “per night" ### E: Make a table about alcohol use in the following story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Use exact words or phrases from the story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Alcohol: "Alcohol" A Status:"no longer drinking" StatusTimeVal:“past" I Alcohol I Amount I Duration I Frequency I History I Type Status I Inference I History: "in 22 months" I Alcohol I unknown I unknown I unknown I in 22 months I unknown I no longer drinking I past I ### ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' A Make a table about alcohol use in the following story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Use exact words or phrases from the story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' h/o EtOH abuse but last drink in 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' I Alcohol I Amount I Duration I Frequency I History I Type I Status I Inference | B end_sequence="###" generator_kwargs = ( "max_new_tokens\':100, ‘T:,d-do, "temperature\':.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='01, \'clean_up_tokenization_spaces\':True, \'do_sample\': True, "early_stopping\': True, "return_full_text\': False, "pad_token_id\':tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='eos_token_id, "eos_token_id: int(tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='convert_tokens_to_ids(end_sequence) res = generator(prompt, **generator_kwargs) print(res) c :ndno :plog I EtOH I unknown I unknown I unknown I in 2001 I unknown I h/o I past |System 2: Fine-tuned GPT-J model With the general availability of medium and large size language models we were curious to explore how much of the social history information extraction task could be performed by leveraging the knowledge encoded in an LLM as opposed to layering additional knowledge on top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' To this end, we attempted to create a system that performed minimal re-representation of the input data in terms of supplementation with linguistic, structural, or clinical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' A toy example of our thought process is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Here we illustrate how few-shotting GPT-J with four brief alcohol related narratives allows the model to correctly generate annotations for an unknown example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Figure 1A demonstrates how we sandwich each narrative with a natural language prompt above and a desired table format below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' For the few-shot examples we include data from the E lines (“Event” annotation in brat, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', a tuple of phrases) and A lines (“Attribute” annotation in brat, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', a label value assigned to a phrase) in the brat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='ann files 11 but re-formatted to fit the table structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The rows with gold annotations are placed beneath the header row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' We chose the table representation as we suspected it would be more “in distribution” (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' out of distribution) for GPT-J than other possibilities, including the raw .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='ann format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' In addition, the table format offers flexibility to encompass all E and A information for a given trigger on a single line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' For the cases where there are multiple triggers of the same type, we simply add additional rows to the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The few-shot example shown in Figure 1A was run with parameters indicated in Figure 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The generated text and gold annotation can be seen in Figure 1C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' As formulated, the few- shot task is essentially asking the LLM to function as both a question/answer language model (for span extraction) and a few-shot classification model (for categorical assignments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' GPT-J performs both tasks admirably in this toy example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' While few-shotting demonstrates the power of models like GPT-J to “learn” from a minimal number of examples, the setup is fragile and does not yield high performance across significant numbers of new inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The context window for GPT-J does not permit enough few-shot samples to represent the range of annotations for a given social history element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Thus, for the actual extraction task we fine-tuned GPT-J using the entirety of the gold .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='ann files provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Fine-tuning was performed on a machine with the following specs: 2 V100 32 Gb graphics cards, Intel Xeon 20 core processor, 11Tb of storage and 512 Gb of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=" The fine- tuning python script was written in-house but calls Hugging Face's highly abstracted API." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' DeepSpeed 20 was used to accomplish offloading as follows: stage 1 shards optimizer states across GPUs, stage 2 adds sharding of gradients, stage 3 adds sharding of model parameters and allows offloading of parameters, weights, and optimizer state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Of note, stage 3 allows offloading to NVMe and CPU + memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' While offloading incurs significant I/O burden, it allows for training arbitrarily large models at the expense of memory, CPU compute, and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Just as in the few-shot examples in Figure 1, input to the fine-tuning procedure was provided as single “sandwiches” of natural language prompt, social history narrative, table format, and table rows generated from the annotations in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='ann files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Specifically, we incorporated unedited narratives stripped of new lines and span annotations and injected only the “knowledge” that the categorical text is an inference of some sort, and the type of social history data we are looking to generate (in the form of our natural language prompt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' In almost every other respect our fine-tuning data is equivalent to using the original files themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' While we performed a few experiments with different natural language prompts, we do not have data on the effectiveness of our chosen verbiage “Make a table about **** in the following story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Use exact words or phrases from the story.” Anecdotally the choice of prompt did not seem to impact performance significantly and in this use case, and possibly others, the prompt may have been superfluous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The LLM demonstrated considerable ability to memorize the training data, achieving 93-94% F1 score when applied to the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' We did not generate exact spans from GPT-J, hypothesizing that would be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Due to time constraints, we created some simple heuristics to map the model evaluation text back to the narrative string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' We did attempt some experiments where we included an additional word on either side of the gold annotation to try and increase specificity in the eventual map back to the narrative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' We do not have results on performance from these experiments but anecdotally it seemed to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The loss of information about spans likely resulted in a decreased recall for our effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' System 3: NLP pipeline reuse In this approach, we regard SDOH information as an event just as the information is annotated and tackle trigger detection, argument detection, and trigger-argument relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' This general framework has been widely used for event extraction 21,22, and subtasks are commonly organized in a pipeline manner, unless they are solved jointly, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', System 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' In System 3, we reuse an in-house NLP pipeline built on the UIMA framework 23 to accommodate these subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The pipeline also provides necessary preprocessing, including tokenization, sentence splitting, part-of-speech tagging, and syntactic parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The pipeline components integrate different methods and software libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' For example, for part-of-speech tagging and syntactic parsing, CoreNLP library 24 is used to derive constituent parse trees and dependency graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' After preprocessing, an existing pipeline component for phrase detection is applied for the extraction of triggers and argument candidates, where trigger phrases are also assigned with the SDOH type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' To this end, a sequence labeling model is trained on trigger and argument phrases annotated in the training corpus using Conditional Random Field (CRF) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Next, a custom component developed for the current task is applied, which links each detected trigger with argument candidates within the same sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' For linking, hand-crafted rules are implemented, which are based on the constituent span, the dependency link, or any selected text pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Rules were developed following the corpus annotation guidelines and provided examples 3,6 and tested on annotations collected from a few notes in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The existing NLP pipeline, which can provide the system framework and reusable preprocessing components, allowed us to put together this layered system quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' During our participation in the challenge, however, we could not allocate sufficient time to write rules for many relations and to test them beyond the few examples used the initial development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' As reported in the next section, the precision and recall were rather low for this reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The performance metric reported on this system, therefore, should be interpreted accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Results The three systems were used in our submission of the Subtask-A in the Track-2, where the training, development and test data set were from MIMIC III 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Table 2 shows the counts of true positives and predicted positives per target type, obtained on the test data using the evaluation script provided by the challenge organizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' As the table rows show, the evaluation script counts triggers and arguments separately, rather than as trigger-argument pairs or trigger-arguments tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Then, it computes the final performance metric from the total counts, which are shown in the first row “OVERALL” in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The span-only arguments are relatively rare, and the performance metric is mostly based on triggers and labeled arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The evaluation results in the two tables show the characteristics of each system as well as that of the evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' System 1 tends to predict more triggers and arguments than the other two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' That would be attributed to multiple models in the system that independently predict targets without considering trigger-argument relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The current scoring metric favors independent prediction because, as stated above, triggers and arguments are counted separately toward the scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' For example, if a trigger is predicted correctly, it is counted as one true positive independent of its arguments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' if an argument is predicted correctly, the argument and the associated trigger are both counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' System 2 achieved a good performance metric, and it may be improved further with a larger model and/or larger data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' It is notable that this system generates a complete table, where many relations must be considered together, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', trigger-argument, argument- argument, and trigger-trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The complete relations among triggers and arguments, though restrictive, must help identify consistent answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' For instance, History is a span-only argument type used for a phrase concerning a patient’s last use of substance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', “7 years ago.” Therefore, it is always related to the Status argument, Status=past, in addition to the trigger, and they should be considered together to report coherent outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Among the three systems, this system achieved good or the best results for the three History arguments as in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The overall performance of System 3 is not as high as the other two systems in Table 1, but the trigger extraction performance is close to the other two in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' In fact, the baseline performance of trigger extraction is high in this task because there is a relatively small number of recurrent and unambiguous trigger phrases, such as “ETOH” (Alcohol), “IVDU” (Drug), “Tobacco” (Tobacco), “works” (Employment), and “lives” (Living Status).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' If a system can memorize those terms, the trigger extraction looks reasonably good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' This suggests that the main interest and challenge in this task is argument detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' A major hurdle for System 3 is that there are so many arguments, and it takes time to manually review relations and develop good rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' When a good rule is created, the precision of the extraction can be high, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', Drug Status=none or Employment Status=retired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Yet, many rules are needed to boost the recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The evaluation results of our three systems and the first rank system on the Subtask A test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' There are 3,471 annotated instances (positives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Subtask True Positives Predicted Positives Precision Recall F1 A System 1 3,070 3,472 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='8842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='8845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='8843 System 2 2,776 3,210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='8648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='7998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='8310 System 3 2,157 3,032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='7114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='6214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='6634 Rank 1 system N/A N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='9093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='9078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='9008 B System 1 18,376 23,261 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='7900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='7477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='7683 Rank 1 system N/A N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='8109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='7703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='7739 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The detailed evaluation results of the three systems on the Subtask A test set.' metadata={'source': 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Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' 2 Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='OVERALL 3471 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='3070 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='2776 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='2157 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='3472 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='Discussion SDOH information in the SHAC corpus is regarded as an “event,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' and it is annotated as a trigger with associated arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' This annotation framework is widely used in event extraction tasks 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Meanwhile, it is reported that “[v]arious versions of the event extraction task exist, depending on the goal” 26 and “[t]he definition of an event varies in granularity depending on the desired application of event extraction” 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' SDOH annotations in the SHAC corpus are particularly unique, in that they are reports on patients’ conditions, rather than event occurrences 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Additionally, the annotations include both direct reports (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', “past smoker” or “unemployed”) and indirect reports, from which patients’ conditions are inferred (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', “He quit smoking” → Smoking Status:past or “former nurse” → Employment Status:unemployed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' All these factors make the current task different from the conventional event extraction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' In the conventional event extraction task, usually, a trigger is a verb, or its nominalization denoting an event occurrence, and arguments are terms syntactically related to the trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' However, triggers in the SHAC corpus are a mixture of clues indicative of SDOH reports, including section headers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', “Tobacco history: …”), verbs or derivative nouns used to state habits or status (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', “smokes” or “smoker”), and any keywords suggestive of SDOH reports (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', “cigarettes” or “ppd” (packs per day)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Then, there are four to seven different kinds of arguments for each of the five SDOH targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Given many kinds of triggers and arguments, relations between them are diverse and complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Compared to the conventional event extraction task, it is particularly difficult to characterize relations between triggers and arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' To mitigate the challenge, System 1 avoided modeling relations and considered independent information extraction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The advantage of this approach is the ease of the complexity in the relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' There are many methods and techniques applicable to the simplified tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The disadvantage is that this approach does not extract phrases and relations as in the corpus annotation guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The system is inherently limited, and it cannot extract two triggers in one sentence or trigger-argument across sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' System 2 does not simplify the task and generate complete structured outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The advantage of this approach is the complete outputs as well as the single end-to-end model dealing with all the relations simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The disadvantage is that the model behavior cannot be easily understood or modified because it is a single end-to-end model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Also, a large computing resource is needed for LLM, while that can help improve the performance further and can be considered an advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' System 3 is based on trigger-argument relation extraction, conforming to the corpus annotation guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The advantage of this approach is the transparency of the extraction procedure and the interpretability of outputs owing to the pipeline architecture and human- readable rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The disadvantage is that, given many relations in the task, it is time-consuming to analyze them and write good rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Also, the management of many rules and many pipeline components can be difficult in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' As discussed above, the three task formulations have different advantages and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Notably, though these systems were evaluated in the same challenge, they are not comparable for building an application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' For example, if the goal is to automatically populate a structured database with extracted phrases, System 1, which does not extract trigger phrases and labeled argument phrases, is not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Systems 2 and 3 are applicable to such an application, but the user experience as well as the system maintenance effort is vastly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' If users expect explanation for outputs, a rule-based system like System 2 may be necessary 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' It must be crucial to understand users’ needs and expectation in the application 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Conclusion In this paper, we describe three information extraction systems that we developed for our participation in the Task-A of the Track-2 in the 2022 n2c2 NLP Challenge, extraction of SDOH from clinical narratives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' While the SDOH information is annotated using the event-based annotation framework in the challenge corpus, the meaning of the “trigger” and “argument” is different from the conventional event extraction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' A commonly used approach to event extraction is difficult to apply, due to the diverse and complex relations annotated in this corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' To overcome this challenge, two alternative task formulations are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' These approaches have different advantages and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The practical utility of the approaches depends on the requirements and expectation in the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' This paper focuses on SDOH extraction, but the analysis and discussion are applicable to other information extraction tasks in the clinical NLP domain, where target information is often not an “event,” but patients’ conditions, clinicians’ observations and assessments, or various other properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=', severity of a symptom, laterality of an anatomy, a measurement reported for a lab test or a radiographic study, or their combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' It is desirable if information extraction 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='edu/2022-track-2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Lybarger K, Yetisgen M, Uzuner Ö.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' The 2022 n2c2/UW Shared Task on Extracting Social Determinants of Health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Published online January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Accessed January 25, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='org/abs/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='05571 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Kevin Lybarger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Social Determinants of Health Extraction Challenge - Evaluation Criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Published online February 17, 2022.' metadata={'source': 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+page_content=' Rule-Based Information Extraction is Dead!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Long Live Rule-Based Information Extraction Systems!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' In: Association for Computational Linguistics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' 2013:827- 832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Lederman A, Lederman R, Verspoor K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' Tasks as needs: reframing the paradigm of clinical natural language processing research for real-world decision support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' J Am Med Inform Assoc JAMIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='29(10):1810-1817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFIT4oBgHgl3EQf4ivI/content/2301.11386v1.pdf'} +page_content='1093/jamia/ocac121' 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a/39E0T4oBgHgl3EQfeQA8/content/tmp_files/2301.02387v1.pdf.txt b/39E0T4oBgHgl3EQfeQA8/content/tmp_files/2301.02387v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..448bb7e24ee5cfafd758b22b1f3efe1552ddb967 --- /dev/null +++ b/39E0T4oBgHgl3EQfeQA8/content/tmp_files/2301.02387v1.pdf.txt @@ -0,0 +1,686 @@ +Efficient simulation of multielectron dynamics in +molecules under intense laser pulses: +Implementation of the multiconfiguration +time-dependent Hartree-Fock method based on +the adaptive finite element method +Yuki Orimo,∗,† Takeshi Sato,†,‡,¶ and Kenichi L. Ishikawa†,‡,¶ +†Department of Nuclear Engineering and Management, Graduate School of Engineering, +The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan +‡Research Institute for Photon Science and Laser Technology, The University of Tokyo, +7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033 Japan +¶Photon Science Center, Graduate School of Engineering, The University of Tokyo, 7-3-1 +Hongo, Bunkyo-ku, Tokyo 113-8656, Japan +E-mail: ykormhk@atto.t.u-tokyo.ac.jp +Abstract +We present an implementation of the multiconfiguration time-dependent Hartree- +Fock method based on the adaptive finite element method for molecules under intense +laser pulses. For efficient simulations, orbital functions are propagated by a stable prop- +agator using the short iterative Arnoldi scheme and our implementation is parallelized +for distributed memory computing. This is demonstrated by simulating high-harmonic +generation from a water molecule and achieves a simulation of multielectron dynamics +with overwhelmingly less computational time, compared to our previous work. +1 +arXiv:2301.02387v1 [quant-ph] 6 Jan 2023 + +Keywords +Ab initio simulation, multielectron dynamics in molecules, intense laser field, TD-MCSCF +method +Introduction +Multielectron dynamics studied in strong-field physics and attosecond science is a com- +plicated phenomenon, which includes non-perturbative and nonlinear effects, and multiple +states or paths excited by ultrashort pulses.1–3 Ab initio simulations have important roles +to understand and predict these physics. Although solving the time-dependent Schr¨odinger +equation (TDSE) gives an exact description of the dynamics in the non-relativistic regime, +it is almost impossible to directly solve TDSE for many-body systems due to the exponen- +tial growth of the computational cost. The time-dependent multiconfiguration self-consistent +field methods (TD-MCSCF) have been developed to overcome this problem.4–13 In the meth- +ods, the total wave function is expressed by the configuration interaction (CI) expansion with +time-dependent orbital functions, whose flexibility effectively reduces the required number of +configurations. The multiconifguration time-dependent Hartree-Fock (MCTDHF) method5–7 +is the most general approach for fermionic systems. It considers all the possible configurations +for a given number of orbital functions. As further developed methods, the time-dependent +complete-active-space self-consistent field method,10 the time-dependent restricted-active- +space self-consistent field method9 and the time-dependent occupation-restricted multiple +active-space method13 have also been proposed. They can significantly reduce the num- +ber of configurations by classifying orbital functions and making restrictions on electronic +excitation. Today, we can accurately simulate atoms containing several tens of electrons +under intense/ultrashort laser pulses thanks to an efficient description of wave functions by +TD-MCSCF methods.14 +However, it is still difficult to handle molecular systems since simple and efficient dis- +2 + +cretization of the three-dimensional space such as the polar coordinate for atomic systems +is not allowed without relying on the symmetries of the systems. One of the elaborated +discretizations to simulate molecules without prohibitive computational cost is using mul- +tiresolution grids. The concept of the method is to discretize only a region near nuclei with +fine grids and the other regions with grids coarse yet sufficiently fine to describe ionizing +wave packets. We have previously implemented the MCTDHF method based on a multires- +olution Cartesian grid and successfully computed high-harmonic generation from a water +molecule.15 +In this study, we further extend our previous work to implement the MCTDHF method +with a finite element method on an adaptively generated multiresolution mesh (adaptive +finite element method). As well as our previous implementation, only the center parts of +the mesh are refined for sharp changes in wave functions and it gradually becomes coarse in +the outer region such as Fig. 1. We can also easily control the order of accuracy since finite +element basis functions are used in each cell. Furthermore, we introduce a highly stable +propagator based on the short iterative Lanczos/Arnoldi propagator16 to address instability +arising from high spatial resolution. +Our simulation code is parallelized for distributed +memory environments, and consequently, achieved over a hundred times faster simulations. +This paper is organized as follows. In section II, our problem setting is defined and the +MCTDHF method is briefly reviewed. In section III, we describe our implementation of +spatial discretization using the adaptive finite element method, the time evolution of wave +functions with the short iterative Arnoldi propagator, and parallelization. In section IV, we +show a numerical result of high-harmonic generation from a water molecule. Conclusions +are given in section V. Hereafter, we use atomic units unless otherwise indicated. +3 + +Figure 1: A part of an adaptive finite element mesh for a hydrogen molecule. The red spheres +show positions of the nuclei and cell colors are electron density. +Molecular system and the MCTDHF method +The Hamiltonian of electrons in a molecule under a laser field can be described as follows. +H = +� +i +H1(ri) + 1 +2 +� +i̸=j +H2(ri, rj) +(1) +H1(ri) = −1 +2∆i − +� +a +Za +|ri − ra| − iA(t) · ∇i +(2) +H2(ri, rj) = +1 +|ri − rj| +(3) +where ri and ra are the positions of the ith electron and the ath nucleus and Za is the charge +of the ath nucleus. A(t) = − +� t +∞ E(t′)dt′ denotes the vector potential of a laser field applied +to the simulated systems, where E(t) is the electric field of it. +Electronic wave functions are modeled by the multiconfiguration time-dependent Hartree- +Fock (MCTDHF) method.5–7 Here, we just briefly reviews the method and show the equation +of motions (EOMs). The detailed descriptions and derivation of EOMs can be found in the +reference.10 +The MCTDHF method expresses a multielectron wave function |Ψ⟩ with a super position +of all the possible Slater determinants composed of a given time-dependent spatial orbital +4 + +set {φp}. +|Ψ⟩ = +� +I +CI(t) |I⟩ +(4) +CI(t) is a configuration interaction (CI) coefficient and |I⟩ is an electronic configuration +(Slater determinant) composed of orbitals. The equation of motion to variationally evolve +the MCTDHF wave function can be derived from the time-dependent variational principle.17 +The time-dependent variational principle requires that the action integral S[Ψ], +S[Ψ] = +� t1 +t0 +dt ⟨Ψ| ˆH − i ∂ +∂t |Ψ⟩ , +(5) +is stationary to an arbitrary infinitesimal wave function variation δΨ, +δS +δΨ = 0. +(6) +As a solution of the stationary condition (Eq. (6)), the equations of motion (EOMs) for CI +coefficients and orbitals are given as follows. +i ˙CI = +� +J +⟨I| ˆH − i ˆX|J⟩ CJ +(7) +i | ˙φp⟩ = ˆQ +� +ˆH1 |φp⟩ + +� +oqrs +(D−1)o +pP qs +or ˆW r +s |φq⟩ +� ++ i +� +q +|φq⟩ Xq +p +(8) +ˆX is an arbitrary anti-Hermitian operator, which can be determined as +ˆX = +� +pq +Xp +q +� +σ +ˆa† +qσˆapσ, +(9) +where apσ(a† +pσ) is the annihilation (creation) operator for a spatial orbital φp with σ spin +(up-spin or down-spin), Xp +q is an arbitrary anti-Hermitian matrix. In this work, we set Xp +q +to be zero. ˆQ is a projection operator onto the orthogonal complement of occupied orbitals, +5 + +ˆQ = 1 − +� +q +|φq⟩⟨φq| . +(10) +D and P are one-body and two-body reduced density matrices, whose matrix elements are +defined as +Dp +q = +� +σ +⟨Ψ|ˆa† +qσˆapσ|Ψ⟩ +(11) +P pq +sr = +� +στ +⟨Ψ|ˆa† +sσˆa† +rτˆaqτˆapσ|Ψ⟩ . +(12) +ˆW r +s is the inter-electronic mean-field potential given by +W r +s (r) = +� +dr′φ∗ +r(r′)φs(r′) +|r − r′| +. +(13) +Implementation +This section shows the implementation of our simulation code developed in this work to +solve Eqs. (7) and (8) as an initial value problem. Simulations of molecular systems require +efficient spatial discretization so that we can simulate electronic dynamics keeping accuracy +with realistic computational cost. We employ the adaptive finite element method18,19 for the +efficient discretization of orbitals based on an open-source finite element library deal.II.20,21 +As described below, while the adaptive finite element method realizes locally high spatial +resolution, time evolution could be unstable due to it. To stably propagate wave functions +for a long period, We employ the short iterative Arnoldi propagator. Although the short +iterative Lanczos propagator is often used in many applications,16,22–24 since the system +matrix is not Hermitian, the Arnodi algorithm is used instead of the Lanczos algorithm +in this application. Applying this scheme to all orbitals at once, we have enabled more +stable time evolution. These numerical computation schemes are described in the rest of +this section. +6 + +Adaptive finite element method +The adaptive finite element (AFEM) used in this work is an approach to improve the accuracy +of simulations requiring locally high resolution by using a multiresolution mesh generated +by local mesh refinement. A finite element mesh is generated by first discretizing the whole +simulation box with coarse uniform cubic cells, and then dividing these cells into half the size +in regions requiring higher resolution. We can generate an adaptive multiresolution mesh by +repeating the second process. Once the multiresolution mesh and cells are generated, most +of the rest of the processes fall into the usual finite element method. +The mesh sizes are determined to make an error in each cell, which is given by Kelly’s +error indicator 25 to estimate the error in each cell from the jump of the gradient of a target +function, less than a threshold. This work adopts the Coulomb potential of the nuclei in a +molecule as the target function for the error estimation. We also limit the minimum and +maximum mesh sizes to avoid generating extremely small and large cells. +The basis functions located in each cell are direct products of the one-dimensional La- +grange polynomials passing through the Gauss-Lobatto quadrature points in each cell. The +quadrature points in each cell are also constructed as the direct product of one-dimensional +Gauss-Lobatto quadrature points. This basis can be considered to be the three-dimensional +version of the finite element discrete variable representation (FEDVR) basis.26,27 +Let us define fI,i(r) as the i th basis function in the I th cell, and LI,jx(x), LI,jy(y) and +LI,jz(z) the (jx, jy, jz) th Lagrange polynomials in each dimension in the I th cell. Then, the +function fI,i(r) is given by +fI,i(r) = LI,jx(x)LI,jy(y)LI,jz(z). +(14) +These functions are defined only in the I th cell and have zero values in other region than +that. +The finite element basis set {bk(r)} is constructed by the basis functions fI,i(r) which +7 + +have zero-value on the boundary of each cell and bridged functions that combine two bases +with nonzero values at the quadrature point shared by two cells on the boundary of adjacent +cells. The bridged functions are required to ensure the continuity of discretized functions. +The mesh generation and construction of the basis are carried out by using deal.II functions. +An arbitrary function h(r) is discretized with this finite element basis as follows. +h(r) = +� +k +ckbk(r) +(15) +ck = +� +l +( ˜ +M −1)k,l +� +drbl(r)h(r) +(16) +The matrix ˜ +M is the overlap matrix of the basis set {bk(r)}, called the mass matrix in the +finite element method, defined as +˜ +Mk,l = +� +drbk(r)bl(r). +(17) +All the spatial integrals are approximated with Gauss-Lobatto quadrature as follows. +� +drh(x, y, z) ≃ +� +I +� +jx,jy,jz +wx +I,jxwy +I,jywz +I,jzh(xI,jx, yI,jy, zI,jz), +(18) +where wd +I,jd (d = x, y, z) and (xI,jx, yI,jy, zI,jz) are the quadrature weights and points of the +I th cell. +Based on this discretization scheme, the equation of motion (Eq. (8)) is converted into a +matrix-vector equation, +i ˜ +M ˙cp = (1 − ˜ +M +� +q +cqc† +q) +� +˜H1cp + ˜ +M +� +oqrs +(D−1)o +pP qs +or W r +s ◦ cq +� ++ i ˜ +M +� +q +cqXq +p +(19) +8 + +where cp denotes a coefficient vector of orbital φp(r) given by, +(cp)k = +� +drbk(r)φp(r) +(20) +and the matrices ˜H1 is defined as the matrix form of the operator ˆH1, +( ˜H1)k,l = +� +drbk(r)H1(r)bl(r). +(21) +W r +s is a coefficient vector of the mean-field potential W r +s (r) and the element-wise product +is denoted by “◦”. +We compute the mean-field potential by solving the following Poisson’s equation, instead +of directly calculating Eq. (13), +∆W r +s (r) = −4πφ∗ +r(r)φs(r) +(22) +with a boundary condition +W r +s (r) +��� +r∈Ω = +� +dr′φ∗ +r(r′)φs(r′) +|r − r′| +, +(23) +where Ω denotes the boundary of a simulation box. We solve this equation by the conjugate +gradient method with algebraic multigrid preconditioning implemented in an open-source +parallel linear algebra library Trilinos28 interfaced on deal.II. +Short iterative Arnoldi propagator +The short iterative Lanczos/Arnoldi propagator is a time evolution method, which approx- +imates a Hamiltonian in a Krylov subspace by the Lanczos/Arnoldi algorithm and iterates +short-time propagation of wave functions in the subspace.16 This approach conserves the +norm of a wave function when a Hamiltonian is Hermitian and enables unconditionally sta- +9 + +ble time evolution. It is also possible to use an adaptive time step or a variable Krylov +subspace dimension based on the error estimation16 However, we cannot straightforwardly +apply it to the equation of motion of orbitals, since it is only applicable to linear equations. +Although some applications of the MCSCF methods, where the EOM of orbitals is non- +linear, use exponential integrators29–31 to enjoy the stability of the short iterative Lanc- +zos/Arnoldi propagator even only for linear parts of the EOM, in our application, we found +that the explicit time propagation of the nonlinear parts causes numerical instability prob- +ably due to the quite fine mesh of AFEM. To avoid this problem, in this work, we propose +an approach to apply the short iterative Lanczos/Arnoldi propagator by approximately re- +garding the whole of the EOMs for all orbitals as one linear system. +The equations of motion for all orbitals (Eq. (8)) can be packed into a matrix-vector +form, whose elements are operators and ket-vectors. +i ∂ +∂tφ = ˆGφ +(24) +φ = +� +�������� +|φ1⟩ +|φ2⟩ +... +|φn⟩ +� +�������� +, +G = +� +�������� +ˆG11 +ˆG12 +· · · +ˆG1N +ˆG21 +ˆG22 +· · · +ˆG2N +... +... +... +ˆGN1 +ˆGN2 +· · · +ˆGNN +� +�������� +(25) +The matrix element ˆGij is an operator defined as +ˆGij = δi +j ˆH1 + +� +osr +(D−1)o +iP js +or ˆW r +s − ⟨φj| +� +ˆH1 |φi⟩ + +� +oqrs +(D−1)o +iP js +or ˆW r +s |φq⟩ +� ++ iXj +i . +(26) +The equation (24) is approximately linear if we can assume that orbitals in the operators +are invariable within a short time ∆t, and then time evolution of orbitals can be described +as +φ(t + ∆t) = exp +� +−i ˆG∆t +� +φ(t). +(27) +10 + +We achieve this time evolution by the short iterative Arnoldi scheme. Although this scheme +has first-order accuracy since the time-dependency of the operator ˆG in a time step ∆t is +not considered, it enables highly stable propagation including nonlinear parts and fits our +implementation based on the AFEM using a fine mesh. The Krylov subspace dimension of +the Arnoldi algorithm is determined so that errors estimated by the method found in the +references16,23 are less than a threshold, which is set to be 10−10 in this work. We also adjust +a time-step size, which is fixed during our simulations, to make the dimension 10-15 at a +maximum. +Parallelization +The spatial discretization and time evolution discussed above are devised to efficiently sim- +ulate multielectron dynamics in a laser field. +Nevertheless, computational costs for the +three-space to describe laser-induced ionization are huge , and distributed memory parallel +computing is essential. The total number of degrees of freedom (DOF) NDOF in our simula- +tion can simply be written as NDOF = Norbital × Nspace + NCI, where Norbital, Nspace and NCI +are the numbers of orbitals, DOF associated with spatial discretization and CI coefficients, +respectively. Norbital is typically from several to several tens, and Nspace usually increases up +to several millions. NCI significantly changes depending on a problem since it exponentially +increases to the numbers of electrons and orbitals. Our strategy to make efficient use of +many processors in many situations is parallelizing orbital functions with respect to both +the number of orbitals and the number of degrees of freedom in the AFEM. +We divide the orbital function set {|φp⟩} by K and create K MPI groups to deal with +them. Each MPI group has L independent processes that are used to distribute a simulation +box by using deal.II functions. Distribution of a simulation box and DOFs accompanying it is +carried out by p4est,20,32 an open-source library to distribute octree structures across multiple +processors, interfaced to deal.II. This addresses load balancing and optimal distribution of +the simulation box to reduce MPI communications among the processors (Fig. 2). +11 + +Figure 2: An example of a divided simulation box. The color-coded cells correspond the +distribution to MPI processes. +Applications +We simulate high harmonic generation from a water molecule to demonstrate the efficiency +of our implementation by comparing our previous work.15 For atomic positions of a water +molecule, two hydrogen atoms of a water molecule are located at (±1.42994, 1.10718, 0) and +an oxygen atom is located at the origin. +The laser pulse used in this simulation has a +wavelength of 2πc/ω = 400nm (c is the speed of light in vacuum) and a peak intensity of +I0 = 8 × 1014 W/cm2, and is linearly polarized along with x-axis. The pulse duration is 2 +optical cycles with a triangular envelope. The shape of the electric field of the laser pulse is +defined as, (see also Fig. 3), +E(t) = E0fenv(t) sin(ωt) +(28) +fenv(t) = +� +� +� +� +� +� +� +� +� +ωt +2π +(0 ≤ ωt ≤ 2π) +4π − ωt +2π +(2π ≤ ωt ≤ 4π) +, +(29) +where E0 is the peak electric field derived from the peak intensity. The time-step size for +real-time evolution is 0.01 a.u.. +12 + +Figure 3: The electric field of the laser pulse used in this simulation. +The simulation box is a cuboid defined within a region [−70, 70]×[−30, 30]×[−30, 30]. We +apply the exterior complex scaling (ECS) as an absorbing boundary in the outside of a region +[−35, 35] × [−10, 10] × [−10, 10]. The details of the ECS can be found in the references.33–35 +The finite element mesh is generated to satisfy that the error in each cell is less than +0.005, which has 6 different sizes between 0.125 a.u. and 4.0 a.u.. At the most distant region +from the molecule, the largest elements, which are cubes with 4.0 a.u long sides, are used +to describe sufficiently absorbed orbital functions and the smallest elements, whose edge +length is 0.125 a.u., are used in the vicinity of the molecule. Figure 4 displays the finite +element mesh used in this simulation. The finite element basis is constructed from first- +order Lagrange polynomials, and thus there are 8 quadrature points in a finite element cell. +While it is possible to dynamically adapt a mesh to time-dependent orbital functions, we +avoid such approaches due to additional computational costs. This would be helpful to gain +computational efficiency if our problem was a larger system. +For the beginning of the simulation, we computed a ground state by imaginary-time +evolution, whose electronic energy was -76.905 a.u. In figure. 5, we compare our simulation +result with the previously calculated one. These spectra do not perfectly agree with each +other since it is extremely difficult to achieve perfect convergence for spatial resolutions in +3D systems, Nevertheless, overall spectral shapes are quite similar. As well as the previous +calculations, the simulations with 5 orbitals and 6 orbitals give almost the same spectra. +13 + +0.1 +0.0 +-0.1 +0 +25 +50 +75 +100Figure 4: Adaptively generated finite element mesh for a water molecule. The largest element +is a cube of edge length 4.0 a.u. used to discretize the outer region, and the smallest one is +a cube of edge length 0.125 a.u. used only in the vicinity of nuclei. +The simulation using 6 orbitals of present work took 6.5 hours with 240 cores (6 nodes, +2 Intel Xeon Gold 2.40GHz processors with 20 cores in a node). Remarkably, it is about +100 times faster than the previous work which took 28 days to finish the simulation. One +of our achievements is successful distributed parallel computing using the 20 times larger +resource. In addition to this, at least 5 times acceleration was gained by factors except for +parallelization. The development of a highly stable propagator mainly contributes to this +speed-up, which enables time evolution with 4 times as large a time-step size as the previous +one. +Conclusion +We have implemented the MCTDHF method based on the adaptive finite element method +to simulate multielectron dynamics in molecules under laser fields. A further sophisticated +discretization is realized by using the multiresolution grid used in our previous implementa- +tion in the frame of the finite element method. Thanks to the finite element method, we can +automatically generate an adaptive mesh using Kelly’s error indicator and easily control the +order of accuracy by changing the polynomial order of basis functions. While locally refined +meshes enable efficient and accurate simulations, they possibly make time evolution unstable. +14 + +140 a.u. +60 a.u. +60 a.u.Figure 5: High harmonic spectra of a water molecule exposed to a laser pulse with a wave- +length of 400nm and a peak intensity of 8 × 1014 W/cm2. (a) The spectrum taken from +Ref.15 The data is normalized for the maximum to be unity. (b) The spectra computed by +the present work. +We developed a more stable propagator based on the short iterative Arnoldi scheme than +exponential integrators. This propagator evolves all orbital functions together as a vector +by using the short iterative Arnoldi scheme. In addition, our simulation code is parallelized +for distributed memory computing, which handles both the orbital set and spatial degrees +of freedom in parallel. +We have applied the present implementation to a simulation of high-harmonic generation +from a water molecule in an intense visible laser pulse to compare with our previous work,15 +and obtained the spectra showing a good agreement with overwhelmingly less computational +time. Parallelization has made the greatest contribution to this reduction in computation +time, and in this study, we were able to successfully use 20 times larger computational +resources than in the past. It is also important to note that we were able to use a 4 times +larger time-step size thanks to the stable propagator. +This study prepared the adaptive mesh based on the discretization error of the Coulomb +15 + +(a) +() +6 orbitals +6 orbitals +10-1 +10-1 +5 orbitals +Intensity (a.u.) +Intensity (a.u.) +10-3 +10-3 +10-5 +10-5 +10-7 +10-7 +10-9 +10-9 +10 +20 +30 +0 +10 +20 +0 +30 +Harmonic order +Harmonic orderpotential of the nuclei, therefore the mesh is fixed during simulations, but it is possible +to dynamically adapt the mesh to a wave function or nuclear positions at each time step. +We consider that it brings efficiency when a larger simulation box is needed or when the +nuclei can move. In future works, we will present ab initio simulations of more complicated +molecular systems and simulations considering nuclear dynamics in a combination of this +development and more advanced theories such as the TD-ORMAS method13 and the time- +dependent coupled cluster theory.36 +Data availability +The data and source code used in this study are available upon reasonable request. +Competing interests +The authors declare there are no competing interests. +Funding information +This research was supported in part by a Grant-in-Aid for Scientific Research (Grants No. +JP19H00869, No. JP21K18903, and No. JP22H05025) and a Grant-in-Aid for Early-Career +Scientists (Grant No. JP22K14616) from the Ministry of Education, Culture, Sports, Sci- +ence and Technology (MEXT) of Japan. +This research was also partially supported by +JST CREST (Grant No. JPMJCR15N1) and by MEXT Quantum Leap Flagship Program +(MEXT Q-LEAP) Grant Number JPMXS0118067246. +References +(1) Brabec, T.; Krausz, F. 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In Large-Scale Scientific Computing; +Lirkov, I., Margenov, S., Eds.; Springer International Publishing: Cham, 2020; pp +557–565. +(31) G´omez Pueyo, A.; Marques, M. A. L.; Rubio, A.; Castro, A. Journal of Chemical +Theory and Computation 2018, 14, 3040–3052. +(32) Burstedde, C.; Wilcox, L. C.; Ghattas, O. SIAM Journal on Scientific Computing 2011, +33, 1103–1133. +(33) McCurdy, C. W.; Stroud, C. K.; Wisinski, M. K. Phys. Rev. A 1991, 43, 5980–5990. +(34) Scrinzi, A. Phys. Rev. A 2010, 81, 053845. +(35) Orimo, Y.; Sato, T.; Scrinzi, A.; Ishikawa, K. L. Phys. Rev. A 2018, 97, 023423. +(36) Sato, T.; Pathak, H.; Orimo, Y.; Ishikawa, K. L. The Journal of Chemical Physics +2018, 148, 051101. +19 + diff --git a/39E0T4oBgHgl3EQfeQA8/content/tmp_files/load_file.txt b/39E0T4oBgHgl3EQfeQA8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e1a5804dc98681ea2c2a54ef0e49f7267181a6e3 --- /dev/null +++ b/39E0T4oBgHgl3EQfeQA8/content/tmp_files/load_file.txt @@ -0,0 +1,568 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf,len=567 +page_content='Efficient simulation of multielectron dynamics in molecules under intense laser pulses: Implementation of the multiconfiguration time-dependent Hartree-Fock method based on the adaptive finite element method Yuki Orimo,∗,† Takeshi Sato,†,‡,¶ and Kenichi L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Ishikawa†,‡,¶ †Department of Nuclear Engineering and Management, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan ‡Research Institute for Photon Science and Laser Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033 Japan ¶Photon Science Center, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan E-mail: ykormhk@atto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='jp Abstract We present an implementation of the multiconfiguration time-dependent Hartree- Fock method based on the adaptive finite element method for molecules under intense laser pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' For efficient simulations, orbital functions are propagated by a stable prop- agator using the short iterative Arnoldi scheme and our implementation is parallelized for distributed memory computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' This is demonstrated by simulating high-harmonic generation from a water molecule and achieves a simulation of multielectron dynamics with overwhelmingly less computational time, compared to our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='02387v1 [quant-ph] 6 Jan 2023 Keywords Ab initio simulation, multielectron dynamics in molecules, intense laser field, TD-MCSCF method Introduction Multielectron dynamics studied in strong-field physics and attosecond science is a com- plicated phenomenon, which includes non-perturbative and nonlinear effects, and multiple states or paths excited by ultrashort pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='1–3 Ab initio simulations have important roles to understand and predict these physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Although solving the time-dependent Schr¨odinger equation (TDSE) gives an exact description of the dynamics in the non-relativistic regime, it is almost impossible to directly solve TDSE for many-body systems due to the exponen- tial growth of the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The time-dependent multiconfiguration self-consistent field methods (TD-MCSCF) have been developed to overcome this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='4–13 In the meth- ods, the total wave function is expressed by the configuration interaction (CI) expansion with time-dependent orbital functions, whose flexibility effectively reduces the required number of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The multiconifguration time-dependent Hartree-Fock (MCTDHF) method5–7 is the most general approach for fermionic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' It considers all the possible configurations for a given number of orbital functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' As further developed methods, the time-dependent complete-active-space self-consistent field method,10 the time-dependent restricted-active- space self-consistent field method9 and the time-dependent occupation-restricted multiple active-space method13 have also been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' They can significantly reduce the num- ber of configurations by classifying orbital functions and making restrictions on electronic excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Today, we can accurately simulate atoms containing several tens of electrons under intense/ultrashort laser pulses thanks to an efficient description of wave functions by TD-MCSCF methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='14 However, it is still difficult to handle molecular systems since simple and efficient dis- 2 cretization of the three-dimensional space such as the polar coordinate for atomic systems is not allowed without relying on the symmetries of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' One of the elaborated discretizations to simulate molecules without prohibitive computational cost is using mul- tiresolution grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The concept of the method is to discretize only a region near nuclei with fine grids and the other regions with grids coarse yet sufficiently fine to describe ionizing wave packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' We have previously implemented the MCTDHF method based on a multires- olution Cartesian grid and successfully computed high-harmonic generation from a water molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='15 In this study, we further extend our previous work to implement the MCTDHF method with a finite element method on an adaptively generated multiresolution mesh (adaptive finite element method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' As well as our previous implementation, only the center parts of the mesh are refined for sharp changes in wave functions and it gradually becomes coarse in the outer region such as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' We can also easily control the order of accuracy since finite element basis functions are used in each cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Furthermore, we introduce a highly stable propagator based on the short iterative Lanczos/Arnoldi propagator16 to address instability arising from high spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Our simulation code is parallelized for distributed memory environments, and consequently, achieved over a hundred times faster simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' In section II, our problem setting is defined and the MCTDHF method is briefly reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' In section III, we describe our implementation of spatial discretization using the adaptive finite element method, the time evolution of wave functions with the short iterative Arnoldi propagator, and parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' In section IV, we show a numerical result of high-harmonic generation from a water molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Conclusions are given in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Hereafter, we use atomic units unless otherwise indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 3 Figure 1: A part of an adaptive finite element mesh for a hydrogen molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The red spheres show positions of the nuclei and cell colors are electron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Molecular system and the MCTDHF method The Hamiltonian of electrons in a molecule under a laser field can be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' H = � i H1(ri) + 1 2 � i̸=j H2(ri, rj) (1) H1(ri) = −1 2∆i − � a Za |ri − ra| − iA(t) · ∇i (2) H2(ri, rj) = 1 |ri − rj| (3) where ri and ra are the positions of the ith electron and the ath nucleus and Za is the charge of the ath nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' A(t) = − � t ∞ E(t′)dt′ denotes the vector potential of a laser field applied to the simulated systems, where E(t) is the electric field of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Electronic wave functions are modeled by the multiconfiguration time-dependent Hartree- Fock (MCTDHF) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='5–7 Here, we just briefly reviews the method and show the equation of motions (EOMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The detailed descriptions and derivation of EOMs can be found in the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='10 The MCTDHF method expresses a multielectron wave function |Ψ⟩ with a super position of all the possible Slater determinants composed of a given time-dependent spatial orbital 4 set {φp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' |Ψ⟩ = � I CI(t) |I⟩ (4) CI(t) is a configuration interaction (CI) coefficient and |I⟩ is an electronic configuration (Slater determinant) composed of orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The equation of motion to variationally evolve the MCTDHF wave function can be derived from the time-dependent variational principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='17 The time-dependent variational principle requires that the action integral S[Ψ], S[Ψ] = � t1 t0 dt ⟨Ψ| ˆH − i ∂ ∂t |Ψ⟩ , (5) is stationary to an arbitrary infinitesimal wave function variation δΨ, δS δΨ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (6) As a solution of the stationary condition (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (6)), the equations of motion (EOMs) for CI coefficients and orbitals are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' i ˙CI = � J ⟨I| ˆH − i ˆX|J⟩ CJ (7) i | ˙φp⟩ = ˆQ � ˆH1 |φp⟩ + � oqrs (D−1)o pP qs or ˆW r s |φq⟩ � + i � q |φq⟩ Xq p (8) ˆX is an arbitrary anti-Hermitian operator, which can be determined as ˆX = � pq Xp q � σ ˆa† qσˆapσ, (9) where apσ(a† pσ) is the annihilation (creation) operator for a spatial orbital φp with σ spin (up-spin or down-spin), Xp q is an arbitrary anti-Hermitian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' In this work, we set Xp q to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' ˆQ is a projection operator onto the orthogonal complement of occupied orbitals, 5 ˆQ = 1 − � q |φq⟩⟨φq| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (10) D and P are one-body and two-body reduced density matrices, whose matrix elements are defined as Dp q = � σ ⟨Ψ|ˆa† qσˆapσ|Ψ⟩ (11) P pq sr = � στ ⟨Ψ|ˆa† sσˆa† rτˆaqτˆapσ|Ψ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (12) ˆW r s is the inter-electronic mean-field potential given by W r s (r) = � dr′φ∗ r(r′)φs(r′) |r − r′| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (13) Implementation This section shows the implementation of our simulation code developed in this work to solve Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (7) and (8) as an initial value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Simulations of molecular systems require efficient spatial discretization so that we can simulate electronic dynamics keeping accuracy with realistic computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' We employ the adaptive finite element method18,19 for the efficient discretization of orbitals based on an open-source finite element library deal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='20,21 As described below, while the adaptive finite element method realizes locally high spatial resolution, time evolution could be unstable due to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' To stably propagate wave functions for a long period, We employ the short iterative Arnoldi propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Although the short iterative Lanczos propagator is often used in many applications,16,22–24 since the system matrix is not Hermitian, the Arnodi algorithm is used instead of the Lanczos algorithm in this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Applying this scheme to all orbitals at once, we have enabled more stable time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' These numerical computation schemes are described in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 6 Adaptive finite element method The adaptive finite element (AFEM) used in this work is an approach to improve the accuracy of simulations requiring locally high resolution by using a multiresolution mesh generated by local mesh refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' A finite element mesh is generated by first discretizing the whole simulation box with coarse uniform cubic cells, and then dividing these cells into half the size in regions requiring higher resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' We can generate an adaptive multiresolution mesh by repeating the second process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Once the multiresolution mesh and cells are generated, most of the rest of the processes fall into the usual finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The mesh sizes are determined to make an error in each cell, which is given by Kelly’s error indicator 25 to estimate the error in each cell from the jump of the gradient of a target function, less than a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' This work adopts the Coulomb potential of the nuclei in a molecule as the target function for the error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' We also limit the minimum and maximum mesh sizes to avoid generating extremely small and large cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The basis functions located in each cell are direct products of the one-dimensional La- grange polynomials passing through the Gauss-Lobatto quadrature points in each cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The quadrature points in each cell are also constructed as the direct product of one-dimensional Gauss-Lobatto quadrature points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' This basis can be considered to be the three-dimensional version of the finite element discrete variable representation (FEDVR) basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='26,27 Let us define fI,i(r) as the i th basis function in the I th cell, and LI,jx(x), LI,jy(y) and LI,jz(z) the (jx, jy, jz) th Lagrange polynomials in each dimension in the I th cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Then, the function fI,i(r) is given by fI,i(r) = LI,jx(x)LI,jy(y)LI,jz(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (14) These functions are defined only in the I th cell and have zero values in other region than that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The finite element basis set {bk(r)} is constructed by the basis functions fI,i(r) which 7 have zero-value on the boundary of each cell and bridged functions that combine two bases with nonzero values at the quadrature point shared by two cells on the boundary of adjacent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The bridged functions are required to ensure the continuity of discretized functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The mesh generation and construction of the basis are carried out by using deal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='II functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' An arbitrary function h(r) is discretized with this finite element basis as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' h(r) = � k ckbk(r) (15) ck = � l ( ˜ M −1)k,l � drbl(r)h(r) (16) The matrix ˜ M is the overlap matrix of the basis set {bk(r)}, called the mass matrix in the finite element method, defined as ˜ Mk,l = � drbk(r)bl(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (17) All the spatial integrals are approximated with Gauss-Lobatto quadrature as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' � drh(x, y, z) ≃ � I � jx,jy,jz wx I,jxwy I,jywz I,jzh(xI,jx, yI,jy, zI,jz), (18) where wd I,jd (d = x, y, z) and (xI,jx, yI,jy, zI,jz) are the quadrature weights and points of the I th cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Based on this discretization scheme, the equation of motion (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (8)) is converted into a matrix-vector equation, i ˜ M ˙cp = (1 − ˜ M � q cqc† q) � ˜H1cp + ˜ M � oqrs (D−1)o pP qs or W r s ◦ cq � + i ˜ M � q cqXq p (19) 8 where cp denotes a coefficient vector of orbital φp(r) given by, (cp)k = � drbk(r)φp(r) (20) and the matrices ˜H1 is defined as the matrix form of the operator ˆH1, ( ˜H1)k,l = � drbk(r)H1(r)bl(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (21) W r s is a coefficient vector of the mean-field potential W r s (r) and the element-wise product is denoted by “◦”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' We compute the mean-field potential by solving the following Poisson’s equation, instead of directly calculating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (13), ∆W r s (r) = −4πφ∗ r(r)φs(r) (22) with a boundary condition W r s (r) ��� r∈Ω = � dr′φ∗ r(r′)φs(r′) |r − r′| , (23) where Ω denotes the boundary of a simulation box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' We solve this equation by the conjugate gradient method with algebraic multigrid preconditioning implemented in an open-source parallel linear algebra library Trilinos28 interfaced on deal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Short iterative Arnoldi propagator The short iterative Lanczos/Arnoldi propagator is a time evolution method, which approx- imates a Hamiltonian in a Krylov subspace by the Lanczos/Arnoldi algorithm and iterates short-time propagation of wave functions in the subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='16 This approach conserves the norm of a wave function when a Hamiltonian is Hermitian and enables unconditionally sta- 9 ble time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' It is also possible to use an adaptive time step or a variable Krylov subspace dimension based on the error estimation16 However, we cannot straightforwardly apply it to the equation of motion of orbitals, since it is only applicable to linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Although some applications of the MCSCF methods, where the EOM of orbitals is non- linear, use exponential integrators29–31 to enjoy the stability of the short iterative Lanc- zos/Arnoldi propagator even only for linear parts of the EOM, in our application, we found that the explicit time propagation of the nonlinear parts causes numerical instability prob- ably due to the quite fine mesh of AFEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' To avoid this problem, in this work, we propose an approach to apply the short iterative Lanczos/Arnoldi propagator by approximately re- garding the whole of the EOMs for all orbitals as one linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The equations of motion for all orbitals (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (8)) can be packed into a matrix-vector form, whose elements are operators and ket-vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' i ∂ ∂tφ = ˆGφ (24) φ = � �������� |φ1⟩ |φ2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' |φn⟩ � �������� , G = � �������� ˆG11 ˆG12 · · ˆG1N ˆG21 ˆG22 · · ˆG2N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' ˆGN1 ˆGN2 · · ˆGNN � �������� (25) The matrix element ˆGij is an operator defined as ˆGij = δi j ˆH1 + � osr (D−1)o iP js or ˆW r s − ⟨φj| � ˆH1 |φi⟩ + � oqrs (D−1)o iP js or ˆW r s |φq⟩ � + iXj i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (26) The equation (24) is approximately linear if we can assume that orbitals in the operators are invariable within a short time ∆t, and then time evolution of orbitals can be described as φ(t + ∆t) = exp � −i ˆG∆t � φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (27) 10 We achieve this time evolution by the short iterative Arnoldi scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Although this scheme has first-order accuracy since the time-dependency of the operator ˆG in a time step ∆t is not considered, it enables highly stable propagation including nonlinear parts and fits our implementation based on the AFEM using a fine mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The Krylov subspace dimension of the Arnoldi algorithm is determined so that errors estimated by the method found in the references16,23 are less than a threshold, which is set to be 10−10 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' We also adjust a time-step size, which is fixed during our simulations, to make the dimension 10-15 at a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Parallelization The spatial discretization and time evolution discussed above are devised to efficiently sim- ulate multielectron dynamics in a laser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Nevertheless, computational costs for the three-space to describe laser-induced ionization are huge , and distributed memory parallel computing is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The total number of degrees of freedom (DOF) NDOF in our simula- tion can simply be written as NDOF = Norbital × Nspace + NCI, where Norbital, Nspace and NCI are the numbers of orbitals, DOF associated with spatial discretization and CI coefficients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Norbital is typically from several to several tens, and Nspace usually increases up to several millions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' NCI significantly changes depending on a problem since it exponentially increases to the numbers of electrons and orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Our strategy to make efficient use of many processors in many situations is parallelizing orbital functions with respect to both the number of orbitals and the number of degrees of freedom in the AFEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' We divide the orbital function set {|φp⟩} by K and create K MPI groups to deal with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Each MPI group has L independent processes that are used to distribute a simulation box by using deal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='II functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Distribution of a simulation box and DOFs accompanying it is carried out by p4est,20,32 an open-source library to distribute octree structures across multiple processors, interfaced to deal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' This addresses load balancing and optimal distribution of the simulation box to reduce MPI communications among the processors (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 11 Figure 2: An example of a divided simulation box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The color-coded cells correspond the distribution to MPI processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Applications We simulate high harmonic generation from a water molecule to demonstrate the efficiency of our implementation by comparing our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='15 For atomic positions of a water molecule, two hydrogen atoms of a water molecule are located at (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='42994, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='10718, 0) and an oxygen atom is located at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The laser pulse used in this simulation has a wavelength of 2πc/ω = 400nm (c is the speed of light in vacuum) and a peak intensity of I0 = 8 × 1014 W/cm2, and is linearly polarized along with x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The pulse duration is 2 optical cycles with a triangular envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The shape of the electric field of the laser pulse is defined as, (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 3), E(t) = E0fenv(t) sin(ωt) (28) fenv(t) = � � � � � � � � � ωt 2π (0 ≤ ωt ≤ 2π) 4π − ωt 2π (2π ≤ ωt ≤ 4π) , (29) where E0 is the peak electric field derived from the peak intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The time-step size for real-time evolution is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='01 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='. 12 Figure 3: The electric field of the laser pulse used in this simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The simulation box is a cuboid defined within a region [−70, 70]×[−30, 30]×[−30, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' We apply the exterior complex scaling (ECS) as an absorbing boundary in the outside of a region [−35, 35] × [−10, 10] × [−10, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The details of the ECS can be found in the references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='33–35 The finite element mesh is generated to satisfy that the error in each cell is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='005, which has 6 different sizes between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='125 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='. At the most distant region from the molecule, the largest elements, which are cubes with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u long sides, are used to describe sufficiently absorbed orbital functions and the smallest elements, whose edge length is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='125 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=', are used in the vicinity of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Figure 4 displays the finite element mesh used in this simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The finite element basis is constructed from first- order Lagrange polynomials, and thus there are 8 quadrature points in a finite element cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' While it is possible to dynamically adapt a mesh to time-dependent orbital functions, we avoid such approaches due to additional computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' This would be helpful to gain computational efficiency if our problem was a larger system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' For the beginning of the simulation, we computed a ground state by imaginary-time evolution, whose electronic energy was -76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='905 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' In figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 5, we compare our simulation result with the previously calculated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' These spectra do not perfectly agree with each other since it is extremely difficult to achieve perfect convergence for spatial resolutions in 3D systems, Nevertheless, overall spectral shapes are quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' As well as the previous calculations, the simulations with 5 orbitals and 6 orbitals give almost the same spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='1 0 25 50 75 100Figure 4: Adaptively generated finite element mesh for a water molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The largest element is a cube of edge length 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' used to discretize the outer region, and the smallest one is a cube of edge length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='125 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' used only in the vicinity of nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The simulation using 6 orbitals of present work took 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='5 hours with 240 cores (6 nodes, 2 Intel Xeon Gold 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='40GHz processors with 20 cores in a node).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Remarkably, it is about 100 times faster than the previous work which took 28 days to finish the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' One of our achievements is successful distributed parallel computing using the 20 times larger resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' In addition to this, at least 5 times acceleration was gained by factors except for parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' The development of a highly stable propagator mainly contributes to this speed-up, which enables time evolution with 4 times as large a time-step size as the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Conclusion We have implemented the MCTDHF method based on the adaptive finite element method to simulate multielectron dynamics in molecules under laser fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' A further sophisticated discretization is realized by using the multiresolution grid used in our previous implementa- tion in the frame of the finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Thanks to the finite element method, we can automatically generate an adaptive mesh using Kelly’s error indicator and easily control the order of accuracy by changing the polynomial order of basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' While locally refined meshes enable efficient and accurate simulations, they possibly make time evolution unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 14 140 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 60 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 60 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='Figure 5: High harmonic spectra of a water molecule exposed to a laser pulse with a wave- length of 400nm and a peak intensity of 8 × 1014 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (a) The spectrum taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='15 The data is normalized for the maximum to be unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (b) The spectra computed by the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' We developed a more stable propagator based on the short iterative Arnoldi scheme than exponential integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' This propagator evolves all orbital functions together as a vector by using the short iterative Arnoldi scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' In addition, our simulation code is parallelized for distributed memory computing, which handles both the orbital set and spatial degrees of freedom in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' We have applied the present implementation to a simulation of high-harmonic generation from a water molecule in an intense visible laser pulse to compare with our previous work,15 and obtained the spectra showing a good agreement with overwhelmingly less computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Parallelization has made the greatest contribution to this reduction in computation time, and in this study, we were able to successfully use 20 times larger computational resources than in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' It is also important to note that we were able to use a 4 times larger time-step size thanks to the stable propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' This study prepared the adaptive mesh based on the discretization error of the Coulomb 15 (a) () 6 orbitals 6 orbitals 10-1 10-1 5 orbitals Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=') Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=') 10-3 10-3 10-5 10-5 10-7 10-7 10-9 10-9 10 20 30 0 10 20 0 30 Harmonic order Harmonic orderpotential of the nuclei, therefore the mesh is fixed during simulations, but it is possible to dynamically adapt the mesh to a wave function or nuclear positions at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' We consider that it brings efficiency when a larger simulation box is needed or when the nuclei can move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' In future works, we will present ab initio simulations of more complicated molecular systems and simulations considering nuclear dynamics in a combination of this development and more advanced theories such as the TD-ORMAS method13 and the time- dependent coupled cluster theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content='36 Data availability The data and source code used in this study are available upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Competing interests The authors declare there are no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Funding information This research was supported in part by a Grant-in-Aid for Scientific Research (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' JP19H00869, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' JP21K18903, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' JP22H05025) and a Grant-in-Aid for Early-Career Scientists (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' JP22K14616) from the Ministry of Education, Culture, Sports, Sci- ence and Technology (MEXT) of Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' This research was also partially supported by JST CREST (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' JPMJCR15N1) and by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number JPMXS0118067246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' References (1) Brabec, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Krausz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 2000, 72, 545–591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 16 (2) Chang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Fundamentals of attosecond optics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' CRC Press, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' (3) Calegari, F.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Kitzler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Koch, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' Kreuzer, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=';' metadata={'source': 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Journal of Chemical Physics 2018, 148, 051101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfeQA8/content/2301.02387v1.pdf'} diff --git a/4tFAT4oBgHgl3EQfmB2k/content/tmp_files/2301.08621v1.pdf.txt b/4tFAT4oBgHgl3EQfmB2k/content/tmp_files/2301.08621v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bc6dd24510f68d2d889e994d401c59d32009c2a6 --- /dev/null +++ b/4tFAT4oBgHgl3EQfmB2k/content/tmp_files/2301.08621v1.pdf.txt @@ -0,0 +1,732 @@ +arXiv:2301.08621v1 [quant-ph] 20 Jan 2023 +Improved Real-time Post-Processing for Qantum Random Number Generators +Qian Li,1 Xiaoming Sun,1 Xingjian Zhang,2 and Hongyi Zhou1, ∗ +1State Key Lab of Processors, Institute of Computing Technology, +Chinese Academy of Sciences, 100190, Beijing, China. +2Center for Qantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China +Randomness extraction is a key problem in cryptography and theoretical computer science. With the recent +rapid development of quantum cryptography, quantum-proof randomness extraction has also been widely +studied, addressing the security issues in the presence of a quantum adversary. In contrast with conventional +quantum-proof randomness extractors characterizing the input raw data as min-entropy sources, we find +that the input raw data generated by a large class of trusted-device quantum random number generators +can be characterized as the so-called reverse block source. Tis fact enables us to design improved extractors. +Specifically, we propose two novel quantum-proof randomness extractors for reverse block sources that realize +real-time block-wise extraction. In comparison with the general min-entropy randomness extractors, our +designs achieve a significantly higher extraction speed and a longer output data length with the same seed +length. In addition, they enjoy the property of online algorithms, which process the raw data on the fly without +waiting for the entire input raw data to be available. Tese features make our designs an adequate choice for +the real-time post-processing of practical quantum random number generators. Applying our extractors to the +raw data of the fastest known quantum random number generator, we achieve a simulated extraction speed +as high as 374 Gbps. +∗ zhouhongyi@ict.ac.cn + +I. +INTRODUCTION +Randomness extraction aims at distilling uniform randomness from a weak random source [1], which is widely +applied ranging from cryptography to distributed algorithms. Recently, quantum cryptography [2] has been devel- +oped rapidly, whose security is guaranteed by the fundamental principle of quantum mechanics [3, 4]. Compared +with its classical counterpart, the unique characteristic of intrinsic randomness enables quantum cryptography with +the feasibility of secure communication regardless of the computation power of the eavesdroppers. Randomness +extraction also serves as a key step, privacy amplification [5, 6], in the post-processing of quantum cryptography, +eliminating the side information possessed by a quantum adversary. +Randomness extraction is realized by various extractors that are mainly composed of a family of hashing func- +tions. In quantum cryptography, only quantum-proof extractors can provide information-theoretic security in the +presence of a quantum adversary [7]. Among all the quantum-proof extractors, Trevisan’s extractor [8, 9] and the +Toeplitz-hashing extractor [10] are two of the most popular choices. Both of them are strong extractors [11], which +means that the extracted randomness is independent of the seed. Trevisan’s extractor requires a seed length of only +polylogarithmic scaling of the input length, which is much lower than the linear scaling for Toeplitz-hashing ex- +tractor. On the other hand, the output speed is much lower than that of the Toeplitz-hashing extractor [12]. Te +Toeplitz-hashing extractor is constructed from a cyclic matrix, which is easy to implement and can be accelerated +by the Fast Fourier Transformation (FFT) [13]. +In the practical implementation of an extractor, there are some subtle practical issues. One is the security of the +block-wise post-processing adopted by most implementations of quantum random number generators (QRNGs). Te +seed used in the extractor is a limited resource. To save the seed and accelerate the extraction speed, the raw data +is divided into multiple small blocks. Ten a small length of seed is repeatedly used for each block. Tis method +assumes the small blocks of the raw data to be mutually independent. However, this is not satisfied by general input +raw data and results in security issues. Another problem is that the current extraction speed is much lower than +the raw data generation speed in most QRNG implementations [14]. Te real-time generation speed of the extracted +quantum random numbers is restricted by the extraction speed in the post-processing, which becomes a botleneck +in the applications of quantum random numbers in quantum communication tasks. Finally, for some commercial +QRNGs where the seed cannot be updated, the output data length is limited by the seed length. We want to extract +as many random numbers as possible with a limited seed length. +To deal with the practical issues above, we consider properties of some specific randomness sources beyond the +conventional min-entropy source characterization. In this work, we design two novel quantum-proof randomness +extraction algorithms for a large class of QRNGs where the raw data can be described as the reverse block source +[15]. Our results are inspired by the extractors designed in [16, 17] for block sources and Santha-Vazirani sources, +respectively. Both of our two extractors are online algorithms: as the input raw data arrives in real time, the two +extractors proceed on-the-fly input raw data piece-by-piece, where the processing is independent of the data in the +future. In fact, our two extractors are block-wise extractors. Tat is, they partition the input raw data into blocks +serially in the time order the input arrives, and then apply a min-entropy extractor to each block. Moreover, suppose +the input raw data is of length 푁, then our first extractor requires a seed length of 푂(log 푁) and takes equipartition +of the block lengths, and our second extractor requires only a seed length of 푂(1) and incremental block lengths. +Compared to the first extractor, the second one scarifies the extraction speed while enjoys a seed length independent +of the input length. As a result, this extractor can deal with infinite raw data, without the need of updating the seed +and determining the raw data length prior to extraction. For both extractors, the output length is a constant fraction +of the min-entropy of the raw data, which indicate that our extractors are quite efficient. To show the performances +of the extractors, we make a simulation estimating the output speed. It turns out that the extraction speed is adequate +for the post-processing of the fastest known implementations of QRNGs [18–20]. +II. +PRELIMINARIES +Troughout the paper, we use capital and lowercase leters to represent random variables and their assignments, +respectively. We use 푈푚 to represent the perfectly uniform random variable on 푚-bit strings and 휌푈푚 to represent +the 푚-dimensional maximally mixed state. +Definition II.1 (Conditional min-entropy). Let 푌 be a classical random variable that takes value 푦 with probability +2 + +푝푦 and E be a quantum system. Te state of the composite system can be writen as 휌푌 E = � +푦 푝푦|푦⟩⟨푦| ⊗ 휌푦 +E, where +{|푦⟩}푦 forms an orthonormal basis. Te conditional min-entropy of 푌 given E is 퐻min(푌 |E)휌푌 E = − log2 푝guess(푌 |E), +where 푝guess(푌 |E) is the maximum average probability of guessing 푌 given the quantum system E. Tat is, +푝guess(푌 |E) = max +{퐸푦 +E }푦 +�� +푦 +푝푦Tr +� +퐸푦 +E휌푦 +E +�� +, +(1) +where the maximization is taken over all positive operator-valued measures (POVMs) {퐸푦 +E}푦 on E. +When system 푌 is decoupled from E, where 휌푌 E = � +푦 푝푦|푦⟩⟨푦| ⊗ 휌E, the conditional min-entropy of 푌 given E +reduces to the classical min-entropy, 퐻min(푌) = − log2 max푦 푝푦. When 휌푌 E is clear from the context, we will denote +the conditional min-entropy as 퐻min(푌 |E) for brevity. +In this paper, we call the raw data generated by a QRNG as a random source. A general random source is the +min-entropy source, where the conditional min-entropy is lower bounded. +Definition II.2 (Min-entropy quantum-proof extractor). A function Ext : {0, 1}푛 × {0, 1}푑 → {0, 1}푚 is a (훿푛,휖) +min-entropy quantum-proof extractor, if for every random source 푌 and quantum system E satisfying 퐻min(푌 |E) ≥ 훿푛, +we have +1 +2 ∥휌Ext(푌,푆) E − 휌푈푚 ⊗ 휌E∥ ≤ 휖, +(2) +where 푆 is called the seed, which is a perfectly uniform random variable on 푑-bit strings independent of the system 푌 E +and ∥ · ∥ denotes the trace norm defined by ∥퐴∥ = Tr +√ +퐴†퐴. An extractor Ext is said to be strong if +1 +2 ∥휌Ext(푌,푆)푆 E − 휌푈푚 ⊗ 휌푈푑 ⊗ 휌E∥ ≤ 휖. +(3) +We call the concatenation of the output string of a strong extractor with the seed as an expansion, denoted as the tuple +(Ext(푦,푠),푠). +It is straightforward to see that an expansion is a standard (훿푛,휖) min-entropy quantum-proof extractor. If the +output of an extractor satisfies Eq. (2) or (3), we say that the output is 휖-close to a uniform distribution. +A widely used randomness extractor is the Toeplitz-hashing extractor. +Definition II.3 (Toeplitz-hashing extractor). A 푢 × 푛 matrix 푇 is a Toeplitz matrix if 푇 푖 푗 = 푇 푖+1,푗+1 = 푠푗−푖 for all +푖 = 1, · · · ,푢 − 1 and 푗 = 1, · · · ,푛 − 1. A Toeplitz matrix over the finite field 퐺퐹 (2), 푇푠, can be specified by a bit string +푠 = (푠1−푢,푠2−푢, · · · ,푠푛−1) of length 푢 + 푛 − 1.Given any 푛,푑 ∈ N+ where 푑 ≥ 푛, define the Toeplitz-hashing extractor +Ext푛,푑 +푇 +: {0, 1}푛 × {0, 1}푑 → {0, 1}푑−푛+1 as Ext푛,푑 +푇 (푦,푠) = 푇푠 · 푦, and define the expanded Toeplitz-hashing extractor +Ext푛,푑 +푇 ′ : {0, 1}푛 × {0, 1}푑 → {0, 1}2푑−푛+1 as Ext푛,푑 +푇 ′ (푦,푠) = (푇푠 ·푦,푠), where the matrix product operation · is calculated +over the field 퐺퐹 (2). +Since {푇푠 · 푦|푠 ∈ {0, 1}푢+푛−1} is a family of pairwise independent hashing functions [21, 22], according to the +quantum Lefover Hash Lemma [23], we can prove that the Toeplitz-hashing extractor is a min-entropy quantum- +proof strong extractor. +Lemma II.4 ([23]). For every 푛 ∈ N+ and 훿 > 0, Ext푛,푑 +푇 +is a (훿푛, 휖) min-entropy quantum-proof strong extractor, where +휖 = 2−(훿푛+푛−푑−1)/2. Equivalently, Ext푛,푑 +푇 ′ is a (훿푛,휖) min-entropy quantum-proof extractor. +Note that the output of Ext푛,푑 +푇 ′ has 2푑 − 푛 + 1 bits, where the last 푑 bits form the seed. We remark that though the +Toeplitz-hashing extractor Ext푛 +푇 is strong, the expanded Toeplitz-hashing extractor Ext푛 +푇 ′ is not. +III. +MAIN RESULT +We use 푋 = 푋1푋2 · · ·푋푁 ∈ ({0, 1}푏)푁 to denote the raw data generated by a QRNG, which contains 푁 samples +each of 푏 bits. For a set 퐼 ⊂ N+, we write 푋퐼 for the restriction of 푋 to the samples determined by 퐼. For example, if +퐼 = {2, 3, 5}, then 푋퐼 = 푋2푋3푋5. We use E to denote the quantum system possessed by the quantum adversary. +3 + +A. +Reverse block source +Intuitively, for a given randomness extractor, there is a trade-off between its performance and the generality of the +random sources it applies to. Te more special random sources the extractor works for, the beter performance the +extractor may achieve. In this paper, the notion of reverse block sources, which are more special than the min-entropy +sources, plays a critical role in the sense that (i) raw data of a large class of QRNGs can be described as a reverse +block source and (ii) prety good quantum-proof extractors for reverse block sources exist. A quantum version of +the reverse block source is defined below. +Definition III.1 (Reverse block source, adapted from Definition 1 in [15]). A string of random variables 푋 = +푋1 · · ·푋푁 ∈ ({0, 1}푏)푁 is a (푏, 푁,훿)-reverse block source given a quantum system E if for every 1 ≤ 푖 ≤ 푁 and every +푥푖+1,푥푖+2, · · · ,푥푁 , +퐻min(푋푖|푋푖+1 = 푥푖+1, · · · ,푋푁 = 푥푁, E) ≥ 훿 · 푏. +(4) +As shown later in Sec. III B, we introduce the reverse block source for the convenience of designing an online +algorithm. A reverse block source can be intuitively understood as a time-reversed block source 퐻min(푋푖|푋푖−1 = +푥푖−1, · · · ,푋1 = 푥1, E) where a new sample can not be completely predicted by the samples that already exists, i.e., +the net randomness increment is non-zero. For a reverse block source, these specific samples are from the future. Ac- +tually, for QRNGs where the raw data are mutually independent, such as the ones based on single photon detection +[24–26], vacuum fluctuations [27–29], and photon arrival time [30–32], the min-entropy source automatically satis- +fies Eq. (4), where ∀푥푖+1, · · · ,푥푁 , 퐻min(푋푖|E) = 퐻min(푋푖|푋푖+1 = 푥푖+1, · · · ,푋푁 = 푥푁, E), hence is also a reverse block +source. For QRNGs with correlated raw data, one can construct appropriate physical models to check whether Eq. (4) +is satisfied. In Appendix A, we take the fastest implementation, the one based on phase fluctuation of spontaneous +emission [18, 19], as an example to prove Eq. (4). +We also consider a smoothed version of the reverse block source. For 푋 = 푋1 · · ·푋푁 ∈ ({0, 1}푏)푁 , Denote the +underlying joint quantum state of the random source over the systems 푋 and E as 휌푋 E. We call 푋 a (푏, 푁,훿, 휖s)- +smoothed reverse block source, if there exists a state 휌∗ +푋 E that is 휀푠-close to 휌푋 E, +1 +2 ∥휌∗ +푋 E − 휌푋 E∥ ≤ 휀푠, +(5) +such that 휌∗ +푋 E is a reverse block source. Te smoothed reverse block source is in the same spirit of the smooth +conditional entropy [33]. Te motivation of introducing a smoothed version is to exclude singular points or region +in a probability distribution, which will help extract more randomness. +Here we remark that we mainly consider the trusted-device QRNGs where the extraction speed is a botleneck. For +QRNGs with a higher security level, such as the semi-device-independent QRNGs and device-independent QRNGs, +the reverse block source property is not satisfied in general. Tese types of QRNGs requires fewer assumptions and +characterizations on the devices at the expense of relatively low randomness generation rates of raw data. Ten their +real-time randomness generation rates are not limited by the extraction speed. +B. +Extractors for reverse block sources +In this section, we design two online quantum-proof extractors that can both extract a constant fraction of the +min-entropy from reverse block sources. Te two extractors both proceed in the following fashion: partition the +input raw data into blocks, and apply a min-entropy quantum-proof extractor to each block using part of the output +of the previous block as the seed. Te basic building block of our designs is family of (훿푛,휖푛) min-entropy quantum- +proof extractors, denoted by Ext푔 = {Ext푛 +푔 : 푛 ∈ N+}. Here, the subscript of Ext푔 means “gadget”. Let 푑푛 and 푚푛 +denote the seed length and output length of Ext푛 +푔, respectively, i.e., Ext푛 +푔 : {0, 1}푛 × {0, 1}푑푛 → {0, 1}푚푛. We require +the gadget Ext푛 +푔 to satisfy: (i) 휖푛 is exponentially small, for instance, 휖푛 = 2−훿푛/4, which aims at that the summation +�∞ +푛=1 휖푛 converges; and (ii) 푚푛 − 푑푛 = Ω(훿푛), which means that Ext푛 +푔 extracts a constant fraction of min-entropy +from the raw data. As an explicit construction, Ext푛 +푔 can be specified as the expanded Toeplitz-hashing extractor +Ext푛,푑푛 +푇 ′ +where 푑푛 = (1 +훿/2)푛 − 1, then 푚푛 −푑푛 = 훿푛/2, 푚푛 = (1 +훿)푛 − 1, and 휖푛 = 2−훿푛/4. In the rest of this paper, +4 + +we abbreviate Ext푛,푑푛 +푇 +and Ext푛,푑푛 +푇 ′ +where 푑푛 = (1 + 훿/2)푛 − 1 to Ext푛 +푇 and Ext푛 +푇 ′, respectively. Besides, to simplify +our discussion, we assume 휖푛 = 2−훿푛/4, and the analysis for other exponentially small values of 휖푛 is similar. +Te two extractors are described in Algorithms 1 and 2, respectively. Given an input raw data of length 푁, the +first extractor, named Ext푒푞 +푟푏푠, evenly partitions the input raw data into blocks each of size 푂(log 푁) and requires +푂(log 푁) random bits as the initial seed. In particular, if the min-entropy quantum-proof extractor Ext푔 in use is the +expansion of a strong extractor such as the expanded Toeplitz-hashing extractor, then Ext푒푞 +푟푏푠 degenerates exactly +to the following naive extractor: partition the input raw data into equal-sized blocks and apply the corresponding +strong extractor to each block separately with the same seed. Te second extractor, named Ext푛푒푞 +푟푏푠 , is inspired by +the extractor that can extract randomness from Santha-Vazirani sources using a seed of constant length [17]. It uses +only 푂(1) random bits as the initial seed and requires incremental block lengths. Compared to the first extractor, +the second one is less hardware-friendly and sacrifices the extraction speed in general. On the other hand, it enjoys +the property that the seed length is independent of the input length. As a result, this extractor can deal with infinite +raw data, without the need of determining the raw data length prior to extraction. In other words, one does not need +to update the seed in practical implementations. We remark that the initial seed is indispensable because there does +not exist any nontrivial deterministic extractor for reverse block sources. Te proof is presented in Appendix B. +Algorithm 1: Ext푒푞 +푟푏푠 +1 Input: 푏 ∈ N+ and 0 < 휖, 훿 < 1. A string 푥 = 푥1, 푥2, · · · , 푥푁 sampled from a (푏, 푁,훿)-reverse block source; +2 Let 푖 := 1 and 푛 := +� +4 +훿푏 · log +� +푁 +휖 +�� +; +3 Sample a uniform random bit string 푠 (1) of length 푑푏푛; +4 for ℓ = 1 to 푁 /푛 do +5 +Let 퐼ℓ := [푖,푖 + 푛 − 1]; +6 +Compute 푧 (ℓ) := Ext푏푛 +푔 (푥퐼ℓ ,푠 (ℓ)); +7 +Let 푖 := 푖 + 푛; +8 +Cut 푧 (ℓ) into two substrings, denoted by 푟 (ℓ) and 푠 (ℓ+1), of size 푚푏푛 − 푑푏푛 and 푑푏푛 respectively; +9 +Output 푟 (ℓ). +Algorithm 2: Ext푛푒푞 +푟푏푠 +1 Input: 푏 ∈ N+ and 0 < 훿 < 1. A string 푥 = 푥1,푥2, · · · sampled from a (푏, ∞,훿)-reverse block source; +2 Parameter: 푛1, Δ ∈ N+; +3 Let 푖 := 1; +4 Sample a uniform random bit string 푠 (1) of length 푑푏푛1; +5 for ℓ = 1 to ∞ do +6 +Let 퐼ℓ := [푖,푖 + 푛ℓ − 1]; +7 +Compute 푧 (ℓ) := Ext푏푛ℓ +푔 +(푥퐼ℓ ,푠 (ℓ)); +8 +Let 푖 := 푖 + 푛ℓ and 푛ℓ+1 := 푛ℓ + Δ; +9 +Cut 푧 (ℓ) into two substrings, denoted by 푟 (ℓ) and 푠 (ℓ+1), of size 푚푏푛ℓ − 푑푏푛ℓ+1 and 푑푏푛ℓ+1 respectivelya; +10 +Output 푟 (ℓ). +a We should choose the parameters 푛1 and Δ properly to have 푚푏푛ℓ − 푑푏푛ℓ+1 ≥ 1. +Note that if we impose Δ = 0 and let 푛1 = +� 4 +훿푏 · log � 푁 +휖 +�� +, then Ext푛푒푞 +푟푏푠 becomes Ext푒푞 +푟푏푠. We first analyze the second +extractor Ext푛푒푞 +푟푏푠 . +Teorem III.2. Te extractor Ext푛푒푞 +푟푏푠 satisfies the following properties: +(a) It uses a seed of 푑푏푛1 length. +5 + +FIG. 1: Illustration of Ext푛푒푞 +푟푏푠 . +(b) For any 푘 ∈ N, we have +1 +2 ∥휌푟 (1)◦푟 (2)◦···◦푟 (푘) E − 휌푈휂푘 ⊗ 휌E∥ ≤ +푘 +� +ℓ=1 +2−훿푏푛ℓ/4 < +2−훿푏푛1/4 +1 − 2−훿푏Δ/4, +(6) +where 휂푘 = �푘 +ℓ=1(푚푏푛ℓ − 푑푏푛ℓ+1). +Proof. Te nontrivial part is Part (b). For convenience of presentation, we use 퐼푘:∞ to represent 퐼푘 ∪ 퐼푘+1 ∪ · · · ∪ 퐼∞. +In fact, we will prove that for any 푘 ∈ N it has +1 +2 +���휌푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1) ◦푋퐼푘+1:∞ E − 휌푈휂푘 +푑푏푛푘+1 ⊗ 휌푋퐼푘+1:∞ E +��� ≤ +푘 +� +ℓ=1 +2−훿푏푛ℓ/4, +(7) +which implies Part (b) immediately. +Te proof is by an induction on 푘. Te base case when 푘 = 0 is trivial. Te induction proceeds as follows. Suppose +Eq. (7) is true. Ten due to the contractivity of trace-preserving quantum operations, we have +1 +2 +����휌푟 (1)◦푟 (2)◦···◦푟 (푘)◦Ext푔(푋퐼푘+1,푠 (푘+1))◦푋퐼푘+2:∞ E − 휌푈휂푘 ⊗ 휌Ext푔 +� +푋퐼푘+1,푈푑푏푛푘+1 +� +◦푋퐼푘+2:∞ E +���� ≤ +푘 +� +ℓ=1 +2−훿푏푛ℓ/4. +(8) +On the other hand, by Definition III.1, for any assignment 푥퐼푘+2:∞ of 푋퐼푘+2:∞, we have +퐻min(푋퐼푘+1 | 푋퐼푘+2:∞ = 푥퐼푘+2:∞, E) ≥ 훿푏푛푘+1. +(9) +Ten, recalling that Ext푔 is a min-entropy quantum-proof extractor, it follows that +1 +2 +����휌Ext푔 +� +푋퐼푘+1,푈푑푏푛푘+1 +� +◦푋퐼푘+2:∞ E − 휌푈푚푏푛푘+1 ⊗ 휌푋퐼푘+2:∞ E +���� ≤ 2−훿푏푛푘+1/4. +(10) +Tus, +1 +2 +����휌푈휂푘 ⊗ 휌Ext푔 +� +푋퐼푘+1,푈푑푏푛푘+1 +� +◦푋퐼푘+2:∞ E − 휌푈휂푘 ⊗ 휌푈푚푏푛푘+1 ⊗ 휌푋퐼푘+2:∞ E +���� ≤ 2−훿푏푛푘+1/4. +(11) +Finally, combining inequalities (8) and (11) and applying the triangle inequality, we conclude that +1 +2 +���휌푟 (1)◦푟 (2)◦···◦푟 (푘+1)◦푠 (푘+2)◦푋퐼푘+2:∞ E − 휌푈휂푘+1+푑푏푛푘+2 ⊗ 휌푋퐼푘+2:∞ E +��� ≤ +푘+1 +� +ℓ=1 +2−훿푏푛ℓ/4. +(12) +By using the summation formula for the geometric progression, the above inequality is further upper bounded by +2−훿푏푛1/4 +1−2−훿푏Δ/4 . +□ +6 + +n1 +n1 + ( - 1) +n1 + △ +x +X1 +X +Xe +5(1 +Ext +Ext +Ext, +6 +9 +9 +r(1) +r(2) +r(t)As can be seen from the proof, the parameter Δ of Ext푛푒푞 +푟푏푠 must be strictly positive, since the upper bound +�푘 +ℓ=1 2−훿푏푛ℓ/4 on the error converges if and only if Δ > 0. Teorem III.2 implies that the (infinitely long) output +string 푟 (1) ◦ 푟 (2) ◦ · · · can be arbitrarily close to the uniform distribution by choosing a sufficiently large constant +푛1. As an explicit construction, suppose Ext푛푒푞 +푟푏푠 adopts the expanded Toeplitz-hashing extractor Ext푛 +푇 ′ as the gadget +Ext푛 +푔, where 푑푛 = (1 +훿/2)푛 − 1 and 푚푛 = (1 +훿)푛 − 1. We further set Δ = 1. We require that 푛1 ≥ 4/훿 + 1 such that +푚푏푛ℓ −푑푏푛ℓ+1 = 푏(훿푛ℓ/2 − 1 −훿/2) ≥ 1 for any ℓ. Ten Ext푛푒푞 +푟푏푠 extracts (1 +훿)푏(푛1 + ℓ − 1) − 1 random bits from the +ℓ-th block 푥퐼ℓ , outputs the first 푚푏푛ℓ − 푑푏푛ℓ+1 ≈ 훿푏푛ℓ/2 bits, and then uses the last (1 + 훿/2)푏(푛1 + ℓ) − 1 bits as the +seed of the next block. +Via a similar argument as in Teorem III.2, we have the following result for Ext푒푞 +푟푏푠. +Teorem III.3. Te extractor Ext푒푞 +푟푏푠 uses 푑푏푛 random bits as a seed and outputs a string 푟 (1) ◦ 푟 (2) ◦ · · · ◦ 푟 (푁/푛) +satisfying that +1 +2 ∥휌푟 (1)◦푟 (2)◦···◦푟 (푁 /푛)◦E − 휌푈휂 ⊗ 휌E∥ ≤ 푁 +푛 · 2−훿푏푛/4 ≤ 휖, +(13) +where 휂 := 푁 +푛 · (푚푏푛 − 푑푏푛). +In particular, suppose Ext푒푞 +푟푏푠 adopts Ext푛 +푇 ′ as the gadget Ext푛 +푔. Ten Ext푒푞 +푟푏푠 extracts (1+훿)푏푛−1 random bits from +the ℓ-th block 푥퐼ℓ and outputs the first 푚푏푛 −푑푏푛 = 훿푏푛/2 bits. Te last 푑푏푛 = (1+훿/2)푏푛 −1 bits, which is exactly the +seed used in this block, will be reused as the seed in the next block. Terefore, Ext푒푞 +푟푏푠 uses 푑푏푛 ≈ (4/훿 + 2) log (푁/휖) +random bits as seed, and outputs (푁/푛) · (푚푏푛 − 푑푏푛) ≈ 훿푏푁/2 bits in total. Tough Ext푒푞 +푟푏푠 uses more seed than +Ext푛푒푞 +푟푏푠 , it is more hardware-friendly and can achieve much higher extraction speed. +Corollary III.4. Suppose the random sources in Algorithms 1 and 2 are replaced with the (푏, 푁,훿,휖s) and (푏, ∞,훿,휖s)- +smoothed reverse block sources, respectively. By using the same processing procedures, the output state of the extractor +Ext푛푒푞 +푟푏푠 satisfies +1 +2 ∥휌푟 (1)◦푟 (2)◦···◦푟 (푘) E − 휌푈휂푘 ⊗ 휌E∥ < +2−훿푏푛1/4 +1 − 2−훿푏Δ/4 + 2휖푠, +(14) +and the output state of the extractor Ext푒푞 +푟푏푠 satisfies +1 +2 ∥휌푟 (1)◦푟 (2)◦···◦푟 (푁 /푛)◦E − 휌푈휂 ⊗ 휌E∥ ≤ 휖 + 2휖푠. +(15) +Proof. We prove the smoothed version of Algorithm 2, and the proof for the smoothed version of Algorithm 1 follows +essentially the same procedures. For brevity, denote +2−훿푏푛1/4 +1−2−훿푏Δ/4 := 휖. According to the definition of the smoothed +reverse block source, there exists a state 휌∗ +푋 E that is 휖푠-close to the real output state 휌푋 E such that 휌∗ +푋 E determines +a (푏, ∞,훿) reverse block source. Using the result in Tm. III.2, +1 +2 +���휌푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1)◦푋퐼푘+1:∞ E − 휌푈휂푘 +푑푏푛푘+1 ⊗ 휌푋퐼푘+1:∞ E +��� +≤1 +2 +���휌푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1)◦푋퐼푘+1:∞ E − 휌∗ +푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1) ◦푋퐼푘+1:∞ E +��� ++ 1 +2 +���휌∗ +푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1) ◦푋퐼푘+1:∞ E − 휌푈휂푘 +푑푏푛푘+1 ⊗ 휌∗ +푋퐼푘+1:∞ E +��� ++ 1 +2 +���휌푈휂푘 +푑푏푛푘+1 ⊗ 휌∗ +푋퐼푘+1:∞ E − 휌푈휂푘 +푑푏푛푘+1 ⊗ 휌푋퐼푘+1:∞ E +��� +≤휖s + 휖 + 휖s = 휖 + 2휖s, +(16) +where the first inequality is due to the contractivity of trace-preserving quantum operations while the last inequality +comes from the fact that the purified distance is an upper bound of the trace distance. +□ +Tis result implies that the output for a smoothed reverse block source can also be arbitrarily close to a uniform +distributed sequence. +7 + +IV. +SIMULATIONS OF THE REAL-TIME RANDOMNESS GENERATION RATE +In this section, we make a simulation estimating the extraction speed of the first extractor Ext푒푞 +푟푏푠 implemented +in the Xilinx Kintex-7 XC7K480T Field Programmable Gate Array (FPGA), a common application in industry. Here, +we adopt the expanded Toeplitz-hashing extractor Ext푛 +푇 ′ as the gadget Ext푛 +푔. We use the raw data from Ref. [20] as +the random source, which is a reverse block source with parameters 푏 = 8 and 훿 = 0.85. We consider a raw data +length of 푁 = 251 bits and a total security parameter 휖 = 10−10, which means the final output data is 10−10-close to +a uniform distribution. Correspondingly, the block length is 푛 = +� 4 +훿푏 · log � 푁 +휖 +�� += 50. +According to the discussion afer Teorem III.3, Ext푒푞 +푟푏푠 will use a seed of (1 + 훿/2)푏푛 − 1 = 569 bits and output +about 0.85 PB random bits. Precisely, Ext푒푞 +푟푏푠 simply divides the input raw data into blocks with length 50, where +each block contains 푏 × 푛 = 400 bits, and multiplies a 170 × 400 Toeplitz matrix by a 400-dimension vector over +퐺퐹 (2). Since the addition 푎 + 푏 and multiplication 푎 · 푏 over 퐺퐹 (2) are exactly 푎 ⊕ 푏 and 푎 ∧ 푏, respectively, which +are both basic logical operations, the matrix multiplication involves 170 × 400 = 68000 ‘∧’ operations and 68000 ‘⊕’ +operations. +Te parameters of the Xilinx Kintex-7 XC7K480T FPGA are as follows. Te clock rate is set to be 200 MHz; the +number of Look-Up-Tables (LUTs) is 3 × 105; each LUT can perform 5 basic logical operations simultaneously. To +make full use of the FPGA, we can perform the matrix multiplications of ⌊3 × 105 × 5/(2 ∗ 68000)⌋ = 11 blocks in +parallel. Terefore, the extraction speed of Ext푒푞 +푟푏푠 is 200 × 106 × 11 × 170 = 374 Gbps, which is improved by one +order of magnitude compared to the state-of-the-art result [34]. As a result, the extraction speed is not a botleneck +for the high-speed QRNGs any more; hence our online extraction is adequate for the post-processing of the fastest +known implementation of QRNGs [18, 19]. +V. +CONCLUSION +In conclusion, we design two novel quantum randomness extractors based on the reverse-block-source property +that is satisfied by a large class of trusted-device QRNGs. Tese results provide theoretical supports to the current +real-time block-wise post-processing widely applied in experiments and industry. Te first extractor improves the +real-time extraction speed while the second one can extract infinite raw data with only a constant seed length. In +particular, the first extractor is easy to be implemented in a FPGA. Te real-time extraction speed with a common +FPGA is high enough for the real-time post-processing of the fastest known QRNGs. +For future work, it is interesting to explore other properties beyond the general min-entropy source to improve +the post-processing. Te improvement may come from boosting the extraction speed or saving the seed length. On +the other hand, randomness extraction with an imperfect seed or even without seed is also a practical and promising +direction. An interesting open question is whether randomness can still be extracted online without the seed for +certain non-trivial random sources. +ACKNOWLEDGMENTS +We thank Y. Nie and B. Bai for enlightening discussions, and Salil Vadhan for telling us the details of the extractor +which extracts randomness from Santha-Vazirani sources using a seed of constant length. Tis work was supported +in part by the National Natural Science Foundation of China Grants No. 61832003, 61872334, 61801459, 62002229, +1217040781, and the Strategic Priority Research Program of Chinese Academy of Sciences Grant No. XDB28000000. +Appendix A: QRNG raw data as a reverse block source +We focus on the QRNG based on measuring the phase fluctuation of a laser, which is the fastest and most widely +applied implementation. Te details of the QRNG design and randomness quantification are given in Ref. [35]. Here, +we make a brief summary. A laser wave with a random phase 휙(푡) passes through an interferometer with time delay +휏푙. Afer the interference, the random phase fluctuation is then transformed into an intensity fluctuation and can be +sampled by an analog-to-digital converter (ADC) with a sampling frequency 1/휏푠. Te laser intensity fluctuation is +8 + +transformed into voltage signal 푉 in ADC. When the sampled voltage signal falls in some interval of the ADC, it will +generate a sequence of corresponding random numbers. Te sequence length determines the resolution of the ADC. +For example, an 8-bit ADC will be divided into 28 intervals. We illustrate the physical seting in Fig. 2. +FIG. 2: Typical seting of a QRNG based on phase fluctuation. MZI: Mach-Zehnder interferometer; PD: photo +detector; ADC: analog-to-digital converter. +According to Ref. [35], the voltage will be proportional to the phase difference Δ휙(휏) = 휙(푡 + 휏) − 휙(푡), i.e., +푉 = 푘Δ휙(휏푙), where 푘 is a constant. We assume that the spontaneous emission leads to a differential random +phase characterized by a Gaussian white noise in time domain 퐹sp(푡), whose expectation and variance are given by +E[퐹sp(푡)] = 0 and Var(퐹sp(푡)) = 휎2, respectively. Te phase difference comes from the integration of the differential +random phase, +Δ휙(휏) = +∫ 푡0+휏 +푡=푡0 +퐹sp(푡)푑푡. +(A1) +Ten, Δ휙(휏) follows a Gaussian distribution 퐺(0, +� +휎2휏푙) with zero mean and variance 휏휎2 due to the property of +Gaussian white noise. Te voltage also follows a Gaussian distribution, 퐺(0, +� +푘2휎2휏푙). When 휏푠 > 휏푙, the raw data +generated by each sample will be independent. In the asymptotic limit, the conditional min-entropy per sample is +given by [35] +퐻min(푉푖|E) ≥ − log2 +� +max +푗 +∫ +푉푖 ∈푆푗 +퐺(0, +� +푘2휎2휏푙)푑푉푖 +� +, +(A2) +where 푉푖 is the 푖-th sample and 푆푗 is the 푗-th interval of the analog-to-digital converter (ADC). While when 휏푠 ≤ +휏푙, there will be correlations between samples and randomness quantification will be different. Without loss of +generality, we consider 휏푙 = 푞휏푠 with some integer 푞, as shown in Fig. 3. Comparing the adjacent two samples, for +FIG. 3: Illustration of the phase interference. +example, the first two samples, we have +Sample1 : +Δ휙1(휏푙) = 휙(푡 = (푞휏푠)) − 휙(푡 = 0) = +∫ 푡=휏푠 +푡=0 +퐹sp(푡)푑푡 + +∫ 푡=푞휏푠 +푡=휏푠 +퐹sp(푡)푑푡, +Sample2 : +Δ휙2(휏푙) = 휙(푡 = (푞 + 1)휏푠) − 휙(푡 = 휏푠) = +∫ 푡=푞휏푠 +푡=휏푠 +퐹sp(푡)푑푡 + +∫ 푡=(푞+1)휏푠 +푡=푞휏푠 +퐹sp(푡)푑푡. +(A3) +9 + +Ti +中1 Φ2 +Φ3 +Φn-2 Φn-1 Φn Φn+1 +t +TsMZI +Laser +PD +ADCSuppose a 푏-bit ADC is applied in the experiment. Ten 푏-bit raw data can be generated per sample, with a min- +entropy lower bound +퐻min(푋푖|E) ≥ − log2 +� +max +푗 +∫ +푉푖 ∈푆푗 +퐺(0, +� +푘2휎2휏푙)푑푉푖 +� +, +(A4) +where 푋푖 ∈ {0, 1}푏 is the sequence of raw random numbers corresponding to the voltage signal 푉푖. +Te conditional min-entropy 퐻min(푋푖|푋푖+1 = 푥푖+1, · · · , 푋∞ = 푥∞, E) will be less than 퐻min(푉푖). Te net increment +randomness comes from the integration of random phase in one time bin of an interval 휏푠, which corresponds to an +effective variance of the voltage +Var(푉)eff = Var(푉)휏푠 +휏푙 +. +(A5) +Ten we have +퐻min(푋푖|푋푖+1 = 푥푖+1, · · · , 푋∞ = 푥∞, E) ≥ − log2 +� +max +푗 +∫ +푉푖 ∈푆푗 +퐺(0, +� +푘2휎2휏푠)푑푉푗 +� +:= 훿푏, +(A6) +which forms a (푏, ∞,훿)-reverse block source. +Appendix B: Initial seed is indispensable for reverse block sources +Te following theorem claims that there does not exist any deterministic extractor that can extract even one bit +of almost uniformly-distributed random number from every reverse block source. Te proof is essentially the same +as that for Santha-Vazirani sources [36, 37]. +Teorem B.1. For all 푏, 푁 ∈ N+ and 0 < 훿 < 1, and any function Ext : {0, 1}푏푁 → {0, 1}, there exists a (푏, 푁,훿)- +reverse block source 푋 such that either P[Ext(푋) = 1] ≥ 2−훿 or P[Ext(푋) = 1] ≤ 1 − 2−훿. +Proof. 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Wigderson, “A note on extracting randomness from santha-vazirani sources,” (2004), unpub- +lished. +11 + diff --git a/4tFAT4oBgHgl3EQfmB2k/content/tmp_files/load_file.txt b/4tFAT4oBgHgl3EQfmB2k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f88eaecc672f5f9ef6be69df0b6b4249ca76da47 --- /dev/null +++ b/4tFAT4oBgHgl3EQfmB2k/content/tmp_files/load_file.txt @@ -0,0 +1,557 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf,len=556 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='08621v1 [quant-ph] 20 Jan 2023 Improved Real-time Post-Processing for Qantum Random Number Generators Qian Li,1 Xiaoming Sun,1 Xingjian Zhang,2 and Hongyi Zhou1, ∗ 1State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences, 100190, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 2Center for Qantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China Randomness extraction is a key problem in cryptography and theoretical computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' With the recent rapid development of quantum cryptography, quantum-proof randomness extraction has also been widely studied, addressing the security issues in the presence of a quantum adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In contrast with conventional quantum-proof randomness extractors characterizing the input raw data as min-entropy sources, we find that the input raw data generated by a large class of trusted-device quantum random number generators can be characterized as the so-called reverse block source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Tis fact enables us to design improved extractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Specifically, we propose two novel quantum-proof randomness extractors for reverse block sources that realize real-time block-wise extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In comparison with the general min-entropy randomness extractors, our designs achieve a significantly higher extraction speed and a longer output data length with the same seed length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In addition, they enjoy the property of online algorithms, which process the raw data on the fly without waiting for the entire input raw data to be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Tese features make our designs an adequate choice for the real-time post-processing of practical quantum random number generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Applying our extractors to the raw data of the fastest known quantum random number generator, we achieve a simulated extraction speed as high as 374 Gbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' ∗ zhouhongyi@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='cn I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' INTRODUCTION Randomness extraction aims at distilling uniform randomness from a weak random source [1], which is widely applied ranging from cryptography to distributed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Recently, quantum cryptography [2] has been devel- oped rapidly, whose security is guaranteed by the fundamental principle of quantum mechanics [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Compared with its classical counterpart, the unique characteristic of intrinsic randomness enables quantum cryptography with the feasibility of secure communication regardless of the computation power of the eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Randomness extraction also serves as a key step, privacy amplification [5, 6], in the post-processing of quantum cryptography, eliminating the side information possessed by a quantum adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Randomness extraction is realized by various extractors that are mainly composed of a family of hashing func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In quantum cryptography, only quantum-proof extractors can provide information-theoretic security in the presence of a quantum adversary [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Among all the quantum-proof extractors, Trevisan’s extractor [8, 9] and the Toeplitz-hashing extractor [10] are two of the most popular choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Both of them are strong extractors [11], which means that the extracted randomness is independent of the seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Trevisan’s extractor requires a seed length of only polylogarithmic scaling of the input length, which is much lower than the linear scaling for Toeplitz-hashing ex- tractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' On the other hand, the output speed is much lower than that of the Toeplitz-hashing extractor [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te Toeplitz-hashing extractor is constructed from a cyclic matrix, which is easy to implement and can be accelerated by the Fast Fourier Transformation (FFT) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In the practical implementation of an extractor, there are some subtle practical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' One is the security of the block-wise post-processing adopted by most implementations of quantum random number generators (QRNGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te seed used in the extractor is a limited resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' To save the seed and accelerate the extraction speed, the raw data is divided into multiple small blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Ten a small length of seed is repeatedly used for each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Tis method assumes the small blocks of the raw data to be mutually independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' However, this is not satisfied by general input raw data and results in security issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Another problem is that the current extraction speed is much lower than the raw data generation speed in most QRNG implementations [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te real-time generation speed of the extracted quantum random numbers is restricted by the extraction speed in the post-processing, which becomes a botleneck in the applications of quantum random numbers in quantum communication tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Finally, for some commercial QRNGs where the seed cannot be updated, the output data length is limited by the seed length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We want to extract as many random numbers as possible with a limited seed length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' To deal with the practical issues above, we consider properties of some specific randomness sources beyond the conventional min-entropy source characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In this work, we design two novel quantum-proof randomness extraction algorithms for a large class of QRNGs where the raw data can be described as the reverse block source [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Our results are inspired by the extractors designed in [16, 17] for block sources and Santha-Vazirani sources, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Both of our two extractors are online algorithms: as the input raw data arrives in real time, the two extractors proceed on-the-fly input raw data piece-by-piece, where the processing is independent of the data in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In fact, our two extractors are block-wise extractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Tat is, they partition the input raw data into blocks serially in the time order the input arrives, and then apply a min-entropy extractor to each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Moreover, suppose the input raw data is of length 푁, then our first extractor requires a seed length of 푂(log 푁) and takes equipartition of the block lengths, and our second extractor requires only a seed length of 푂(1) and incremental block lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Compared to the first extractor, the second one scarifies the extraction speed while enjoys a seed length independent of the input length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' As a result, this extractor can deal with infinite raw data, without the need of updating the seed and determining the raw data length prior to extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' For both extractors, the output length is a constant fraction of the min-entropy of the raw data, which indicate that our extractors are quite efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' To show the performances of the extractors, we make a simulation estimating the output speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' It turns out that the extraction speed is adequate for the post-processing of the fastest known implementations of QRNGs [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' PRELIMINARIES Troughout the paper, we use capital and lowercase leters to represent random variables and their assignments, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We use 푈푚 to represent the perfectly uniform random variable on 푚-bit strings and 휌푈푚 to represent the 푚-dimensional maximally mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='1 (Conditional min-entropy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Let 푌 be a classical random variable that takes value 푦 with probability 2 푝푦 and E be a quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te state of the composite system can be writen as 휌푌 E = � 푦 푝푦|푦⟩⟨푦| ⊗ 휌푦 E, where {|푦⟩}푦 forms an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te conditional min-entropy of 푌 given E is 퐻min(푌 |E)휌푌 E = − log2 푝guess(푌 |E), where 푝guess(푌 |E) is the maximum average probability of guessing 푌 given the quantum system E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Tat is, 푝guess(푌 |E) = max {퐸푦 E }푦 �� 푦 푝푦Tr � 퐸푦 E휌푦 E �� , (1) where the maximization is taken over all positive operator-valued measures (POVMs) {퐸푦 E}푦 on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' When system 푌 is decoupled from E, where 휌푌 E = � 푦 푝푦|푦⟩⟨푦| ⊗ 휌E, the conditional min-entropy of 푌 given E reduces to the classical min-entropy, 퐻min(푌) = − log2 max푦 푝푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' When 휌푌 E is clear from the context, we will denote the conditional min-entropy as 퐻min(푌 |E) for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In this paper, we call the raw data generated by a QRNG as a random source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' A general random source is the min-entropy source, where the conditional min-entropy is lower bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='2 (Min-entropy quantum-proof extractor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' A function Ext : {0, 1}푛 × {0, 1}푑 → {0, 1}푚 is a (훿푛,휖) min-entropy quantum-proof extractor, if for every random source 푌 and quantum system E satisfying 퐻min(푌 |E) ≥ 훿푛, we have 1 2 ∥휌Ext(푌,푆) E − 휌푈푚 ⊗ 휌E∥ ≤ 휖, (2) where 푆 is called the seed, which is a perfectly uniform random variable on 푑-bit strings independent of the system 푌 E and ∥ · ∥ denotes the trace norm defined by ∥퐴∥ = Tr √ 퐴†퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' An extractor Ext is said to be strong if 1 2 ∥휌Ext(푌,푆)푆 E − 휌푈푚 ⊗ 휌푈푑 ⊗ 휌E∥ ≤ 휖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (3) We call the concatenation of the output string of a strong extractor with the seed as an expansion, denoted as the tuple (Ext(푦,푠),푠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' It is straightforward to see that an expansion is a standard (훿푛,휖) min-entropy quantum-proof extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' If the output of an extractor satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (2) or (3), we say that the output is 휖-close to a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' A widely used randomness extractor is the Toeplitz-hashing extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='3 (Toeplitz-hashing extractor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' A 푢 × 푛 matrix 푇 is a Toeplitz matrix if 푇 푖 푗 = 푇 푖+1,푗+1 = 푠푗−푖 for all 푖 = 1, · · · ,푢 − 1 and 푗 = 1, · · · ,푛 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' A Toeplitz matrix over the finite field 퐺퐹 (2), 푇푠, can be specified by a bit string 푠 = (푠1−푢,푠2−푢, · · · ,푠푛−1) of length 푢 + 푛 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='Given any 푛,푑 ∈ N+ where 푑 ≥ 푛, define the Toeplitz-hashing extractor Ext푛,푑 푇 : {0, 1}푛 × {0, 1}푑 → {0, 1}푑−푛+1 as Ext푛,푑 푇 (푦,푠) = 푇푠 · 푦, and define the expanded Toeplitz-hashing extractor Ext푛,푑 푇 ′ : {0, 1}푛 × {0, 1}푑 → {0, 1}2푑−푛+1 as Ext푛,푑 푇 ′ (푦,푠) = (푇푠 ·푦,푠), where the matrix product operation · is calculated over the field 퐺퐹 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Since {푇푠 · 푦|푠 ∈ {0, 1}푢+푛−1} is a family of pairwise independent hashing functions [21, 22], according to the quantum Lefover Hash Lemma [23], we can prove that the Toeplitz-hashing extractor is a min-entropy quantum- proof strong extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Lemma II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='4 ([23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' For every 푛 ∈ N+ and 훿 > 0, Ext푛,푑 푇 is a (훿푛, 휖) min-entropy quantum-proof strong extractor, where 휖 = 2−(훿푛+푛−푑−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Equivalently, Ext푛,푑 푇 ′ is a (훿푛,휖) min-entropy quantum-proof extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Note that the output of Ext푛,푑 푇 ′ has 2푑 − 푛 + 1 bits, where the last 푑 bits form the seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We remark that though the Toeplitz-hashing extractor Ext푛 푇 is strong, the expanded Toeplitz-hashing extractor Ext푛 푇 ′ is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' MAIN RESULT We use 푋 = 푋1푋2 · · ·푋푁 ∈ ({0, 1}푏)푁 to denote the raw data generated by a QRNG, which contains 푁 samples each of 푏 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' For a set 퐼 ⊂ N+, we write 푋퐼 for the restriction of 푋 to the samples determined by 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' For example, if 퐼 = {2, 3, 5}, then 푋퐼 = 푋2푋3푋5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We use E to denote the quantum system possessed by the quantum adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Reverse block source Intuitively, for a given randomness extractor, there is a trade-off between its performance and the generality of the random sources it applies to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te more special random sources the extractor works for, the beter performance the extractor may achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In this paper, the notion of reverse block sources, which are more special than the min-entropy sources, plays a critical role in the sense that (i) raw data of a large class of QRNGs can be described as a reverse block source and (ii) prety good quantum-proof extractors for reverse block sources exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' A quantum version of the reverse block source is defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Definition III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='1 (Reverse block source, adapted from Definition 1 in [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' A string of random variables 푋 = 푋1 · · ·푋푁 ∈ ({0, 1}푏)푁 is a (푏, 푁,훿)-reverse block source given a quantum system E if for every 1 ≤ 푖 ≤ 푁 and every 푥푖+1,푥푖+2, · · · ,푥푁 , 퐻min(푋푖|푋푖+1 = 푥푖+1, · · · ,푋푁 = 푥푁, E) ≥ 훿 · 푏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (4) As shown later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' III B, we introduce the reverse block source for the convenience of designing an online algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' A reverse block source can be intuitively understood as a time-reversed block source 퐻min(푋푖|푋푖−1 = 푥푖−1, · · · ,푋1 = 푥1, E) where a new sample can not be completely predicted by the samples that already exists, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=', the net randomness increment is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' For a reverse block source, these specific samples are from the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Ac- tually, for QRNGs where the raw data are mutually independent, such as the ones based on single photon detection [24–26], vacuum fluctuations [27–29], and photon arrival time [30–32], the min-entropy source automatically satis- fies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (4), where ∀푥푖+1, · · · ,푥푁 , 퐻min(푋푖|E) = 퐻min(푋푖|푋푖+1 = 푥푖+1, · · · ,푋푁 = 푥푁, E), hence is also a reverse block source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' For QRNGs with correlated raw data, one can construct appropriate physical models to check whether Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (4) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In Appendix A, we take the fastest implementation, the one based on phase fluctuation of spontaneous emission [18, 19], as an example to prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We also consider a smoothed version of the reverse block source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' For 푋 = 푋1 · · ·푋푁 ∈ ({0, 1}푏)푁 , Denote the underlying joint quantum state of the random source over the systems 푋 and E as 휌푋 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We call 푋 a (푏, 푁,훿, 휖s)- smoothed reverse block source, if there exists a state 휌∗ 푋 E that is 휀푠-close to 휌푋 E, 1 2 ∥휌∗ 푋 E − 휌푋 E∥ ≤ 휀푠, (5) such that 휌∗ 푋 E is a reverse block source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te smoothed reverse block source is in the same spirit of the smooth conditional entropy [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te motivation of introducing a smoothed version is to exclude singular points or region in a probability distribution, which will help extract more randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Here we remark that we mainly consider the trusted-device QRNGs where the extraction speed is a botleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' For QRNGs with a higher security level, such as the semi-device-independent QRNGs and device-independent QRNGs, the reverse block source property is not satisfied in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Tese types of QRNGs requires fewer assumptions and characterizations on the devices at the expense of relatively low randomness generation rates of raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Ten their real-time randomness generation rates are not limited by the extraction speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Extractors for reverse block sources In this section, we design two online quantum-proof extractors that can both extract a constant fraction of the min-entropy from reverse block sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te two extractors both proceed in the following fashion: partition the input raw data into blocks, and apply a min-entropy quantum-proof extractor to each block using part of the output of the previous block as the seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te basic building block of our designs is family of (훿푛,휖푛) min-entropy quantum- proof extractors, denoted by Ext푔 = {Ext푛 푔 : 푛 ∈ N+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Here, the subscript of Ext푔 means “gadget”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Let 푑푛 and 푚푛 denote the seed length and output length of Ext푛 푔, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=', Ext푛 푔 : {0, 1}푛 × {0, 1}푑푛 → {0, 1}푚푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We require the gadget Ext푛 푔 to satisfy: (i) 휖푛 is exponentially small, for instance, 휖푛 = 2−훿푛/4, which aims at that the summation �∞ 푛=1 휖푛 converges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' and (ii) 푚푛 − 푑푛 = Ω(훿푛), which means that Ext푛 푔 extracts a constant fraction of min-entropy from the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' As an explicit construction, Ext푛 푔 can be specified as the expanded Toeplitz-hashing extractor Ext푛,푑푛 푇 ′ where 푑푛 = (1 +훿/2)푛 − 1, then 푚푛 −푑푛 = 훿푛/2, 푚푛 = (1 +훿)푛 − 1, and 휖푛 = 2−훿푛/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In the rest of this paper, 4 we abbreviate Ext푛,푑푛 푇 and Ext푛,푑푛 푇 ′ where 푑푛 = (1 + 훿/2)푛 − 1 to Ext푛 푇 and Ext푛 푇 ′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Besides, to simplify our discussion, we assume 휖푛 = 2−훿푛/4, and the analysis for other exponentially small values of 휖푛 is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te two extractors are described in Algorithms 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Given an input raw data of length 푁, the first extractor, named Ext푒푞 푟푏푠, evenly partitions the input raw data into blocks each of size 푂(log 푁) and requires 푂(log 푁) random bits as the initial seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In particular, if the min-entropy quantum-proof extractor Ext푔 in use is the expansion of a strong extractor such as the expanded Toeplitz-hashing extractor, then Ext푒푞 푟푏푠 degenerates exactly to the following naive extractor: partition the input raw data into equal-sized blocks and apply the corresponding strong extractor to each block separately with the same seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te second extractor, named Ext푛푒푞 푟푏푠 , is inspired by the extractor that can extract randomness from Santha-Vazirani sources using a seed of constant length [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' It uses only 푂(1) random bits as the initial seed and requires incremental block lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Compared to the first extractor, the second one is less hardware-friendly and sacrifices the extraction speed in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' On the other hand, it enjoys the property that the seed length is independent of the input length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' As a result, this extractor can deal with infinite raw data, without the need of determining the raw data length prior to extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In other words, one does not need to update the seed in practical implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We remark that the initial seed is indispensable because there does not exist any nontrivial deterministic extractor for reverse block sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te proof is presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Algorithm 1: Ext푒푞 푟푏푠 1 Input: 푏 ∈ N+ and 0 < 휖, 훿 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' A string 푥 = 푥1, 푥2, · · · , 푥푁 sampled from a (푏, 푁,훿)-reverse block source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 2 Let 푖 := 1 and 푛 := � 4 훿푏 · log � 푁 휖 �� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 3 Sample a uniform random bit string 푠 (1) of length 푑푏푛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 4 for ℓ = 1 to 푁 /푛 do 5 Let 퐼ℓ := [푖,푖 + 푛 − 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 6 Compute 푧 (ℓ) := Ext푏푛 푔 (푥퐼ℓ ,푠 (ℓ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 7 Let 푖 := 푖 + 푛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 8 Cut 푧 (ℓ) into two substrings, denoted by 푟 (ℓ) and 푠 (ℓ+1), of size 푚푏푛 − 푑푏푛 and 푑푏푛 respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 9 Output 푟 (ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Algorithm 2: Ext푛푒푞 푟푏푠 1 Input: 푏 ∈ N+ and 0 < 훿 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' A string 푥 = 푥1,푥2, · · · sampled from a (푏, ∞,훿)-reverse block source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 2 Parameter: 푛1, Δ ∈ N+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 3 Let 푖 := 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 4 Sample a uniform random bit string 푠 (1) of length 푑푏푛1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 5 for ℓ = 1 to ∞ do 6 Let 퐼ℓ := [푖,푖 + 푛ℓ − 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 7 Compute 푧 (ℓ) := Ext푏푛ℓ 푔 (푥퐼ℓ ,푠 (ℓ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 8 Let 푖 := 푖 + 푛ℓ and 푛ℓ+1 := 푛ℓ + Δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 9 Cut 푧 (ℓ) into two substrings, denoted by 푟 (ℓ) and 푠 (ℓ+1), of size 푚푏푛ℓ − 푑푏푛ℓ+1 and 푑푏푛ℓ+1 respectivelya;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 10 Output 푟 (ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' a We should choose the parameters 푛1 and Δ properly to have 푚푏푛ℓ − 푑푏푛ℓ+1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Note that if we impose Δ = 0 and let 푛1 = � 4 훿푏 · log � 푁 휖 �� , then Ext푛푒푞 푟푏푠 becomes Ext푒푞 푟푏푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We first analyze the second extractor Ext푛푒푞 푟푏푠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Teorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te extractor Ext푛푒푞 푟푏푠 satisfies the following properties: (a) It uses a seed of 푑푏푛1 length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 1: Illustration of Ext푛푒푞 푟푏푠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (b) For any 푘 ∈ N, we have 1 2 ∥휌푟 (1)◦푟 (2)◦···◦푟 (푘) E − 휌푈휂푘 ⊗ 휌E∥ ≤ 푘 � ℓ=1 2−훿푏푛ℓ/4 < 2−훿푏푛1/4 1 − 2−훿푏Δ/4, (6) where 휂푘 = �푘 ℓ=1(푚푏푛ℓ − 푑푏푛ℓ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te nontrivial part is Part (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' For convenience of presentation, we use 퐼푘:∞ to represent 퐼푘 ∪ 퐼푘+1 ∪ · · · ∪ 퐼∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In fact, we will prove that for any 푘 ∈ N it has 1 2 ���휌푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1) ◦푋퐼푘+1:∞ E − 휌푈휂푘 +푑푏푛푘+1 ⊗ 휌푋퐼푘+1:∞ E ��� ≤ 푘 � ℓ=1 2−훿푏푛ℓ/4, (7) which implies Part (b) immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te proof is by an induction on 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te base case when 푘 = 0 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te induction proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Suppose Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (7) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Ten due to the contractivity of trace-preserving quantum operations, we have 1 2 ����휌푟 (1)◦푟 (2)◦···◦푟 (푘)◦Ext푔(푋퐼푘+1,푠 (푘+1))◦푋퐼푘+2:∞ E − 휌푈휂푘 ⊗ 휌Ext푔 � 푋퐼푘+1,푈푑푏푛푘+1 � 푋퐼푘+2:∞ E ���� ≤ 푘 � ℓ=1 2−훿푏푛ℓ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (8) On the other hand, by Definition III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='1, for any assignment 푥퐼푘+2:∞ of 푋퐼푘+2:∞, we have 퐻min(푋퐼푘+1 | 푋퐼푘+2:∞ = 푥퐼푘+2:∞, E) ≥ 훿푏푛푘+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (9) Ten, recalling that Ext푔 is a min-entropy quantum-proof extractor, it follows that 1 2 ����휌Ext푔 � 푋퐼푘+1,푈푑푏푛푘+1 � 푋퐼푘+2:∞ E − 휌푈푚푏푛푘+1 ⊗ 휌푋퐼푘+2:∞ E ���� ≤ 2−훿푏푛푘+1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (10) Tus, 1 2 ����휌푈휂푘 ⊗ 휌Ext푔 � 푋퐼푘+1,푈푑푏푛푘+1 � 푋퐼푘+2:∞ E − 휌푈휂푘 ⊗ 휌푈푚푏푛푘+1 ⊗ 휌푋퐼푘+2:∞ E ���� ≤ 2−훿푏푛푘+1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (11) Finally, combining inequalities (8) and (11) and applying the triangle inequality, we conclude that 1 2 ���휌푟 (1)◦푟 (2)◦···◦푟 (푘+1)◦푠 (푘+2)◦푋퐼푘+2:∞ E − 휌푈휂푘+1+푑푏푛푘+2 ⊗ 휌푋퐼푘+2:∞ E ��� ≤ 푘+1 � ℓ=1 2−훿푏푛ℓ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (12) By using the summation formula for the geometric progression, the above inequality is further upper bounded by 2−훿푏푛1/4 1−2−훿푏Δ/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' □ 6 n1 n1 + ( - 1) n1 + △ x X1 X Xe 5(1 Ext Ext Ext, 6 9 9 r(1) r(2) r(t)As can be seen from the proof, the parameter Δ of Ext푛푒푞 푟푏푠 must be strictly positive, since the upper bound �푘 ℓ=1 2−훿푏푛ℓ/4 on the error converges if and only if Δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Teorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='2 implies that the (infinitely long) output string 푟 (1) ◦ 푟 (2) ◦ · · · can be arbitrarily close to the uniform distribution by choosing a sufficiently large constant 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' As an explicit construction, suppose Ext푛푒푞 푟푏푠 adopts the expanded Toeplitz-hashing extractor Ext푛 푇 ′ as the gadget Ext푛 푔, where 푑푛 = (1 +훿/2)푛 − 1 and 푚푛 = (1 +훿)푛 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We further set Δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We require that 푛1 ≥ 4/훿 + 1 such that 푚푏푛ℓ −푑푏푛ℓ+1 = 푏(훿푛ℓ/2 − 1 −훿/2) ≥ 1 for any ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Ten Ext푛푒푞 푟푏푠 extracts (1 +훿)푏(푛1 + ℓ − 1) − 1 random bits from the ℓ-th block 푥퐼ℓ , outputs the first 푚푏푛ℓ − 푑푏푛ℓ+1 ≈ 훿푏푛ℓ/2 bits, and then uses the last (1 + 훿/2)푏(푛1 + ℓ) − 1 bits as the seed of the next block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Via a similar argument as in Teorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='2, we have the following result for Ext푒푞 푟푏푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Teorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te extractor Ext푒푞 푟푏푠 uses 푑푏푛 random bits as a seed and outputs a string 푟 (1) ◦ 푟 (2) ◦ · · · ◦ 푟 (푁/푛) satisfying that 1 2 ∥휌푟 (1)◦푟 (2)◦···◦푟 (푁 /푛)◦E − 휌푈휂 ⊗ 휌E∥ ≤ 푁 푛 · 2−훿푏푛/4 ≤ 휖, (13) where 휂 := 푁 푛 · (푚푏푛 − 푑푏푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In particular, suppose Ext푒푞 푟푏푠 adopts Ext푛 푇 ′ as the gadget Ext푛 푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Ten Ext푒푞 푟푏푠 extracts (1+훿)푏푛−1 random bits from the ℓ-th block 푥퐼ℓ and outputs the first 푚푏푛 −푑푏푛 = 훿푏푛/2 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te last 푑푏푛 = (1+훿/2)푏푛 −1 bits, which is exactly the seed used in this block, will be reused as the seed in the next block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Terefore, Ext푒푞 푟푏푠 uses 푑푏푛 ≈ (4/훿 + 2) log (푁/휖) random bits as seed, and outputs (푁/푛) · (푚푏푛 − 푑푏푛) ≈ 훿푏푁/2 bits in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Tough Ext푒푞 푟푏푠 uses more seed than Ext푛푒푞 푟푏푠 , it is more hardware-friendly and can achieve much higher extraction speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Corollary III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Suppose the random sources in Algorithms 1 and 2 are replaced with the (푏, 푁,훿,휖s) and (푏, ∞,훿,휖s)- smoothed reverse block sources, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' By using the same processing procedures, the output state of the extractor Ext푛푒푞 푟푏푠 satisfies 1 2 ∥휌푟 (1)◦푟 (2)◦···◦푟 (푘) E − 휌푈휂푘 ⊗ 휌E∥ < 2−훿푏푛1/4 1 − 2−훿푏Δ/4 + 2휖푠, (14) and the output state of the extractor Ext푒푞 푟푏푠 satisfies 1 2 ∥휌푟 (1)◦푟 (2)◦···◦푟 (푁 /푛)◦E − 휌푈휂 ⊗ 휌E∥ ≤ 휖 + 2휖푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We prove the smoothed version of Algorithm 2, and the proof for the smoothed version of Algorithm 1 follows essentially the same procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' For brevity, denote 2−훿푏푛1/4 1−2−훿푏Δ/4 := 휖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' According to the definition of the smoothed reverse block source, there exists a state 휌∗ 푋 E that is 휖푠-close to the real output state 휌푋 E such that 휌∗ 푋 E determines a (푏, ∞,훿) reverse block source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Using the result in Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 1 2 ���휌푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1)◦푋퐼푘+1:∞ E − 휌푈휂푘 +푑푏푛푘+1 ⊗ 휌푋퐼푘+1:∞ E ��� ≤1 2 ���휌푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1)◦푋퐼푘+1:∞ E − 휌∗ 푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1) ◦푋퐼푘+1:∞ E ��� + 1 2 ���휌∗ 푟 (1)◦푟 (2)◦···◦푟 (푘)◦푠 (푘+1) ◦푋퐼푘+1:∞ E − 휌푈휂푘 +푑푏푛푘+1 ⊗ 휌∗ 푋퐼푘+1:∞ E ��� + 1 2 ���휌푈휂푘 +푑푏푛푘+1 ⊗ 휌∗ 푋퐼푘+1:∞ E − 휌푈휂푘 +푑푏푛푘+1 ⊗ 휌푋퐼푘+1:∞ E ��� ≤휖s + 휖 + 휖s = 휖 + 2휖s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (16) where the first inequality is due to the contractivity of trace-preserving quantum operations while the last inequality comes from the fact that the purified distance is an upper bound of the trace distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' □ Tis result implies that the output for a smoothed reverse block source can also be arbitrarily close to a uniform distributed sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 7 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' SIMULATIONS OF THE REAL-TIME RANDOMNESS GENERATION RATE In this section, we make a simulation estimating the extraction speed of the first extractor Ext푒푞 푟푏푠 implemented in the Xilinx Kintex-7 XC7K480T Field Programmable Gate Array (FPGA), a common application in industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Here, we adopt the expanded Toeplitz-hashing extractor Ext푛 푇 ′ as the gadget Ext푛 푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We use the raw data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' [20] as the random source, which is a reverse block source with parameters 푏 = 8 and 훿 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We consider a raw data length of 푁 = 251 bits and a total security parameter 휖 = 10−10, which means the final output data is 10−10-close to a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Correspondingly, the block length is 푛 = � 4 훿푏 · log � 푁 휖 �� = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' According to the discussion afer Teorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='3, Ext푒푞 푟푏푠 will use a seed of (1 + 훿/2)푏푛 − 1 = 569 bits and output about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='85 PB random bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Precisely, Ext푒푞 푟푏푠 simply divides the input raw data into blocks with length 50, where each block contains 푏 × 푛 = 400 bits, and multiplies a 170 × 400 Toeplitz matrix by a 400-dimension vector over 퐺퐹 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Since the addition 푎 + 푏 and multiplication 푎 · 푏 over 퐺퐹 (2) are exactly 푎 ⊕ 푏 and 푎 ∧ 푏, respectively, which are both basic logical operations, the matrix multiplication involves 170 × 400 = 68000 ‘∧’ operations and 68000 ‘⊕’ operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te parameters of the Xilinx Kintex-7 XC7K480T FPGA are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te clock rate is set to be 200 MHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' the number of Look-Up-Tables (LUTs) is 3 × 105;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' each LUT can perform 5 basic logical operations simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' To make full use of the FPGA, we can perform the matrix multiplications of ⌊3 × 105 × 5/(2 ∗ 68000)⌋ = 11 blocks in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Terefore, the extraction speed of Ext푒푞 푟푏푠 is 200 × 106 × 11 × 170 = 374 Gbps, which is improved by one order of magnitude compared to the state-of-the-art result [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' As a result, the extraction speed is not a botleneck for the high-speed QRNGs any more;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' hence our online extraction is adequate for the post-processing of the fastest known implementation of QRNGs [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' CONCLUSION In conclusion, we design two novel quantum randomness extractors based on the reverse-block-source property that is satisfied by a large class of trusted-device QRNGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Tese results provide theoretical supports to the current real-time block-wise post-processing widely applied in experiments and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te first extractor improves the real-time extraction speed while the second one can extract infinite raw data with only a constant seed length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In particular, the first extractor is easy to be implemented in a FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te real-time extraction speed with a common FPGA is high enough for the real-time post-processing of the fastest known QRNGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' For future work, it is interesting to explore other properties beyond the general min-entropy source to improve the post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te improvement may come from boosting the extraction speed or saving the seed length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' On the other hand, randomness extraction with an imperfect seed or even without seed is also a practical and promising direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' An interesting open question is whether randomness can still be extracted online without the seed for certain non-trivial random sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Nie and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Bai for enlightening discussions, and Salil Vadhan for telling us the details of the extractor which extracts randomness from Santha-Vazirani sources using a seed of constant length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Tis work was supported in part by the National Natural Science Foundation of China Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 61832003, 61872334, 61801459, 62002229, 1217040781, and the Strategic Priority Research Program of Chinese Academy of Sciences Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' XDB28000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Appendix A: QRNG raw data as a reverse block source We focus on the QRNG based on measuring the phase fluctuation of a laser, which is the fastest and most widely applied implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te details of the QRNG design and randomness quantification are given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Here, we make a brief summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' A laser wave with a random phase 휙(푡) passes through an interferometer with time delay 휏푙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Afer the interference, the random phase fluctuation is then transformed into an intensity fluctuation and can be sampled by an analog-to-digital converter (ADC) with a sampling frequency 1/휏푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te laser intensity fluctuation is 8 transformed into voltage signal 푉 in ADC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' When the sampled voltage signal falls in some interval of the ADC, it will generate a sequence of corresponding random numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te sequence length determines the resolution of the ADC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' For example, an 8-bit ADC will be divided into 28 intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We illustrate the physical seting in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 2: Typical seting of a QRNG based on phase fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' MZI: Mach-Zehnder interferometer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' PD: photo detector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' ADC: analog-to-digital converter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' According to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' [35], the voltage will be proportional to the phase difference Δ휙(휏) = 휙(푡 + 휏) − 휙(푡), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=', 푉 = 푘Δ휙(휏푙), where 푘 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' We assume that the spontaneous emission leads to a differential random phase characterized by a Gaussian white noise in time domain 퐹sp(푡), whose expectation and variance are given by E[퐹sp(푡)] = 0 and Var(퐹sp(푡)) = 휎2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te phase difference comes from the integration of the differential random phase, Δ휙(휏) = ∫ 푡0+휏 푡=푡0 퐹sp(푡)푑푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (A1) Ten, Δ휙(휏) follows a Gaussian distribution 퐺(0, � 휎2휏푙) with zero mean and variance 휏휎2 due to the property of Gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te voltage also follows a Gaussian distribution, 퐺(0, � 푘2휎2휏푙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' When 휏푠 > 휏푙, the raw data generated by each sample will be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In the asymptotic limit, the conditional min-entropy per sample is given by [35] 퐻min(푉푖|E) ≥ − log2 � max 푗 ∫ 푉푖 ∈푆푗 퐺(0, � 푘2휎2휏푙)푑푉푖 � , (A2) where 푉푖 is the 푖-th sample and 푆푗 is the 푗-th interval of the analog-to-digital converter (ADC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' While when 휏푠 ≤ 휏푙, there will be correlations between samples and randomness quantification will be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Without loss of generality, we consider 휏푙 = 푞휏푠 with some integer 푞, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Comparing the adjacent two samples, for FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 3: Illustration of the phase interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' example, the first two samples, we have Sample1 : Δ휙1(휏푙) = 휙(푡 = (푞휏푠)) − 휙(푡 = 0) = ∫ 푡=휏푠 푡=0 퐹sp(푡)푑푡 + ∫ 푡=푞휏푠 푡=휏푠 퐹sp(푡)푑푡, Sample2 : Δ휙2(휏푙) = 휙(푡 = (푞 + 1)휏푠) − 휙(푡 = 휏푠) = ∫ 푡=푞휏푠 푡=휏푠 퐹sp(푡)푑푡 + ∫ 푡=(푞+1)휏푠 푡=푞휏푠 퐹sp(푡)푑푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (A3) 9 Ti 中1 Φ2 Φ3 Φn-2 Φn-1 Φn Φn+1 t TsMZI Laser PD ADCSuppose a 푏-bit ADC is applied in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Ten 푏-bit raw data can be generated per sample, with a min- entropy lower bound 퐻min(푋푖|E) ≥ − log2 � max 푗 ∫ 푉푖 ∈푆푗 퐺(0, � 푘2휎2휏푙)푑푉푖 � , (A4) where 푋푖 ∈ {0, 1}푏 is the sequence of raw random numbers corresponding to the voltage signal 푉푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te conditional min-entropy 퐻min(푋푖|푋푖+1 = 푥푖+1, · · · , 푋∞ = 푥∞, E) will be less than 퐻min(푉푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te net increment randomness comes from the integration of random phase in one time bin of an interval 휏푠, which corresponds to an effective variance of the voltage Var(푉)eff = Var(푉)휏푠 휏푙 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' (A5) Ten we have 퐻min(푋푖|푋푖+1 = 푥푖+1, · · · , 푋∞ = 푥∞, E) ≥ − log2 � max 푗 ∫ 푉푖 ∈푆푗 퐺(0, � 푘2휎2휏푠)푑푉푗 � := 훿푏, (A6) which forms a (푏, ∞,훿)-reverse block source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Appendix B: Initial seed is indispensable for reverse block sources Te following theorem claims that there does not exist any deterministic extractor that can extract even one bit of almost uniformly-distributed random number from every reverse block source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Te proof is essentially the same as that for Santha-Vazirani sources [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Teorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' For all 푏, 푁 ∈ N+ and 0 < 훿 < 1, and any function Ext : {0, 1}푏푁 → {0, 1}, there exists a (푏, 푁,훿)- reverse block source 푋 such that either P[Ext(푋) = 1] ≥ 2−훿 or P[Ext(푋) = 1] ≤ 1 − 2−훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Because |Ext−1(0)| + |Ext−1(1)| = 2푏푁, either |Ext−1(0)| ≥ 2푏푁−1 or |Ext−1(1)| ≥ 2푏푁−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Without loss of generality, let us assume that |Ext−1(1)| ≥ 2푏푁−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Pick an arbitrary subset 푆 of Ext−1(1) with |푆| = 2푏푁−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Consider the source 푋 that is uniformly distributed on 푆 with probability 2−훿 and is uniformly distributed on {0, 1}푏푁 \\푆 with probability 1 − 2−훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' It is easy to check that P[Ext(푋) = 1] ≥ 2−훿 > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' In addition, we have that P[푋 = 푥]/P[푋 = 푥 ′] ≤ (2−훿)/(1 − 2−훿) for any 푥,푥 ′ ∈ {0, 1}푏푁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Ten by Definition III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content='1, it is straightforward to check that 푋 is a (푏, 푁,훿)-reverse block source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' □ [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Impagliazzo and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Zuckerman, in FOCS, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 30 (1989) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 248–253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' Wigderson, “A note on extracting randomness from santha-vazirani sources,” (2004), unpub- lished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} +page_content=' 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFAT4oBgHgl3EQfmB2k/content/2301.08621v1.pdf'} diff --git a/59AyT4oBgHgl3EQfpfhu/vector_store/index.faiss b/59AyT4oBgHgl3EQfpfhu/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e6a8d5b1a7569e1a4ae4e6cdd9bd95a28578f568 --- /dev/null +++ b/59AyT4oBgHgl3EQfpfhu/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e047e6c3a96b8c38221706b964555111ebbf1c58abf6bd09f84e36ef8831b2f5 +size 4063277 diff --git a/69E1T4oBgHgl3EQfnASE/content/tmp_files/2301.03304v1.pdf.txt b/69E1T4oBgHgl3EQfnASE/content/tmp_files/2301.03304v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e6f660d29e20e4cd1370d00cf60f8464d8a58b1e --- /dev/null +++ b/69E1T4oBgHgl3EQfnASE/content/tmp_files/2301.03304v1.pdf.txt @@ -0,0 +1,958 @@ +Randomization advice and ambiguity aversion∗ +Christoph Kuzmics† +Brian W. Rogers‡ +Xiannong Zhang§ +January 10, 2023 +Abstract +We design and implement lab experiments to evaluate the normative +appeal of behavior arising from models of ambiguity-averse preferences. We +report two main empirical findings. First, we demonstrate that behavior +reflects an incomplete understanding of the problem, providing evidence +that subjects do not act on the basis of preferences alone. Second, additional +clarification of the decision making environment pushes subjects’ choices in +the direction of ambiguity aversion models, regardless of whether or not the +choices are also consistent with subjective expected utility, supporting the +position that subjects find such behavior normatively appealing. +JEL codes: C91, D81 +Keywords: Knightian uncertainty, subjective expected utility, ambiguity aversion, +lab experiment +∗We thank Michael Greinecker, John Nachbar, Paulo Natenzon, Andrea Prat, Todd Sarver, +Tomasz Strzalecki, Peter Wakker and Jonathan Weinstein for insightful comments. The authors +gratefully acknowledge the funding from the Weidenbaum Center on the Economy, Government, +and Public Policy at Washington University in St. Louis. +†University of Graz, Austria, christoph.kuzmics@uni-graz.at +‡Washington University in St. Louis, U.S.A., brogers@wustl.edu +§Corresponding author, Washington University in St. Louis, 1 Brookings Dr. St. Louis, +Missouri, U.S.A. (63130), zhangxiannong@wustl.edu +1 +arXiv:2301.03304v1 [econ.TH] 9 Jan 2023 + +1 +Introduction +Many economic decisions are made under uncertainty that cannot be readily quan- +tified by objective probabilities. Consider saving decisions, that is investing money +into bonds or stocks in the presence of inflation uncertainties and general uncer- +tainties about the economic future. Even in the absence of wars and pandemics +most people find it hard to attach probabilities to the relevant possible events, let +alone agree on such an assessment. +The classical paradigm of rational decision making under such uncertainty is +subjective expected utility (SEU), underpinned by the axiomatic foundation of +Savage [1954]. The experimental designs of Ellsberg [1961], later implemented in +many studies, see e.g., the survey of Trautmann and Van De Kuilen [2015], have +challenged the SEU paradigm. This challenge was mostly on positive, that is, +empirical, grounds in that these experiments found that many people behave in a +way that is inconsistent with subjective expected utility. The fact, however, that +this behavior has been so robust in lab experiments and that many researchers have +developed axiomatic foundations for alternative models of decision making that +admit such behavior, see references below, may be received as posing a normative +challenge to SEU in postulating that a broader set of preference models could be +considered normatively appealing. +In this paper we aim to test the normative appeal of these alternative models +of decision making under uncertainty. We use the term “normative” in the sense of +Ellsberg’s 1962 PhD thesis, see Ellsberg [2001, pages 22-26], and as in the “subjec- +tive” definition of rationality given by Gilboa [2012, p. 5]: We consider a decision +normatively appealing (to the decision maker) if the decision maker (still) makes +this choice after thorough reflection. In all of our treatments, subjects are provided +with a complete, and fairly standard, description of all payoff-relevant aspects of +the environment. We operationalize this reflection in the lab by providing subjects +with supplementary descriptions that, while not payoff-relevant, emphasize certain +ways to think about the environment. The descriptions are provided in the form +of short videos that subjects watch before the elicitation of their choice. +Our findings are relevant to all models of ambiguity aversion that are monotone. +That is, if one act is better than another act in all states, then the inferior act +2 + +cannot be chosen when the superior act is available. We call such models classical +ambiguity aversion (CAA) models.1 +The experimental environment is directly inspired by the hedging argument of +Raiffa [1961] in the context of the Ellsberg [1961] two-color urn experiment. A +risky urn contains 49 White and 51 Red balls.2 An ambiguous urn also contains +100 balls, each of which is either Green or Yellow, but nothing more is known +about the composition of the ambiguous urn. +The decision-maker (DM) must +choose one of three actions, after which the experimenter draws one ball from each +urn Together, these determine the consequence for the DM, which is either “Win” +or “Lose”, with Win being strictly preferred to Lose. We call the choices bets for +White, Green, and Yellow. Each bet Wins if the experimenter draws a ball of the +corresponding color, and loses otherwise. Ellsberg’s key insight is that Bet White, +while commonly chosen, is incompatible with SEU.3 +In this context, the Raiffa [1961] hedge against ambiguity works as follows. +The DM flips a fair coin and bets Green if the coin lands on heads and bets Yellow +if the coin lands on tails. This randomized action provides an (objective) winning +probability of 50%, regardless of the color of the drawn ball from the ambiguous +urn. As this is higher than the 49% winning probability from betting White, this +action strictly dominates the Ellsberg choice.4 +1This category includes most preference models reviewed in the recent survey of Machina +and Siniscalchi [2014], such as the maxmin expected utility model of Gilboa and Schmeidler +[1989], the Choquet expected utility model of Schmeidler [1989], the smooth ambiguity model of +Klibanoff et al. [2005], the variational and multiplier preference models of Maccheroni et al. [2006] +and Hansen and Sargent [2001], confidence function preferences of Chateauneuf and Faro [2009], +uncertainty aversion preferences of Cerreia-Vioglio et al. [2011], and the incomplete preference +model of Bewley [2002]. +2This variation from 50/50 avoids identification complications that can arise from indifference. +3If Bet White is chosen, it must be weakly preferred to Bet Yellow, so that the DM’s subjective +probability of a yellow ball being drawn from the ambiguous urn must be at most 49%. But then +the subjective probability of a green ball being drawn from the ambiguous urn must be at least +51%, so that Bet Green is strictly preferred to Bet White, a contradiction. +4Kuzmics [2017], appealing to results in classical statistical decision theory – in particular the +complete class theorem of Wald [1947], has shown that a DM who can randomize over choices and +commit to following the realized prescription of this randomization can never make choices that +are inconsistent with SEU and at the same time consistent with CAA in any decision problem. +See also Bade [2015], Oechssler and Roomets [2014], Azrieli et al. [2018], Baillon et al. [2022a], +and Baillon et al. [2022b] for similar results and arguments along these lines. One can, in fact, +regard ambiguity aversion as a preference for randomization, see e.g., Eichberger and Kelsey +[1996] and Epstein and Schneider [2010]. +3 + +In light of the Raiffa [1961] argument, what are the possible explanations for +a classically ambiguity-averse DM to nonetheless bet on White in this experiment +even though it is dominated? First, it is possible, even likely, that designing a +random choice does not occur to subjects as an option. Second, even if a subject +recognizes such a possibility, it is possible that they cannot, or choose not to, +go through the required construction and reasoning that would allow them to +see that betting on White is dominated.5 But suppose now that a subject does +recognize the possibility and understands the argument. +A third possibility is +that the subject has no access to a suitable randomization device, nor thinks they +can simulate one. So suppose the subjects does have a fair coin. The fourth and +final explanation is that the subject lacks the ability to commit to the randomized +action. Once the coin flip realizes, the subject could revisit their choice, and if they +are ambiguity averse they will want to flip the coin again, and again ad infinitum, +with one possible outcome that they bet on White in the end after all. +Classical ambiguity aversion models do not allow us to delve into the reasons +behind the choice of White in the presence of the Raiffa [1961] argument, as these +preference models are axiomatized for preferences over the space of all pure acts, +see e.g., Seo [2009], and not over the set of all mixed acts as in Anscombe and +Aumann [1963]. This means, however, that whether or not a classically ambiguity +averse subject who understands the environment may choose to bet on White +depends simply on whether the Raiffa [1961] hedge (including the commitment to +its outcome) is provided as a pure choice or not. If it is given as a pure choice the +subject cannot choose to bet on White. If it is not given, they can. +Motivated by these considerations, our experimental treatments provide differ- +ential supplementary observations that focus on the hedging argument. All of the +supplementary observations are given after a (common to all treatments) standard +and complete description of the environment, and before the elicitation of the sub- +ject’s choice. If the standard and complete description of the environment suffices +to impart a full understanding of the consequences of each possible action, then +5The argument is based on the reduction of compound lotteries and as Halevy [2007] found +many subjects who make Ellsberg choices also fail to reduce compound lotteries. This failure, +however, can hardly be seen as normatively appealing - it is simply a mathematical mistake. Ab- +dellaoui et al. [2015] has found a much weaker association between displayed ambiguity aversion +and a failure to reduce compound lotteries. +4 + +the treatment effects of the supplementary observations will be null. Indeed, there +is no marginal payoff-relevant information in the supplementary observations. +To the contrary, our first main finding is that the commentary significantly +changes behavior. Since the commentary does not affect a subject’s underlying +preferences, it changes behavior through modifying a subjects’ understanding of +the environment. This is only possible if the subject’s understanding after the stan- +dard description was incomplete or erroneous. We conclude that many subjects +indeed have an incomplete or erroneous understanding after being provided with a +standard and complete description of the classical Ellsberg two-urn environment, +so that directly applying the revealed preference toolkit to such data may not be +appropriate. Instead, a given choice may reflect an incomplete comprehension of +the environment, and therefore be viewed as a mistake, or the result of confusion, +rather than as a manifestation of the subject’s preference. +Our second main finding concerns the direction of the effects of the supplemen- +tal observations. In this regard, and to the extent we can infer preferences from +the treatments with commentary, our data supports the broader normative appeal +of ambiguity aversion models over SEU in the following sense: When the Ellsberg +choice (bet White) is compatible with ambiguity averse preferences but not with +SEU, the supplemental hedging observations increase the frequency of Ellsberg be- +havior; when the Ellsberg choice is incompatible with ambiguity aversion (and so +also with SEU), the same observations decrease the frequency of Ellsberg behavior. +The paper proceeds as follows: The experimental design is given in Section 2. +The results are given in Section 3. Section 4 offers a discussion of these results and +Section 5 provides a brief survey of related literature before Section 6 concludes. +Experimental instructions are presented in the Appendix. +2 +Experimental Design +The slight variation of the Ellsberg two-color urn experiment, outlined above, +serves as the baseline and control. We then vary the supplemental observations +that subjects receive about the decision-making environment. +Our treatments +differ from the baseline along two dimensions. First, in addition to the standard +options of betting on White, Yellow, or Green, in some treatments subjects are +5 + +presented with an additional fourth option, which executes a bet on either Green +or Yellow according to the outcome of a fair coin to be tossed by a third party +after the balls are drawn from the urns - the Raiffa hedge. Note that this option +also serves as a commitment device for randomization, as the bet will be executed +by the experimenter on the subject’s behalf. Recall that the compatibility of the +Ellsberg choice (bet White) with classical ambiguity aversion models hinges on the +presence or absence of this option. +Second, after the environment is fully described as transparently as possible, in +some treatments subjects watch short videos before making their (single) choice. +The videos, while all factually correct, emphasize different aspects of the con- +sequences of using the randomization device, in ways that we hypothesize may +change some subjects’ understanding of the merits of the Raiffa hedge. +In the main treatment, the coin flip bet is included as an option, and subjects +are presented with a single video (denoted V 1 and available here) containing sup- +plemental observations that describe the hedging argument of Raiffa [1961].6 It +describes the outcome of betting on the coin conditional on the outcome of the +ball drawn from the ambiguous urn. It states that the winning probability using +that option is 50% in either case (green ball or yellow ball drawn), and concludes +by reminding the subject that betting on White wins with probability 49%. +This video, as well as our videos, does not explicitly advocate for any particular +choice, so that it contains observations rather than advice. The transcripts of the +videos are read by an anonymous (to subjects) third party to avoid a perception +that the experimenters are giving implicit advice. +Partly to control for a possible experimenter demand effect, we designed a +second video (denoted V 2 and available here), in which the structure of the argu- +ment and the language is symmetric to the first video. It describes the outcome +of betting on the coin conditional on the outcome of the coin flip. It states that +no known winning probability can be specified in either case (heads or tails), and +concludes by reminding the subject that betting on White wins with probability +49%. Again, it does not advocate for any particular choice. +We ran treatments utilizing exclusively this second video, as well as treatments +in which subjects were presented with both videos before eliciting their choice (in +6Transcripts of all videos are included in the appendix for an offline audience. +6 + +both orders; there were no order effects).7 +As we want to understand the effect of the supplementary observations inde- +pendently from the effect of presenting the hedging device as an explicit option, we +ran a parallel set of treatments with similar videos but where the available options +were simply bets for White, Green, or Yellow, as in the baseline case, without the +coin flip option. In these treatments, the Ellsberg choice remains compatible with +CAA models. We varied the videos slightly to accommodate the different choice +set. First, as there was no explicit coin, before showing either V 1 or V 2, we showed +a preliminary video (denoted V 0 and available here) in which it was explained that +a subject could imagine a virtual coin toss, and then bet on Green/Yellow accord- +ing to the outcome. Second, in videos V 1 and V 2 the coin toss was referred to as +a virtual coin toss. We refer to the treatment with the explicit hedge/commitment +option as “Coin” and those without it as “No Coin” treatments. +3 +Results +Table 1 summarizes the main findings. +W +G +Y +W +G +Y +C +Baseline +45% +41% +14% +Baseline +37% +26% +11% +26% +(26/58) +(24/58) +(8/58) +(20/54) +(14/54) +(6/54) +(14/54) +V1 +27% +55% +18% +V1 +29% +2% +2% +67% +(12/44) +(24/44) +(8/44) +(14/48) +(1/48) +(1/48) +(32/48) +V2 +- +- +- +V2 +41% +9% +9% +41% +(18/44) +(4/44) +(4/44) +(18/44) +V1+V2 +58% +28% +13% +V1+V2 +23% +13% +9% +55% +(35/60) +(17/60) +(8/60) +(13/56) +(7/56) +(5/56) +(31/56) +Table 1: +Summary of data (left: without a randomization device; right: explicit +randomization option). +7In sessions using both videos, we showed one video first, and the other video next. +We +did this in both orders. Then after both videos were shown, the subjects were provided with +additional time to revisit any portions of either or both of the two videos before proceeding to +enter their bet. During this time subjects could pause, rewind, and switch between videos as +they liked. We thereby tried to minimize possible order effects and, indeed, there is no evidence +that the order of the videos has any effect, and so we have pooled that data in our analysis. +7 + +We summarize the key findings from this data in the following three results. +Result 1 If preferences alone dictate choices, the supplementary observations con- +tained in the videos should have no effect on behavior. For the No Coin (Coin, +resp.) treatments, pooling the data for Green and Yellow (being conservative), the +p-value for the null hypothesis that choice frequencies are the same in the baseline +and the V 1 video treatment is 0.067 (< 0.001, resp.)8, and for the null hypothesis +that choice frequencies are the same in the baseline and the neutral V 1 + V 2 video +treatment is 0.142 (0.007, resp.). +Result 2 The p-value for the null hypothesis that neutral video observations (V 1+ +V 2) does not decrease the choice of White relative to the baseline is ≥ 0.5 for the +No Coin treatments, and 0.057 for the Coin treatments. +Result 3 Without supplemental observations there is no significant difference in +the frequency of Bet White between the Coin and No Coin treatment, p-value 0.402. +With supplemental observations (V 1 + V 2) there is a significant difference in the +frequency of Bet White between the Coin and No Coin treatment, p-value < 0.001. +4 +Discussion +There is firm evidence that behavior across treatments is not a pure consequence +of underlying preferences combined with a complete understanding of the environ- +ment. The observations contained in the videos cannot change preferences as they +do not change any of the payoff-relevant considerations. Rather, any differences +in behavior must come from differences in the subjects’ understanding. +Suppose we adopt the view that choices after studying both videos indeed re- +veal preferences, since subjects may have a more complete understanding of the +environment and the consequences of their choices after considering the observa- +tions contained therein. Even then, 23% of subjects, i.e., those who chose Bet +White in the Coin treatment, have preferences different from any CAA model. +The remaining 77% of subjects make choices in the Coin treatment that can be +explained by CAA as well as SEU. +8All statistical tests performed in this paper are likelihood ratio tests. +8 + +However, the No Coin treatments provide an interesting contrast. In these +treatments, Bet White is undominated and the supplemental observations increase +the frequency of Bet White relative to the baseline description. If, again, we view +the choices after studying both videos as revealing preferences, the choice of Bet +White made by 58% of subjects is inconsistent with SEU but is consistent with +CAA models. +Together, these findings suggest that CAA models have broader normative ap- +peal than (the narrower theory of) SEU despite their descriptive problems in some +environments. Of note, we asked subjects (in a non-incentivized post-experiment +questionnaire) if their preference became more or less clear after watching the +videos. In the No Coin treatments 17% (27 out of 162) of subjects reported that +their preferences became “less clear” after watching the videos, which is much +higher than the 3% (6 out of 202) reporting the same in the Coin treatment +(p < 0.001), calling into question the presumption that behavior directly reveals +preferences, especially in the No Coin treatments. One interpretation is that many +subjects find monotonicity to be a normatively appealing property, yet lack the +sophistication to identify its consequences. +We conclude this discussion by considering preferences that may depend on +the timing of the resolution of uncertainty, as in the models of Seo [2009], Saito +[2015], and Ke and Zhang [2020]. Such models are not classical, as they are not +monotone.9 In the Coin treatments subjects were (truthfully) told, as part of the +baseline description of the environment, that the coin flip would be executed after +the balls were drawn from the urns and revealed. Thus, the choice of Bet White +in the Coin treatments (37% in the baseline and 23% after both videos) is even +inconsistent with these more flexible models. +Roughly, one could categorize our subjects as follows. There is one group of +subjects (≥ 23%) who make choices inconsistent with all models of ambiguity +aversion. There is a second group of subjects (≤ 58% - 23% = 35%) who make +choices consistent with ambiguity aversion but not with SEU. The remainder make +choices consistent with SEU. Subjects in the second group would have a demand +for randomization devices as they cannot, to their satisfaction, create and commit +9Motivated by the state separability embedded in monotonicity, Bommier [2017] provides a +model where monotonicity is relaxed. +9 + +to randomized choices themselves. +5 +Related Literature +A number of papers have studied the consistency of subjects’ choices across decision +problems. These include Binmore et al. [2012], Stahl [2014], Voorhoeve et al. [2016] +and Crockett et al. [2019]. +This literature finds, on the whole, that relatively +few subjects make consistent choices, and those who do tend to be ambiguity- +neutral. The lack of consistency can be interpreted as evidence against people +choosing according to a clear preference. However, ambiguity-averse DMs may +find inconsistent choices to be a useful hedge against ambiguity, see e.g., Kuzmics +[2017] and Azrieli et al. [2018] for more general arguments.10 Our single choice +design is immune to such problems. This is why we constrained our design to a +single incentivized elicitation per subject, even at the cost of forgoing the ability +to conduct within-subject analyses across treatments. +Spears [2009], Dominiak and Schnedler [2011], and Oechssler et al. [2016] study +experiments in which subjects are given the Raiffa hedge as an option, similarly to +our baseline Coin treatment (without supplemental commentary). Generally, they +find very few subjects choosing this option, with more subjects instead choosing a +dominated option. This too is evidence against CAA models. They also find that +subjects do not care about the timing of the resolution of uncertainty, evidence +even against the non-CAA models of Seo [2009], Saito [2015], and Ke and Zhang +[2020]. Our focus is on the possible effects of providing explicit descriptions of the +Raiffa hedge. We thus add to these findings that such observations significantly +influence behavior, and does so in directions that support the appeal of CAA +models. +Finally, several studies test, in various ways, the normative appeal of ambigu- +ity aversion preference models.11 The closest to our design is that of Slovic and +Tversky [1974], who give subjects written advice for and against Allais [1953] and +10This problem persists under many different preference elicitation schemes. +See also e.g., +Baillon et al. [2014], Bade [2015], Oechssler and Roomets [2015], which builds on earlier work +on eliciting non-expected utility preferences under objective uncertainty by e.g., Karni and Safra +[1987]. +11Al-Najjar and Weinstein [2011] provide normative arguments against ambiguity aversion. +10 + +Ellsberg [1961] choices. However, their advice is built around the independence +axiom, and so concerns a quite distinct domain. Jabarian and Lazarus [2022] also +study the effect of a form of advice on subjects’ decisions in a framework with +ambiguity aversion. Their framework involves two independent draws from the +same two-color ambiguous urn (and two draws from a 50-50 risky urn) in which +many subjects make dominated choices, similarly also to Yang and Yao [2017] and +Kuzmics et al. [2022]. Subjects win if they draw two balls of the same color from +the urn that they choose, making betting on the ambiguous urn a (weakly) domi- +nant choice - as the more extreme the ball distribution in the ambiguous urn the +higher the chance of drawing two balls of the same color. Jabarian and Lazarus +[2022] have treatments in which subjects are given additional decision problems +that should help them understand the mechanism why a choice is dominated. They +find that while subjects do seem to understand the mechanism, they, nevertheless, +do not seem to transfer this knowledge to the original problem in which they make +dominated choices regardless. Finally, Keller et al. [2007], Trautmann et al. [2008], +Charness et al. [2013], and Keck et al. [2014] study decision problems with am- +biguity in groups (or under peer observation) and find, on the whole, that group +discussion and related phenomena tend to lead to more ambiguity-neutral choices. +6 +Conclusion +We have subjected classical preference models of ambiguity aversion models to +tests of their normative appeal with experiments that stay close to the original +Ellsberg (two-color urn) design. +We find that subjects’ choices are affected by payoff-irrelevant commentary. +This implies that at least one of the two treatments, without or with commentary, +does not allow the full revelation of subjects’ preferences. +At least some of our subjects do seem to see a certain normative appeal in +the kind of behavior prescribed by classical models of ambiguity aversion and, in +particular, the monotonicity axiom. Giving subjects access to additional commen- +tary, in the form of short video clips, results in behavior that is significantly more +consistent with these models. +The nature of this normative appeal suggests that people, after sufficient re- +11 + +flection, would have a demand for the ability to commit to randomized choices, +a demand which one would surmise should be observable. It would be interest- +ing to identify instruments outside the lab, in the various areas of application of +ambiguity aversion models, that could serve to satisfy this demand. +We also find that our subjects lack a complete and perfect understanding of +their decision environment and how their choices map into final outcomes, in spite +of the fact that we did our best to describe the environment completely and accu- +rately. If this is the case, then there is room for further descriptions to influence +behavior. We have shown that this is indeed readily observable, using the rela- +tively weak instrument of short video clips providing commentary on the hedging +argument of Raiffa. +This means that in classical designs, it may be necessary +to view a given choice as arising from a combination of preferences and how the +subject understands the environment, where the second channel is non-trivial. Ac- +cordingly, a given choice may not provide direct evidence for or against any given +preference model. +A +Experimental Design +A.1 +Experiment details +The experimental sessions took place in April and May of 2018, and February of +2020. The experiment was conducted at the Experimental and Behavioral Eco- +nomics Laboratory (EBEL) at University of California, Santa Barbara. There are +two waves of data collection. In the first wave, 176 students participated in 10 +sessions and the average session length was 60 minutes. In the second wave, 213 +students participated in 12 sessions and the average session length was 60 minutes. +In all sessions, subjects answered exactly one incentivized question, which was re- +lated to guessing the color of a ball. If the guess was correct, the subject received +10 USD, and 0 otherwise. The show-up fee for all sessions was 5 dollars. At the +end of each session we conducted a short questionnaire. The questions were not +incentivized, but we emphasized that answering these questions would be help- +ful for our research. The experiment was programmed using z-Tree [Fischbacher, +2007]. See Figure 2 for a screen shot. +12 + +A.2 +Physical environment +In all sessions, the urns and states were implemented using two cardboard boxes +and colored ping-pong balls. During the experiment (and in what follows), we refer +to the two containers as Box A and Box B. A photo of the boxes can be found in +Figure 1 (a). The protocols we used were guided by the desire to be as clear and +transparent as possible. Box A contained 49 white and 51 red balls. The balls were +displayed in clear plastic tubes at the beginning of the experiment so that subjects +could easily see that there were two more red than white balls. Photos of the tubes +are included as Figure 1 (b). After showing the balls to subjects, they were poured +into Box A. On the other hand, it was important that the exact contents of Box +B were unknown. We therefore informed subjects that Box B contained 100 balls, +each of which was either green or yellow, but we were intentionally not telling them +anything further about the contents. Box B was shaken so that it was credible that +it contained the same number of balls as Box A. After this presentation, subjects +were told that they could inspect all the boxes and balls at the conclusion of the +experiment if they so desired. +In each session, one subject was randomly selected to act as a monitor. The +monitor was the person who physically conducted all draws of balls and displayed +their colors to the other subjects, as well as coin flips, as relevant. +a +b +Figure 1: Boxes +13 + +A +BA.3 +A Screen Shot +Figure 2: +Second experiment: Video review. +After seeing videos and before +making their choices, subjects had the chance to re-visit all videos they watched +before. +A.4 +Questionnaire +Table 2 shows the additional questions that we asked at the end of the experiment. +14 + +Click here to start the first video +Start +Click here to start the second video +Click here to start the third video +Start +Start +Nowyou canreviewall threevideos +You have enough time to watch the full videos more than two times +Proceed to next stageQuestions asked in all groups +Gender (Male, Female, Prefer not to tell) +Major +How many Green balls do you think there are in Box B? +How many Yellow balls do you think there are in Box B? +Was there any part of the experiment that was unclear? +After watching the videos, my preference over choices was: +(More clear, Less clear, Unchanged, I don’t know) +Questions asked when V1 and V2 are both presented +Which video do you think is more compelling? (V1, V2, Equal) +Questions asked in when option Coin is not available +Do you find it difficult to simulate a coin toss in you head? (Yes, No) +Questions asked when subjects chose White ball in Box A +Why did you choose White? +Questions asked when subjects chose Green or Yellow ball in Box B +Why did you choose Green or Yellow? +Questions asked in no-video treatments +Why did you recommend this to your friend? +Table 2: List of questions asked in the questionnaire +In the battery of sessions for the experiment, different treatments required dif- +ferent questions. The first block lists the questions that we asked in all treatments. +The second block lists questions that are asked when both videos are presented. +We denote by V1 the video in favor of the hedging argument and by V2 the video +with the counter-argument. The third block lists questions that are asked when +subjects are offered only three options and must implement the randomization +with a virtual coin on their own. The fourth and fifth blocks list questions contin- +gent on subjects’ choices. In the treatments in which videos are not shown to the +subjects before their incentivized choices, we showed the video before the ques- +tionnaire and asked for their “recommendation” in the questionnaire. We further +asked for their reasoning. This is listed in the last block. +A.5 +Instructions +We attached the instruction of the most comprehensive treatment. In this treat- +ment, we provided the subjects with 4 options and both videos. Instructions for +15 + +all the other treatments are written in a similar fashion. +Instructions +Welcome to the experiment! Please take a seat as directed. Please wait for +instructions and do not touch the computer until you are instructed to do so. +Please put away and silence all personal belongings, especially your phone. We +need your full attention for the entire experiment. +Adjust your chair so that +you can see the screen in the front of the room. +The experiment you will be +participating in today is an experiment in decision making. At the end of the +experiment you will be paid for your participation in cash. +Each of you may +earn different amounts. The amount you earn depends on your decisions and on +chance. You will be using the computer for the experiment, and all decisions will +be made through the computer. DO NOT socialize or talk during the experiment. +All instructions and descriptions that you will be given in this experiment are +factually accurate. According to the policy of this lab, at no point will we attempt +to deceive you in any way. Your payment today will include a $5 show up fee. +One of you will be randomly selected to act as a monitor. The monitor will be +paid a fixed amount for the experiment. The monitor will assist us in running +the experiment and verifying the procedures. If you have any questions about the +description of the experiment, raise your hand and your question will be answered +out loud so everyone can hear. We will not answer any questions about how you +“should” make your choices. +As I said before, do not use the computer until +you are asked to do so. When it is time to use the computer, please follow the +instructions precisely. +We will now explain the experiment. There are two containers on the table that +we will refer to as Box A and Box B. This is Box A. The Box is empty. Box A will +contain 100 ping pong balls. Each of the balls in Box A will be either White, like +this, or Red, like this. Specifically, Box A will contain exactly 49 White balls and +51 Red balls, for a total of 100 balls. You don’t have to remember these numbers. +When it is time to make a decision, we will remind you of these numbers. We +have counted and displayed the balls in these tubes to make it easier to show the +contents of Box A. There are 25 white balls in this tube and 24 in this tube, for +a total of 49 white balls. There are 25 red balls in this tube and 26 red balls in +16 + +this tube, for a total of 51. We will now pour these balls into Box A and shake +it to mix the balls together. This is Box B. We have already filled Box B with +100 ping pong balls. Each ball is either Green, like this, or Yellow, like this. We +will not reveal the exact numbers of Green and Yellow balls. Instead, you know +only that there are 100 balls in total, consisting of some combination of Green and +Yellow balls. We will now shake Box B to mix the balls up. At the end of the +experiment, you will have an opportunity to inspect the Boxes and ping pong balls +if you wish. In a few moments we will ask the Monitor to draw one ball from each +Box for everyone to observe. You will be asked to choose from several options that +correspond to guessing the color of a ball that the Monitor draws. If your guess +matches the result, you will receive 10 dollars in additional to the show up fee. If +your guess does not match, you will receive 0 dollars in addition to the show up +fee. +We will now start the experiment. On the computer desktop you will find a +green icon named zleaf. Double click it now. Now there should be a welcome +screen. Type your name and click the OK button in the welcome screen. One of +you has been randomly selected by the software to serve as the monitor. Please +raise your hand if you are the monitor. Could you please click the OK button +on your screen and come to the front? Now your screen should have changed to +“Please listen to the instructions”. Please leave it like that and do not click OK. +In a few moments the Monitor is going to draw one ball from Box A and one ball +from Box B. We are going to ask you to bet on the outcome of those draws. +Specifically, you will be able to place one of 4 bets. Let me explain three of +these options first. You can bet on White (from Box A), Green (from Box B) or +Yellow (from Box B). Notice that you cannot bet on Red. If you bet on the White +ball from Box A, then your payoff is not related to the draw from Box B. In other +words, if the monitor draws a White ball from Box A, you win. If the monitor +draws a Red ball, you lose. Similarly, if you choose Green or Yellow, your payoff +only depends on the draw from Box B. For the fourth option, your bet will depend +on the outcome of a coin flip. The monitor will flip a coin like this. If the coin +lands on Heads, then we will set your bet to Green. If the coin lands on tails, +we will set your bet to Yellow. To repeat, we will set your bet to either Green +or Yellow, depending on the coin flip result. Again, you don’t have to write this +17 + +down, since we will remind you about all the options when it is time to make your +choice. +Before you make you decision, we are going to provide you with some comments +on the experiment contained in 2 short videos. The videos are in total about 5 +minutes long. After the videos, you will make your choice by selecting one of the +four options. We will proceed like this: You will first watch the two videos. Then, +you will have a chance to review the videos if you like. During the review session, +you can pause or rewind the videos. There will be enough time to watch both +videos more than two times in the review session. After the review session, we are +going to ask for your choice. After you enter your decision, please wait for others +to finish. There will not be any further instructions until all of you make your +decisions. Please follow the instructions on the screen and focus on the videos as +much as possible. If you finish early, please remain quiet since others may still be +watching. Now please put on the headphones provided at your desk and watch +the videos. Once you are ready, please click OK. +The monitor is now going to draw the balls. Please look away and draw a ball +from Box A and show it to everyone. The color is [REALIZED COLOR]. Please +put the ball back. We will write down the result on the blackboard. Now please +look away and draw a ball from Box B and show it to everyone. The color is +[REALIZED COLOR]. Please put the ball back. We will write down the result +on the blackboard. Please toss the coin and announce the result. The result is +[REALIZED SIDE]. Please put the coin down. We will write down the result on +the blackboard. Now please return to your seat and enter these results into your +computer screen, accompanied by an Experimenter. You can now see the outcome +and your earnings on the screen. If you have questions about your payoff, please +raise your hand. +We will now conduct a short questionnaire. Please wait for the questionnaire +to start. The monitor doesn’t have to fill the questionnaire. Please complete the +questionnaire. Please be as specific as you can in your responses. Answering the +question is helpful to our research, but your responses are entirely voluntary. After +you finish, please wait for others. We will call you to the front by your participant +ID to be paid before leaving. Thank you very much for your participation. This +concludes the experiment. We will now begin calling you to the front to be paid +18 + +before leaving. +A.6 +Video scripts +A.6.1 +Names and notations +Recall that we denote the video that explains the Raiffa hedging argument, used +in our main treatment, by V1 and its counter argument by V2. In the treatments +without an explicit coin flip option, the instructions do not describe a coin. In- +stead, we show a short video introducing the hedging idea through the use of an +“imaginary coin.” We call this video V0. V0 is neither in favor of hedging nor +against hedging. It merely states the idea of conditioning one’s bet on the outcome +of a virtual coin flip. We then slightly modified V1 and V2 by changing “the coin +flip option” to “the imaginary coin.” For more details, please see the script below. +To summarize, we have in total five distinct videos, listed in the table below, +where “p” stands for “physical coin” and “v” for “virtual coin.” +V1p +V2p +V0 +V1v +V2v +In favor of hedging? +Yes +No +n/a +Yes +No +Against hedging? +No +Yes +n/a +No +Yes +Describe hedging using a real coin? +Yes +Yes +n/a +No +No +Describe hedging using a virtual coin? +No +No +n/a +Yes +Yes +Description of a virtual coin? +n/a +n/a +Yes +n/a +n/a +A.6.2 +V0 Script +Recall that your three options are to choose: a White Ball from Box A, a Green +Ball from Box B, or a Yellow Ball from Box B. Let me suggest a new method for +choosing how to bet. To use this method, you need to create a random event. +So, imagine you have a coin and you can flip it. The coin lands on Heads with +probability 50% and on Tails with probability 50%. Before the toss, you plan to +bet on a Green Ball from Box B if the coin lands on Heads, and on a Yellow Ball +from Box B if the coin lands on Tails. Using this rule, you will not bet on a White +Ball from Box A. To summarize, you first imagine the outcome of the coin flip. +Then you choose Green Ball from Box B if the coin lands on Heads and you choose +19 + +Yellow Ball from Box B if the coin lands on Tails. Click to view +A.6.3 +V1p Script +Recall that Box A contains 49 white balls and 51 red balls. So, you will win with +probability 49% if you choose the “White Ball from Box A.” Let me describe the +outcome when you choose the “Coin flip for green/yellow ball.” Recall that Box +B contains an unknown combination of 100 Green and Yellow balls. So when the +Monitor draws a ball from Box B there are two possible cases: the ball can either +be Green, or it can be Yellow. Suppose the ball happens to be Green. Now, when +the monitor flips the coin, it will land either on Heads or on Tails. Each case is +equally likely: the probability of Heads is 50% and the probability of Tails is 50%. +If the coin lands on Heads, you would bet on Green and win. If the coin lands +on Tails, you would bet on Yellow and lose. So, what we have observed is that if +the ball from Box B happens to be Green, you would win with probability 50%. +Now suppose that the ball from Box B happens to be Yellow. As before, when +the monitor flips the coin, it will land either on Heads or on Tails. Each case is +equally likely: the probability of Heads is 50% and the probability of Tails is 50%. +If the coin lands on Heads, you would bet on Green and lose. If the coin lands on +Tails, you would bet on Yellow and win. So, what we have observed now is that if +the ball from Box B happens to be Yellow, you would again win with probability +50%. To summarize, if you choose the option “Coin flip for green/yellow ball” +you will win with probability 50% whether the ball from Box B is green or yellow. +Therefore, it does not matter how many of the balls are green and how many are +yellow, since you will win with probability 50% in either case. By betting instead +on a White Ball from Box A, you will win with probability 49%. Click to view +A.6.4 +V2p Script +Recall that Box A contains 49 white balls and 51 red balls. So, you will win with +probability 49% if you choose the “White Ball from Box A.” Let me describe the +outcome when you choose the “Coin flip for green/yellow ball.” If you choose this +option, there are two possibilities: when the monitor flips the coin, it will land +either on Heads or on Tails. Each case is equally likely: the probability of Heads +20 + +is 50% and the probability of Tails is 50%. Suppose the coin happens to land on +Heads. In this case, you would be betting on a Green Ball from Box B. The chance +that betting on a Green Ball from Box B wins depends on how many green balls +are in Box B. Since you are not told how many green balls are in Box B, your +probability of winning is uncertain. So, what we have observed is that if the coin +lands on Heads, your probability of winning is uncertain. Now suppose the coin +happens to land on Tails. In this case, you would be betting on a Yellow Ball from +Box B. The chance that betting on a Yellow Ball from Box B wins depends on +how many yellow balls are in Box B. Since you are not told how many yellow balls +are in Box B, your probability of winning is again uncertain. Click to view +A.6.5 +V1v Script +Recall that Box A contains 49 white balls and 51 red balls. So, you will win with +probability 49% if you choose the “White Ball from Box A.” Let me describe the +outcome when you choose the method based on the coin flip I described before. +Recall that Box B contains an unknown combination of 100 Green and Yellow +balls. So when the Monitor draws a ball from Box B there are two possible cases: +the ball can either be Green, or it can be Yellow. Suppose the ball happens to +be Green. Now, when you imagine flipping the coin, it will land either on Heads +or on Tails. Each case is equally likely: the probability of Heads is 50% and the +probability of Tails is 50%. If the coin lands on Heads, you would bet on Green +and win. If the coin lands on Tails, you would bet on Yellow and lose. So, what +we have observed is that if the ball from Box B happens to be Green, you would +win with probability 50%. Now suppose that the ball from Box B happens to be +Yellow. As before, when you imagine flipping the coin, it will land either on Heads +or on Tails. Each case is equally likely: the probability of Heads is 50% and the +probability of Tails is 50%. If the coin lands on Heads, you would bet on Green +and lose. If the coin lands on Tails, you would bet on Yellow and win. So, what we +have observed now is that if the ball from Box B happens to be Yellow, you would +again win with probability 50%. To summarize, if you use the method based on +the coin flip, you will win with probability 50% whether the ball from Box B is +green or yellow. Therefore, it does not matter how many of the balls are green +21 + +and how many are yellow, since you will win with probability 50% in either case. +By betting instead on a White Ball from Box A, you have a known probability of +winning of 49%. Click to view +A.6.6 +V2v Script +Recall that Box A contains 49 white balls and 51 red balls. So, you will win with +probability 49% if you choose the “White Ball from Box A.” Let me describe the +outcome when you choose the method based on the coin flip I described before. +If you use this method, there are two possibilities: when you imagine flipping the +coin, it will land either on Heads or on Tails. Each case is equally likely: the +probability of Heads is 50% and the probability of Tails is 50%. Suppose the coin +happens to land on Heads. In this case, you would be betting on a Green Ball +from Box B. The chance that betting on a Green Ball from Box B wins depends on +how many green balls are in Box B. Since you are not told how many green balls +are in Box B, your probability of winning is uncertain. So, what we have observed +is that if your coin lands on Heads, your probability of winning is uncertain. Now +suppose your coin happens to land on Tails. In this case, you would be betting on +a Yellow Ball from Box B. The chance that betting on a Yellow Ball from Box B +wins depends on how many yellow balls are in Box B. Since you are not told how +many yellow balls are in Box B, your probability of winning is again uncertain. +So, what we have now observed is that if your coin lands on Tails, your probability +of winning is also uncertain. To summarize, if you use the method based on the +coin flip, your probability of winning is uncertain if the coin lands on Heads and +it is also uncertain if the coin lands on Tails. By betting instead on a White Ball +from Box A, you have a known probability of winning of 49%. Click to view +References +M. Abdellaoui, P. Klibanoff, and L. Placido. Experiments on compound risk in +relation to simple risk and to ambiguity. Management Science, 61(6):1306–1322, +2015. +22 + +N. Al-Najjar and J. L. Weinstein. The ambiguity aversion literature: A critical +assessment. Economics and Philosophy, 25:249–284, 2011. +M. Allais. Le Comportement de l’Homme rationnel devant le Risque, Critique des +Postulats et Axiomes de l’´Ecole Am´ericaine. Econometrica, 21:503–546, 1953. +F. J. Anscombe and R. J. Aumann. A definition of subjective probability. Annals +of Mathematical Statistics, 34:199–205, 1963. +Y. Azrieli, C. P. Chambers, and P. J. Healy. 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Theory and Decision, 81(3):313–337, 2016. +26 + +A. Wald. +An essentially complete class of admissible decision functions. +The +Annals of Mathematical Statistics, 18(4):549–555, 1947. +C.-L. Yang and L. Yao. Testing ambiguity theories with a mean-preserving design. +Quantitative Economics, 8(1):219–238, 2017. +27 + diff --git a/69E1T4oBgHgl3EQfnASE/content/tmp_files/load_file.txt b/69E1T4oBgHgl3EQfnASE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1e8114d070649e26659071db807b1e453209672 --- /dev/null +++ b/69E1T4oBgHgl3EQfnASE/content/tmp_files/load_file.txt @@ -0,0 +1,790 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf,len=789 +page_content='Randomization advice and ambiguity aversion∗ Christoph Kuzmics† Brian W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Rogers‡ Xiannong Zhang§ January 10, 2023 Abstract We design and implement lab experiments to evaluate the normative appeal of behavior arising from models of ambiguity-averse preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We report two main empirical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' First, we demonstrate that behavior reflects an incomplete understanding of the problem, providing evidence that subjects do not act on the basis of preferences alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Second, additional clarification of the decision making environment pushes subjects’ choices in the direction of ambiguity aversion models, regardless of whether or not the choices are also consistent with subjective expected utility, supporting the position that subjects find such behavior normatively appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' JEL codes: C91, D81 Keywords: Knightian uncertainty, subjective expected utility, ambiguity aversion, lab experiment ∗We thank Michael Greinecker, John Nachbar, Paulo Natenzon, Andrea Prat, Todd Sarver, Tomasz Strzalecki, Peter Wakker and Jonathan Weinstein for insightful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The authors gratefully acknowledge the funding from the Weidenbaum Center on the Economy, Government, and Public Policy at Washington University in St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Louis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' †University of Graz, Austria, christoph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='kuzmics@uni-graz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='at ‡Washington University in St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Louis, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=', brogers@wustl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='edu §Corresponding author, Washington University in St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Louis, 1 Brookings Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Louis, Missouri, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' (63130), zhangxiannong@wustl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='03304v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='TH] 9 Jan 2023 1 Introduction Many economic decisions are made under uncertainty that cannot be readily quan- tified by objective probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Consider saving decisions, that is investing money into bonds or stocks in the presence of inflation uncertainties and general uncer- tainties about the economic future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Even in the absence of wars and pandemics most people find it hard to attach probabilities to the relevant possible events, let alone agree on such an assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The classical paradigm of rational decision making under such uncertainty is subjective expected utility (SEU), underpinned by the axiomatic foundation of Savage [1954].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The experimental designs of Ellsberg [1961], later implemented in many studies, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=', the survey of Trautmann and Van De Kuilen [2015], have challenged the SEU paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This challenge was mostly on positive, that is, empirical, grounds in that these experiments found that many people behave in a way that is inconsistent with subjective expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The fact, however, that this behavior has been so robust in lab experiments and that many researchers have developed axiomatic foundations for alternative models of decision making that admit such behavior, see references below, may be received as posing a normative challenge to SEU in postulating that a broader set of preference models could be considered normatively appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In this paper we aim to test the normative appeal of these alternative models of decision making under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We use the term “normative” in the sense of Ellsberg’s 1962 PhD thesis, see Ellsberg [2001, pages 22-26], and as in the “subjec- tive” definition of rationality given by Gilboa [2012, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 5]: We consider a decision normatively appealing (to the decision maker) if the decision maker (still) makes this choice after thorough reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In all of our treatments, subjects are provided with a complete, and fairly standard, description of all payoff-relevant aspects of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We operationalize this reflection in the lab by providing subjects with supplementary descriptions that, while not payoff-relevant, emphasize certain ways to think about the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The descriptions are provided in the form of short videos that subjects watch before the elicitation of their choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Our findings are relevant to all models of ambiguity aversion that are monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' That is, if one act is better than another act in all states, then the inferior act 2 cannot be chosen when the superior act is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We call such models classical ambiguity aversion (CAA) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='1 The experimental environment is directly inspired by the hedging argument of Raiffa [1961] in the context of the Ellsberg [1961] two-color urn experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' A risky urn contains 49 White and 51 Red balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='2 An ambiguous urn also contains 100 balls, each of which is either Green or Yellow, but nothing more is known about the composition of the ambiguous urn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The decision-maker (DM) must choose one of three actions, after which the experimenter draws one ball from each urn Together, these determine the consequence for the DM, which is either “Win” or “Lose”, with Win being strictly preferred to Lose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We call the choices bets for White, Green, and Yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Each bet Wins if the experimenter draws a ball of the corresponding color, and loses otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Ellsberg’s key insight is that Bet White, while commonly chosen, is incompatible with SEU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='3 In this context, the Raiffa [1961] hedge against ambiguity works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The DM flips a fair coin and bets Green if the coin lands on heads and bets Yellow if the coin lands on tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This randomized action provides an (objective) winning probability of 50%, regardless of the color of the drawn ball from the ambiguous urn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' As this is higher than the 49% winning probability from betting White, this action strictly dominates the Ellsberg choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='4 1This category includes most preference models reviewed in the recent survey of Machina and Siniscalchi [2014], such as the maxmin expected utility model of Gilboa and Schmeidler [1989], the Choquet expected utility model of Schmeidler [1989], the smooth ambiguity model of Klibanoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2005], the variational and multiplier preference models of Maccheroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2006] and Hansen and Sargent [2001], confidence function preferences of Chateauneuf and Faro [2009], uncertainty aversion preferences of Cerreia-Vioglio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2011], and the incomplete preference model of Bewley [2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 2This variation from 50/50 avoids identification complications that can arise from indifference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 3If Bet White is chosen, it must be weakly preferred to Bet Yellow, so that the DM’s subjective probability of a yellow ball being drawn from the ambiguous urn must be at most 49%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' But then the subjective probability of a green ball being drawn from the ambiguous urn must be at least 51%, so that Bet Green is strictly preferred to Bet White, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 4Kuzmics [2017], appealing to results in classical statistical decision theory – in particular the complete class theorem of Wald [1947], has shown that a DM who can randomize over choices and commit to following the realized prescription of this randomization can never make choices that are inconsistent with SEU and at the same time consistent with CAA in any decision problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' See also Bade [2015], Oechssler and Roomets [2014], Azrieli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2018], Baillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2022a], and Baillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2022b] for similar results and arguments along these lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' One can, in fact, regard ambiguity aversion as a preference for randomization, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=', Eichberger and Kelsey [1996] and Epstein and Schneider [2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 3 In light of the Raiffa [1961] argument, what are the possible explanations for a classically ambiguity-averse DM to nonetheless bet on White in this experiment even though it is dominated?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' First, it is possible, even likely, that designing a random choice does not occur to subjects as an option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Second, even if a subject recognizes such a possibility, it is possible that they cannot, or choose not to, go through the required construction and reasoning that would allow them to see that betting on White is dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='5 But suppose now that a subject does recognize the possibility and understands the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' A third possibility is that the subject has no access to a suitable randomization device, nor thinks they can simulate one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So suppose the subjects does have a fair coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The fourth and final explanation is that the subject lacks the ability to commit to the randomized action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Once the coin flip realizes, the subject could revisit their choice, and if they are ambiguity averse they will want to flip the coin again, and again ad infinitum, with one possible outcome that they bet on White in the end after all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Classical ambiguity aversion models do not allow us to delve into the reasons behind the choice of White in the presence of the Raiffa [1961] argument, as these preference models are axiomatized for preferences over the space of all pure acts, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=', Seo [2009], and not over the set of all mixed acts as in Anscombe and Aumann [1963].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This means, however, that whether or not a classically ambiguity averse subject who understands the environment may choose to bet on White depends simply on whether the Raiffa [1961] hedge (including the commitment to its outcome) is provided as a pure choice or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If it is given as a pure choice the subject cannot choose to bet on White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If it is not given, they can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Motivated by these considerations, our experimental treatments provide differ- ential supplementary observations that focus on the hedging argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' All of the supplementary observations are given after a (common to all treatments) standard and complete description of the environment, and before the elicitation of the sub- ject’s choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If the standard and complete description of the environment suffices to impart a full understanding of the consequences of each possible action, then 5The argument is based on the reduction of compound lotteries and as Halevy [2007] found many subjects who make Ellsberg choices also fail to reduce compound lotteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This failure, however, can hardly be seen as normatively appealing - it is simply a mathematical mistake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Ab- dellaoui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2015] has found a much weaker association between displayed ambiguity aversion and a failure to reduce compound lotteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 4 the treatment effects of the supplementary observations will be null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Indeed, there is no marginal payoff-relevant information in the supplementary observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' To the contrary, our first main finding is that the commentary significantly changes behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Since the commentary does not affect a subject’s underlying preferences, it changes behavior through modifying a subjects’ understanding of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This is only possible if the subject’s understanding after the stan- dard description was incomplete or erroneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We conclude that many subjects indeed have an incomplete or erroneous understanding after being provided with a standard and complete description of the classical Ellsberg two-urn environment, so that directly applying the revealed preference toolkit to such data may not be appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Instead, a given choice may reflect an incomplete comprehension of the environment, and therefore be viewed as a mistake, or the result of confusion, rather than as a manifestation of the subject’s preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Our second main finding concerns the direction of the effects of the supplemen- tal observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In this regard, and to the extent we can infer preferences from the treatments with commentary, our data supports the broader normative appeal of ambiguity aversion models over SEU in the following sense: When the Ellsberg choice (bet White) is compatible with ambiguity averse preferences but not with SEU, the supplemental hedging observations increase the frequency of Ellsberg be- havior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' when the Ellsberg choice is incompatible with ambiguity aversion (and so also with SEU), the same observations decrease the frequency of Ellsberg behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The paper proceeds as follows: The experimental design is given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The results are given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Section 4 offers a discussion of these results and Section 5 provides a brief survey of related literature before Section 6 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Experimental instructions are presented in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 2 Experimental Design The slight variation of the Ellsberg two-color urn experiment, outlined above, serves as the baseline and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We then vary the supplemental observations that subjects receive about the decision-making environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Our treatments differ from the baseline along two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' First, in addition to the standard options of betting on White, Yellow, or Green, in some treatments subjects are 5 presented with an additional fourth option, which executes a bet on either Green or Yellow according to the outcome of a fair coin to be tossed by a third party after the balls are drawn from the urns - the Raiffa hedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Note that this option also serves as a commitment device for randomization, as the bet will be executed by the experimenter on the subject’s behalf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Recall that the compatibility of the Ellsberg choice (bet White) with classical ambiguity aversion models hinges on the presence or absence of this option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Second, after the environment is fully described as transparently as possible, in some treatments subjects watch short videos before making their (single) choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The videos, while all factually correct, emphasize different aspects of the con- sequences of using the randomization device, in ways that we hypothesize may change some subjects’ understanding of the merits of the Raiffa hedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In the main treatment, the coin flip bet is included as an option, and subjects are presented with a single video (denoted V 1 and available here) containing sup- plemental observations that describe the hedging argument of Raiffa [1961].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='6 It describes the outcome of betting on the coin conditional on the outcome of the ball drawn from the ambiguous urn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' It states that the winning probability using that option is 50% in either case (green ball or yellow ball drawn), and concludes by reminding the subject that betting on White wins with probability 49%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This video, as well as our videos, does not explicitly advocate for any particular choice, so that it contains observations rather than advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The transcripts of the videos are read by an anonymous (to subjects) third party to avoid a perception that the experimenters are giving implicit advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Partly to control for a possible experimenter demand effect, we designed a second video (denoted V 2 and available here), in which the structure of the argu- ment and the language is symmetric to the first video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' It describes the outcome of betting on the coin conditional on the outcome of the coin flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' It states that no known winning probability can be specified in either case (heads or tails), and concludes by reminding the subject that betting on White wins with probability 49%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Again, it does not advocate for any particular choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We ran treatments utilizing exclusively this second video, as well as treatments in which subjects were presented with both videos before eliciting their choice (in 6Transcripts of all videos are included in the appendix for an offline audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 6 both orders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' there were no order effects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='7 As we want to understand the effect of the supplementary observations inde- pendently from the effect of presenting the hedging device as an explicit option, we ran a parallel set of treatments with similar videos but where the available options were simply bets for White, Green, or Yellow, as in the baseline case, without the coin flip option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In these treatments, the Ellsberg choice remains compatible with CAA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We varied the videos slightly to accommodate the different choice set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' First, as there was no explicit coin, before showing either V 1 or V 2, we showed a preliminary video (denoted V 0 and available here) in which it was explained that a subject could imagine a virtual coin toss, and then bet on Green/Yellow accord- ing to the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Second, in videos V 1 and V 2 the coin toss was referred to as a virtual coin toss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We refer to the treatment with the explicit hedge/commitment option as “Coin” and those without it as “No Coin” treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 3 Results Table 1 summarizes the main findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' W G Y W G Y C Baseline 45% 41% 14% Baseline 37% 26% 11% 26% (26/58) (24/58) (8/58) (20/54) (14/54) (6/54) (14/54) V1 27% 55% 18% V1 29% 2% 2% 67% (12/44) (24/44) (8/44) (14/48) (1/48) (1/48) (32/48) V2 V2 41% 9% 9% 41% (18/44) (4/44) (4/44) (18/44) V1+V2 58% 28% 13% V1+V2 23% 13% 9% 55% (35/60) (17/60) (8/60) (13/56) (7/56) (5/56) (31/56) Table 1: Summary of data (left: without a randomization device;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' right: explicit randomization option).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 7In sessions using both videos, we showed one video first, and the other video next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We did this in both orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Then after both videos were shown, the subjects were provided with additional time to revisit any portions of either or both of the two videos before proceeding to enter their bet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' During this time subjects could pause, rewind, and switch between videos as they liked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We thereby tried to minimize possible order effects and, indeed, there is no evidence that the order of the videos has any effect, and so we have pooled that data in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 7 We summarize the key findings from this data in the following three results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Result 1 If preferences alone dictate choices, the supplementary observations con- tained in the videos should have no effect on behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' For the No Coin (Coin, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=') treatments, pooling the data for Green and Yellow (being conservative), the p-value for the null hypothesis that choice frequencies are the same in the baseline and the V 1 video treatment is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='067 (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='001, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' )8, and for the null hypothesis that choice frequencies are the same in the baseline and the neutral V 1 + V 2 video treatment is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='142 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='007, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Result 2 The p-value for the null hypothesis that neutral video observations (V 1+ V 2) does not decrease the choice of White relative to the baseline is ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='5 for the No Coin treatments, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='057 for the Coin treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Result 3 Without supplemental observations there is no significant difference in the frequency of Bet White between the Coin and No Coin treatment, p-value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' With supplemental observations (V 1 + V 2) there is a significant difference in the frequency of Bet White between the Coin and No Coin treatment, p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 4 Discussion There is firm evidence that behavior across treatments is not a pure consequence of underlying preferences combined with a complete understanding of the environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The observations contained in the videos cannot change preferences as they do not change any of the payoff-relevant considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Rather, any differences in behavior must come from differences in the subjects’ understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Suppose we adopt the view that choices after studying both videos indeed re- veal preferences, since subjects may have a more complete understanding of the environment and the consequences of their choices after considering the observa- tions contained therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Even then, 23% of subjects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=', those who chose Bet White in the Coin treatment, have preferences different from any CAA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The remaining 77% of subjects make choices in the Coin treatment that can be explained by CAA as well as SEU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 8All statistical tests performed in this paper are likelihood ratio tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 8 However, the No Coin treatments provide an interesting contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In these treatments, Bet White is undominated and the supplemental observations increase the frequency of Bet White relative to the baseline description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If, again, we view the choices after studying both videos as revealing preferences, the choice of Bet White made by 58% of subjects is inconsistent with SEU but is consistent with CAA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Together, these findings suggest that CAA models have broader normative ap- peal than (the narrower theory of) SEU despite their descriptive problems in some environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Of note, we asked subjects (in a non-incentivized post-experiment questionnaire) if their preference became more or less clear after watching the videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In the No Coin treatments 17% (27 out of 162) of subjects reported that their preferences became “less clear” after watching the videos, which is much higher than the 3% (6 out of 202) reporting the same in the Coin treatment (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='001), calling into question the presumption that behavior directly reveals preferences, especially in the No Coin treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' One interpretation is that many subjects find monotonicity to be a normatively appealing property, yet lack the sophistication to identify its consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We conclude this discussion by considering preferences that may depend on the timing of the resolution of uncertainty, as in the models of Seo [2009], Saito [2015], and Ke and Zhang [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Such models are not classical, as they are not monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='9 In the Coin treatments subjects were (truthfully) told, as part of the baseline description of the environment, that the coin flip would be executed after the balls were drawn from the urns and revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Thus, the choice of Bet White in the Coin treatments (37% in the baseline and 23% after both videos) is even inconsistent with these more flexible models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Roughly, one could categorize our subjects as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' There is one group of subjects (≥ 23%) who make choices inconsistent with all models of ambiguity aversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' There is a second group of subjects (≤ 58% - 23% = 35%) who make choices consistent with ambiguity aversion but not with SEU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The remainder make choices consistent with SEU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Subjects in the second group would have a demand for randomization devices as they cannot, to their satisfaction, create and commit 9Motivated by the state separability embedded in monotonicity, Bommier [2017] provides a model where monotonicity is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 9 to randomized choices themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 5 Related Literature A number of papers have studied the consistency of subjects’ choices across decision problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' These include Binmore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2012], Stahl [2014], Voorhoeve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2016] and Crockett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This literature finds, on the whole, that relatively few subjects make consistent choices, and those who do tend to be ambiguity- neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The lack of consistency can be interpreted as evidence against people choosing according to a clear preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' However, ambiguity-averse DMs may find inconsistent choices to be a useful hedge against ambiguity, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=', Kuzmics [2017] and Azrieli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2018] for more general arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='10 Our single choice design is immune to such problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This is why we constrained our design to a single incentivized elicitation per subject, even at the cost of forgoing the ability to conduct within-subject analyses across treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Spears [2009], Dominiak and Schnedler [2011], and Oechssler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2016] study experiments in which subjects are given the Raiffa hedge as an option, similarly to our baseline Coin treatment (without supplemental commentary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Generally, they find very few subjects choosing this option, with more subjects instead choosing a dominated option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This too is evidence against CAA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' They also find that subjects do not care about the timing of the resolution of uncertainty, evidence even against the non-CAA models of Seo [2009], Saito [2015], and Ke and Zhang [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Our focus is on the possible effects of providing explicit descriptions of the Raiffa hedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We thus add to these findings that such observations significantly influence behavior, and does so in directions that support the appeal of CAA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Finally, several studies test, in various ways, the normative appeal of ambigu- ity aversion preference models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='11 The closest to our design is that of Slovic and Tversky [1974], who give subjects written advice for and against Allais [1953] and 10This problem persists under many different preference elicitation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' See also e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=', Baillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2014], Bade [2015], Oechssler and Roomets [2015], which builds on earlier work on eliciting non-expected utility preferences under objective uncertainty by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=', Karni and Safra [1987].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 11Al-Najjar and Weinstein [2011] provide normative arguments against ambiguity aversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 10 Ellsberg [1961] choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' However, their advice is built around the independence axiom, and so concerns a quite distinct domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Jabarian and Lazarus [2022] also study the effect of a form of advice on subjects’ decisions in a framework with ambiguity aversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Their framework involves two independent draws from the same two-color ambiguous urn (and two draws from a 50-50 risky urn) in which many subjects make dominated choices, similarly also to Yang and Yao [2017] and Kuzmics et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Subjects win if they draw two balls of the same color from the urn that they choose, making betting on the ambiguous urn a (weakly) domi- nant choice - as the more extreme the ball distribution in the ambiguous urn the higher the chance of drawing two balls of the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Jabarian and Lazarus [2022] have treatments in which subjects are given additional decision problems that should help them understand the mechanism why a choice is dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' They find that while subjects do seem to understand the mechanism, they, nevertheless, do not seem to transfer this knowledge to the original problem in which they make dominated choices regardless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Finally, Keller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2007], Trautmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2008], Charness et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2013], and Keck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' [2014] study decision problems with am- biguity in groups (or under peer observation) and find, on the whole, that group discussion and related phenomena tend to lead to more ambiguity-neutral choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 6 Conclusion We have subjected classical preference models of ambiguity aversion models to tests of their normative appeal with experiments that stay close to the original Ellsberg (two-color urn) design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We find that subjects’ choices are affected by payoff-irrelevant commentary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This implies that at least one of the two treatments, without or with commentary, does not allow the full revelation of subjects’ preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' At least some of our subjects do seem to see a certain normative appeal in the kind of behavior prescribed by classical models of ambiguity aversion and, in particular, the monotonicity axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Giving subjects access to additional commen- tary, in the form of short video clips, results in behavior that is significantly more consistent with these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The nature of this normative appeal suggests that people, after sufficient re- 11 flection, would have a demand for the ability to commit to randomized choices, a demand which one would surmise should be observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' It would be interest- ing to identify instruments outside the lab, in the various areas of application of ambiguity aversion models, that could serve to satisfy this demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We also find that our subjects lack a complete and perfect understanding of their decision environment and how their choices map into final outcomes, in spite of the fact that we did our best to describe the environment completely and accu- rately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If this is the case, then there is room for further descriptions to influence behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We have shown that this is indeed readily observable, using the rela- tively weak instrument of short video clips providing commentary on the hedging argument of Raiffa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This means that in classical designs, it may be necessary to view a given choice as arising from a combination of preferences and how the subject understands the environment, where the second channel is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Ac- cordingly, a given choice may not provide direct evidence for or against any given preference model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' A Experimental Design A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='1 Experiment details The experimental sessions took place in April and May of 2018, and February of 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The experiment was conducted at the Experimental and Behavioral Eco- nomics Laboratory (EBEL) at University of California, Santa Barbara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' There are two waves of data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In the first wave, 176 students participated in 10 sessions and the average session length was 60 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In the second wave, 213 students participated in 12 sessions and the average session length was 60 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In all sessions, subjects answered exactly one incentivized question, which was re- lated to guessing the color of a ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If the guess was correct, the subject received 10 USD, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The show-up fee for all sessions was 5 dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' At the end of each session we conducted a short questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The questions were not incentivized, but we emphasized that answering these questions would be help- ful for our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The experiment was programmed using z-Tree [Fischbacher, 2007].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' See Figure 2 for a screen shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='2 Physical environment In all sessions, the urns and states were implemented using two cardboard boxes and colored ping-pong balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' During the experiment (and in what follows), we refer to the two containers as Box A and Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' A photo of the boxes can be found in Figure 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The protocols we used were guided by the desire to be as clear and transparent as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Box A contained 49 white and 51 red balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The balls were displayed in clear plastic tubes at the beginning of the experiment so that subjects could easily see that there were two more red than white balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Photos of the tubes are included as Figure 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' After showing the balls to subjects, they were poured into Box A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' On the other hand, it was important that the exact contents of Box B were unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We therefore informed subjects that Box B contained 100 balls, each of which was either green or yellow, but we were intentionally not telling them anything further about the contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Box B was shaken so that it was credible that it contained the same number of balls as Box A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' After this presentation, subjects were told that they could inspect all the boxes and balls at the conclusion of the experiment if they so desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In each session, one subject was randomly selected to act as a monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The monitor was the person who physically conducted all draws of balls and displayed their colors to the other subjects, as well as coin flips, as relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' a b Figure 1: Boxes 13 A BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='3 A Screen Shot Figure 2: Second experiment: Video review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' After seeing videos and before making their choices, subjects had the chance to re-visit all videos they watched before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='4 Questionnaire Table 2 shows the additional questions that we asked at the end of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 14 Click here to start the first video Start Click here to start the second video Click here to start the third video Start Start Nowyou canreviewall threevideos You have enough time to watch the full videos more than two times Proceed to next stageQuestions asked in all groups Gender (Male, Female, Prefer not to tell) Major How many Green balls do you think there are in Box B?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' How many Yellow balls do you think there are in Box B?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Was there any part of the experiment that was unclear?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' After watching the videos, my preference over choices was: (More clear, Less clear, Unchanged, I don’t know) Questions asked when V1 and V2 are both presented Which video do you think is more compelling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' (V1, V2, Equal) Questions asked in when option Coin is not available Do you find it difficult to simulate a coin toss in you head?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' (Yes, No) Questions asked when subjects chose White ball in Box A Why did you choose White?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Questions asked when subjects chose Green or Yellow ball in Box B Why did you choose Green or Yellow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Questions asked in no-video treatments Why did you recommend this to your friend?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Table 2: List of questions asked in the questionnaire In the battery of sessions for the experiment, different treatments required dif- ferent questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The first block lists the questions that we asked in all treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The second block lists questions that are asked when both videos are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We denote by V1 the video in favor of the hedging argument and by V2 the video with the counter-argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The third block lists questions that are asked when subjects are offered only three options and must implement the randomization with a virtual coin on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The fourth and fifth blocks list questions contin- gent on subjects’ choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In the treatments in which videos are not shown to the subjects before their incentivized choices, we showed the video before the ques- tionnaire and asked for their “recommendation” in the questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We further asked for their reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This is listed in the last block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='5 Instructions We attached the instruction of the most comprehensive treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In this treat- ment, we provided the subjects with 4 options and both videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Instructions for 15 all the other treatments are written in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Instructions Welcome to the experiment!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please take a seat as directed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please wait for instructions and do not touch the computer until you are instructed to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please put away and silence all personal belongings, especially your phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We need your full attention for the entire experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Adjust your chair so that you can see the screen in the front of the room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The experiment you will be participating in today is an experiment in decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' At the end of the experiment you will be paid for your participation in cash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Each of you may earn different amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The amount you earn depends on your decisions and on chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' You will be using the computer for the experiment, and all decisions will be made through the computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' DO NOT socialize or talk during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' All instructions and descriptions that you will be given in this experiment are factually accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' According to the policy of this lab, at no point will we attempt to deceive you in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Your payment today will include a $5 show up fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' One of you will be randomly selected to act as a monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The monitor will be paid a fixed amount for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The monitor will assist us in running the experiment and verifying the procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If you have any questions about the description of the experiment, raise your hand and your question will be answered out loud so everyone can hear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We will not answer any questions about how you “should” make your choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' As I said before, do not use the computer until you are asked to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' When it is time to use the computer, please follow the instructions precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We will now explain the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' There are two containers on the table that we will refer to as Box A and Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This is Box A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The Box is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Box A will contain 100 ping pong balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Each of the balls in Box A will be either White, like this, or Red, like this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Specifically, Box A will contain exactly 49 White balls and 51 Red balls, for a total of 100 balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' You don’t have to remember these numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' When it is time to make a decision, we will remind you of these numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We have counted and displayed the balls in these tubes to make it easier to show the contents of Box A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' There are 25 white balls in this tube and 24 in this tube, for a total of 49 white balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' There are 25 red balls in this tube and 26 red balls in 16 this tube, for a total of 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We will now pour these balls into Box A and shake it to mix the balls together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This is Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We have already filled Box B with 100 ping pong balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Each ball is either Green, like this, or Yellow, like this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We will not reveal the exact numbers of Green and Yellow balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Instead, you know only that there are 100 balls in total, consisting of some combination of Green and Yellow balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We will now shake Box B to mix the balls up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' At the end of the experiment, you will have an opportunity to inspect the Boxes and ping pong balls if you wish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In a few moments we will ask the Monitor to draw one ball from each Box for everyone to observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' You will be asked to choose from several options that correspond to guessing the color of a ball that the Monitor draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If your guess matches the result, you will receive 10 dollars in additional to the show up fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If your guess does not match, you will receive 0 dollars in addition to the show up fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We will now start the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' On the computer desktop you will find a green icon named zleaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Double click it now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Now there should be a welcome screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Type your name and click the OK button in the welcome screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' One of you has been randomly selected by the software to serve as the monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please raise your hand if you are the monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Could you please click the OK button on your screen and come to the front?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Now your screen should have changed to “Please listen to the instructions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please leave it like that and do not click OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In a few moments the Monitor is going to draw one ball from Box A and one ball from Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We are going to ask you to bet on the outcome of those draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Specifically, you will be able to place one of 4 bets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Let me explain three of these options first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' You can bet on White (from Box A), Green (from Box B) or Yellow (from Box B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Notice that you cannot bet on Red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If you bet on the White ball from Box A, then your payoff is not related to the draw from Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In other words, if the monitor draws a White ball from Box A, you win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If the monitor draws a Red ball, you lose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Similarly, if you choose Green or Yellow, your payoff only depends on the draw from Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' For the fourth option, your bet will depend on the outcome of a coin flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The monitor will flip a coin like this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If the coin lands on Heads, then we will set your bet to Green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If the coin lands on tails, we will set your bet to Yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' To repeat, we will set your bet to either Green or Yellow, depending on the coin flip result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Again, you don’t have to write this 17 down, since we will remind you about all the options when it is time to make your choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Before you make you decision, we are going to provide you with some comments on the experiment contained in 2 short videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The videos are in total about 5 minutes long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' After the videos, you will make your choice by selecting one of the four options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We will proceed like this: You will first watch the two videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Then, you will have a chance to review the videos if you like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' During the review session, you can pause or rewind the videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' There will be enough time to watch both videos more than two times in the review session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' After the review session, we are going to ask for your choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' After you enter your decision, please wait for others to finish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' There will not be any further instructions until all of you make your decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please follow the instructions on the screen and focus on the videos as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If you finish early, please remain quiet since others may still be watching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Now please put on the headphones provided at your desk and watch the videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Once you are ready, please click OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The monitor is now going to draw the balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please look away and draw a ball from Box A and show it to everyone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The color is [REALIZED COLOR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please put the ball back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We will write down the result on the blackboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Now please look away and draw a ball from Box B and show it to everyone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The color is [REALIZED COLOR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please put the ball back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We will write down the result on the blackboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please toss the coin and announce the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The result is [REALIZED SIDE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please put the coin down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We will write down the result on the blackboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Now please return to your seat and enter these results into your computer screen, accompanied by an Experimenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' You can now see the outcome and your earnings on the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If you have questions about your payoff, please raise your hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We will now conduct a short questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please wait for the questionnaire to start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The monitor doesn’t have to fill the questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please complete the questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Please be as specific as you can in your responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Answering the question is helpful to our research, but your responses are entirely voluntary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' After you finish, please wait for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We will call you to the front by your participant ID to be paid before leaving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Thank you very much for your participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' This concludes the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We will now begin calling you to the front to be paid 18 before leaving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='6 Video scripts A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='1 Names and notations Recall that we denote the video that explains the Raiffa hedging argument, used in our main treatment, by V1 and its counter argument by V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In the treatments without an explicit coin flip option, the instructions do not describe a coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In- stead, we show a short video introducing the hedging idea through the use of an “imaginary coin.” We call this video V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' V0 is neither in favor of hedging nor against hedging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' It merely states the idea of conditioning one’s bet on the outcome of a virtual coin flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' We then slightly modified V1 and V2 by changing “the coin flip option” to “the imaginary coin.” For more details, please see the script below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' To summarize, we have in total five distinct videos, listed in the table below, where “p” stands for “physical coin” and “v” for “virtual coin.” V1p V2p V0 V1v V2v In favor of hedging?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Yes No n/a Yes No Against hedging?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' No Yes n/a No Yes Describe hedging using a real coin?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Yes Yes n/a No No Describe hedging using a virtual coin?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' No No n/a Yes Yes Description of a virtual coin?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' n/a n/a Yes n/a n/a A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='2 V0 Script Recall that your three options are to choose: a White Ball from Box A, a Green Ball from Box B, or a Yellow Ball from Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Let me suggest a new method for choosing how to bet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' To use this method, you need to create a random event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So, imagine you have a coin and you can flip it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The coin lands on Heads with probability 50% and on Tails with probability 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Before the toss, you plan to bet on a Green Ball from Box B if the coin lands on Heads, and on a Yellow Ball from Box B if the coin lands on Tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Using this rule, you will not bet on a White Ball from Box A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' To summarize, you first imagine the outcome of the coin flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Then you choose Green Ball from Box B if the coin lands on Heads and you choose 19 Yellow Ball from Box B if the coin lands on Tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Click to view A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='3 V1p Script Recall that Box A contains 49 white balls and 51 red balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So, you will win with probability 49% if you choose the “White Ball from Box A.” Let me describe the outcome when you choose the “Coin flip for green/yellow ball.” Recall that Box B contains an unknown combination of 100 Green and Yellow balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So when the Monitor draws a ball from Box B there are two possible cases: the ball can either be Green, or it can be Yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Suppose the ball happens to be Green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Now, when the monitor flips the coin, it will land either on Heads or on Tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Each case is equally likely: the probability of Heads is 50% and the probability of Tails is 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If the coin lands on Heads, you would bet on Green and win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If the coin lands on Tails, you would bet on Yellow and lose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So, what we have observed is that if the ball from Box B happens to be Green, you would win with probability 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Now suppose that the ball from Box B happens to be Yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' As before, when the monitor flips the coin, it will land either on Heads or on Tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Each case is equally likely: the probability of Heads is 50% and the probability of Tails is 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If the coin lands on Heads, you would bet on Green and lose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If the coin lands on Tails, you would bet on Yellow and win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So, what we have observed now is that if the ball from Box B happens to be Yellow, you would again win with probability 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' To summarize, if you choose the option “Coin flip for green/yellow ball” you will win with probability 50% whether the ball from Box B is green or yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Therefore, it does not matter how many of the balls are green and how many are yellow, since you will win with probability 50% in either case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' By betting instead on a White Ball from Box A, you will win with probability 49%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Click to view A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='4 V2p Script Recall that Box A contains 49 white balls and 51 red balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So, you will win with probability 49% if you choose the “White Ball from Box A.” Let me describe the outcome when you choose the “Coin flip for green/yellow ball.” If you choose this option, there are two possibilities: when the monitor flips the coin, it will land either on Heads or on Tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Each case is equally likely: the probability of Heads 20 is 50% and the probability of Tails is 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Suppose the coin happens to land on Heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In this case, you would be betting on a Green Ball from Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The chance that betting on a Green Ball from Box B wins depends on how many green balls are in Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Since you are not told how many green balls are in Box B, your probability of winning is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So, what we have observed is that if the coin lands on Heads, your probability of winning is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Now suppose the coin happens to land on Tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In this case, you would be betting on a Yellow Ball from Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The chance that betting on a Yellow Ball from Box B wins depends on how many yellow balls are in Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Since you are not told how many yellow balls are in Box B, your probability of winning is again uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Click to view A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='5 V1v Script Recall that Box A contains 49 white balls and 51 red balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So, you will win with probability 49% if you choose the “White Ball from Box A.” Let me describe the outcome when you choose the method based on the coin flip I described before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Recall that Box B contains an unknown combination of 100 Green and Yellow balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So when the Monitor draws a ball from Box B there are two possible cases: the ball can either be Green, or it can be Yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Suppose the ball happens to be Green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Now, when you imagine flipping the coin, it will land either on Heads or on Tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Each case is equally likely: the probability of Heads is 50% and the probability of Tails is 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If the coin lands on Heads, you would bet on Green and win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If the coin lands on Tails, you would bet on Yellow and lose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So, what we have observed is that if the ball from Box B happens to be Green, you would win with probability 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Now suppose that the ball from Box B happens to be Yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' As before, when you imagine flipping the coin, it will land either on Heads or on Tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Each case is equally likely: the probability of Heads is 50% and the probability of Tails is 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If the coin lands on Heads, you would bet on Green and lose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If the coin lands on Tails, you would bet on Yellow and win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So, what we have observed now is that if the ball from Box B happens to be Yellow, you would again win with probability 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' To summarize, if you use the method based on the coin flip, you will win with probability 50% whether the ball from Box B is green or yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Therefore, it does not matter how many of the balls are green 21 and how many are yellow, since you will win with probability 50% in either case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' By betting instead on a White Ball from Box A, you have a known probability of winning of 49%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Click to view A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='6 V2v Script Recall that Box A contains 49 white balls and 51 red balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So, you will win with probability 49% if you choose the “White Ball from Box A.” Let me describe the outcome when you choose the method based on the coin flip I described before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' If you use this method, there are two possibilities: when you imagine flipping the coin, it will land either on Heads or on Tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Each case is equally likely: the probability of Heads is 50% and the probability of Tails is 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Suppose the coin happens to land on Heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In this case, you would be betting on a Green Ball from Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The chance that betting on a Green Ball from Box B wins depends on how many green balls are in Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Since you are not told how many green balls are in Box B, your probability of winning is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So, what we have observed is that if your coin lands on Heads, your probability of winning is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Now suppose your coin happens to land on Tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' In this case, you would be betting on a Yellow Ball from Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' The chance that betting on a Yellow Ball from Box B wins depends on how many yellow balls are in Box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Since you are not told how many yellow balls are in Box B, your probability of winning is again uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' So, what we have now observed is that if your coin lands on Tails, your probability of winning is also uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' To summarize, if you use the method based on the coin flip, your probability of winning is uncertain if the coin lands on Heads and it is also uncertain if the coin lands on Tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' By betting instead on a White Ball from Box A, you have a known probability of winning of 49%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Click to view References M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Abdellaoui, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Klibanoff, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Placido.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Experiments on compound risk in relation to simple risk and to ambiguity.' metadata={'source': 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18(4):549–555, 1947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Yang and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Testing ambiguity theories with a mean-preserving design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' Quantitative Economics, 8(1):219–238, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} +page_content=' 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf'} diff --git a/6dA0T4oBgHgl3EQfN__p/content/2301.02156v1.pdf b/6dA0T4oBgHgl3EQfN__p/content/2301.02156v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4c2df015b577e0635b2bb70041abe4ca37807d92 --- /dev/null +++ b/6dA0T4oBgHgl3EQfN__p/content/2301.02156v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:433a021569a0286ee7ee75b1840a8b1752ecb210110b719dedfd15767f39af2c +size 1033685 diff --git a/6dA0T4oBgHgl3EQfN__p/vector_store/index.pkl b/6dA0T4oBgHgl3EQfN__p/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..9b94b477f930e163dfdd0490c7186b42c4c8ba22 --- /dev/null +++ b/6dA0T4oBgHgl3EQfN__p/vector_store/index.pkl @@ -0,0 +1,3 @@ +version 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In this work, we prepared a series of polar molecules +revealing a ferroelectric nematic phase (NF) with a very high dielectric constant (>104). A new +motif, which differs from previously reported molecular structures, was optimized to support +the NF phase. For all homologues the NF phase was observed directly on the cooling from the +isotropic phase and ferroelectric behaviour was investigated by dielectric spectroscopy, second +harmonic generation, polarization current measurements and by analysis of textures in the +polarized light. The presented materials combine ferroelectricity with giant permittivity in a +fluid media at room temperatures, so they appear to be extremely attractive. Polarity of +molecules with the strong susceptibility to the electric field represent high potential for various +applications in energy-efficient memory devices or capacitors. + +1. +Introduction +In thermotropic liquid crystals (LCs) molecules can self-assemble and create intermediate +phases (mesophases) in a certain temperature range between liquid and crystalline phases [1], +combining the fluidity of liquids with the anisotropy characteristic for crystals. Anisotropic +properties of LC medium manifest itself as a result of the anisotropic shape of (partially) +ordered constituent molecules. A large variety of phases and structures can be observed in LCs, +which are susceptible to external field and boundary conditions. Many LC phases reveal a large +electro-optical response, which became a background of mass-production technological +applications (monitors, sensors, etc.). First, the ferroelectricity in LCs was associated with +chirality of the constituent molecules and only a tilted smectic phase formed by chiral rod-like +molecules [1] was considered to feature ferroelectricity (FE) and/or antiferroelectricity (AF). +With the discovery of bent-core materials [2], it was found that non-chiral mesogens may also +form FE and AF phases as the close packing and hindered rotation can lead to the structural +chirality. Nevertheless, due to a higher viscosity, smectic phases never reached such broad +application range as nematics. +Recent discoveries stimulated renewed intensive progress in the field of nematic liquid +crystals. For a conventional nematic phase, the director orientations n and –n are +indistinguishable due to the thermal fluctuations, so they form only non-polar phases. However, + + +2 +as far back as in 1918, Max Born [3] predicted a possibility of a ferroelectric fluid, in which all +the dipoles point in the same direction. In such a nematic ferroelectric state (NF), the dipole +moments μ should be strong enough such that the dipole-dipole interactions overwhelm the +thermal fluctuations. In 2017, a real breakthrough was announced in the development of LCs, +as the first two ferroelectric nematics (denoted RM734 and DIO) were reported simultaneously +by two research teams [4-6]. Both materials reveal extremely high longitudinal dipole moments +(about 10 D), anomalously huge dielectric anisotropy Δε, and a spontaneous polarisation of +about 4 μC/cm2, which is an order of magnitude higher than the previously reported values in +other ferroelectric LC phases. Recently, these materials have been intensively studied [7-17]. +Mandle at al. [9] synthesised a homologue series relevant to the molecular structure of RM734 +and analysed the mesomorphic properties and tendencies leading to the NF phase. The +compounds have been intensively studied by Ljubljana researchers [10-12] and by the Boulder +group [13,14]. The existence of ferroelectric domains with a different macroscopic orientation +of the dipoles in the absence of electric field was reported [10-14]. Details of polar nature of +self-assembly, evolution of topological objects and analysis of their character [12,17,18] are +under intensive research progress. Currently, the research is focused on the preparation and +characterisation of new compounds. Machine learning procedures were applied to predict ideal +conditions for the NF phase presence, including a dipole moment value, aspect ratio, length of +the molecule as well as the dipolar angle [19]. In spite of the fact that these conditions are rather +restrictive, development in the designing of prosperous molecular structures was promoted. +At the moment, microscopic organisation of the polar molecules and the mechanism of +the phase transition to the ferroelectric nematic phase undergo intensive research and +stimulating debates. A theoretical description of the ferroelectric nematic phase has been +proposed [20,21], and chiral analogues of highly polar molecules were developed recently [22]. +Additionally, a possibility of oligomer synthesis was shown [23] and new phases and effects +introduced. In any case, the ferroelectric properties combined with the giant permittivity in a +fluid media represent an attractive rapidly developing subject. Since the discovery of NF phase, +the ongoing research is mostly concentrated on the design of new molecular structures. Up to +now, the library of NF materials is strictly limited to a couple of general structures possessing a +suitable aspect ratio and a large enough dipole moment, which develops due to the effective +electron donating and withdrawing groups within the molecules. +In this contribution, we demonstrate newly designed structural motif (see Fig.1). In +contrast to the previously reported molecular designs [3-5,8-19], which utilise an oxygen-based +electron donating group, we synthesised a series possessing a more efficient nitrogen electron +donating group in the terminal part of the aromatic system. Such a design yields higher dipole +moment along the long molecular axis compared to other published materials. To modify the +lateral interactions, which are strong in highly polar systems, we introduced a lateral alkyl chain +with varied number of carbon atoms from 1 to 6. Based on these considerations, we synthesised +a series of compounds (Fig. 1) which exhibit the NF phase directly below the isotropic liquid +on cooling. By tuning the lateral substitution, we shifted the temperature interval of NF down +to the room temperature (RT), at which it may eventually relax to a stable glassy state preserving +the ferroelectric behaviour. + + + +3 + +Fig. 1. Chemical formula of compounds NFn with n = 1 - 6. + +2. +Materials and methods +Chemical formula of the studied compounds is presented in Fig. 1. Synthesis of materials +started from commercial 4-aminosalicylic acid (1, see Scheme 1). Its amino group was +protected by acetylation and the carboxylic group was protected by alkylative esterification by +methyl iodide, so as neither of the two groups interfere with the alkylation of phenolic hydroxyl. +Protected derivative 2 was then alkylated by 1-bromoalkanes to get a series of alkyl homologues +3-n. In the next steps, the acetyl group was cleaved by acidic hydrolysis under mild conditions +and the liberated amino group was alkylated by dimethyl sulphate yielding the key intermediate, +acid 4-n. The lowest alkyl homologue (4-1) was synthesised directly from acid 1 by alkylation +with the excess of dimethyl sulphate. The second part of the molecular core was synthesised +from 4-hydroxybenzoic acid (5), which was protected by the reaction with 3,4-dihydro-2H- +pyrane and reacted with 4-nitrophenol in a DCC-mediated esterification. The protected +hydroxyl group was then liberated by the treatment with p-toluenesulfonic acid. The final step +of the synthesis was esterification of acids 4-n with phenol 6 mediated by EDC. +Differential scanning calorimetry (DSC) measurements were performed to acquire +thermal properties. For electro-optical studies, a polarising optical microscope was used, +equipped with a heating/cooling stage. Details about the compound characterisation and +experimental apparatus are in Supplemental file. + +Scheme 1. +Synthesis of the studied polar nematogens denoted NFn with n varying from 1 +to 6. + + +4 +3. +Results +We studied all newly synthesised homologues by DSC and observed textures and their +changes in polarising microscope to assess the phase behaviour. We performed DSC +measurements in a broad temperature range. We established the melting point (m.p.) from the +first heating run, during which we observed a direct transformation from the crystalline to the +isotropic (Iso.) phase. After the first heating of the fresh sample, we followed with a cooling +run from the Iso phase down to -25°C. On the cooling run, the compounds transformed to the +liquid crystalline state at a significantly lower temperature, Tiso. Under the polarising +microscope, we observed characteristic textures in the LC state, which were previously ascribed +to the ferroelectric nematic phase, NF [12-18]. In the following description, the properties of NF +phase are systematically uncovered. +The analysed DSC data are summarised in Table 1. Compounds NF1, NF2, and NF3 +did not crystallise during the cooling run, however, they crystallised during the subsequent +heating. These homologues revealed the ferroelectric nematic phase only during the cooling of +the sample; temperature stabilisation or heating of the sample caused the crystallisation. The +homologue NF4 revealed the shortest temperature range of the NF phase and crystallised at +about 74°C. On the other hand, the longest homologues NF5 and NF6 did not crystallise during +the DSC measurements at all. For these homologues, the NF phase persisted during the second +and third cooling-heating DSC cycles. The stability of the NF phase for these two homologues +was confirmed during electro-optical measurements: the NF phase was stable for several hours +at RT. The DSC thermograph for the homologue NF6 is demonstrated in Fig. 2. For the first +heating of the sample, the melting point (m.p.) was established; for the second heating run, the +NF phase melted at a temperature corresponding to Tiso. A glassy transition was clearly +distinguishable and its temperature, Tg, was determined from the onset calculated at a half heat +capacity, cp, see Table 1. Glassy properties and ability to form fibres from the melted compound +NF5 is demonstrated in Supplemental file (Fig. S2). + +Fig. 2. DSC thermograph detected for NF6 during the first and second heating and cooling +runs. + +8 +- the first heating +- the second heating +6 +- the cooling +Heat flow ( mW) + glassy state +N. +Iso. +-20 +0 +20 +40 +60 +80 +100 +120 +T(C) +5 +Table 1. +Calorimetric data taken from DSC measurements: melting point, m.p., detected +at the first heating run, the NF-Iso phase transition temperature, Tiso, and the glassy transition +temperature, Tg. All temperatures are presented in °C, and enthalpy changes, H, in J/g, are in +square brackets at the corresponding temperatures. + +m.p. +H (J/g) +Tiso (C) +H (J/g) +Tg +H (J/g) +NF1 +188 [+98.6] +170 [-2.73] +24 [+0.47] +NF2 +150 [+71.3 +136 [-7.59] +30 [+0.42] +NF3 +156 [+73.8] +116 [-7.13] +15 [+0.27] +NF4 +144 [+80.2] +96 [-6.11] +- +NF5 +120 [+55.1] +82 [-4.86] +-9 [+0.44] +NF6 +104 [+51.4] +65 [-3.33] +4 [+0.28] + +In the polarising microscope, we observed various textural features in different +commercial or home-made cells. There are two basic geometries for rod-shaped liquid +crystalline molecules: in HG cells, the molecules are oriented along the cell surface, and in the +HT cell, a homeotropic anchoring ensured molecular orientation perpendicular to this direction. +In the HG cell, the alignment is provided by rubbed polyimide layers with a small pretilt to +arrange defect-free textures. The pretilt results in nonzero polar surface energy as was pointed +out by Chen et al. [14]. Two kinds of HG cells were available, with parallel (HG-P) or +antiparallel (HG-A) rubbing directions on opposite glass surfaces. +Let us start with HG-A cells and compare the results for various cell thicknesses. In this +geometry, we observed two kinds of domains. The texture in 5m HG-A cell for the studied +homologue NF6 is shown in Fig. 3. The dominating type of domains are twisted domains, which +were described for the NF phase in literature [12]. The twisted domains are recognisable when +slightly uncrossing the analyser from the crossed position. Another type of domains can be +observed in less extensive areas of the HG-A samples. In the upper right part of Fig. 3, we found +“red-colour” domains with characteristic borderline approximately parallel to the rubbing +direction. The red colour was typical for these domains in 5 m HG-A cell, see Fig. S2-S5 in +Supplemental for other homologues. Extinction position in these domains are not easy to be +established and the colour of these domains changes when rotating the sample with respect to +the polariser position. We did not find twisted domains in HG-P cells with parallel alignment. +In this geometry, we observed homogeneously aligned area as well as “red” domains, as is +demonstrated for NF6 in Fig. S6 in Supplemental file. +We concentrate on twisted domains, which are very frequent in the HG-A geometry. +We found that for very thin HG-A sample, the twisted domain can be extended for a large area +by quick cooling from the isotropic phase (with a rate 20 K/min). Rather big twisted domains +separated by a zig-zag borderline are demonstrated in Fig. 4(a) for NF6 in 1.6 m HG-A cell. +One can see that the borderline between the twisted domains is oriented approximately +perpendicularly to the rubbing direction. When we turn the analyser from the crossed position +by an angle of ~ 20 degrees, we clearly observe two kinds of domains (see insets in Fig. 4); the +sense of twist is opposite for two neighbouring domains and they are separated by 2 + + +6 +disclination line. In the paper by Sebastian [12], similar domains were observed for another +type of ferroelectric nematogen and designated “sierra-domains”. In our particular case, these +twisted domains reveal sharper contour and can be renamed as “shark-domains”. For the +homologue NF5, the twisted domains are demonstrated in Fig. S4 in Supplemental. Schematic +picture of molecular twist between surfaces with antiparallel alignment is shown in Fig. 4(b). + +Fig. 3. +Microphotograph of NF6 homologue in 5 m HG-A cell. The red arrow (R) +marks the rubbing direction, the white arrows show the polariser (P) / analyser (A) directions. + + +R +50um +7 +Fig. 4. +Textures of NF6 in 1.6 m HG-A cell under a polarizing microscope (a) between +crossed polarisers, the red arrow marks the rubbing direction, R, the orientation of the analyser +(A) and the polariser (P) is schematically shown by white arrows. In the figure (b) there is a +schematic arrangement of molecules in neighbouring twisted domains between glass surfaces +with antiparallel rubbing. The part of the figure (a) marked by white lines is shown in (c) and +(d) when A is rotated by an angle of about 20 degrees counterclockwise or clockwise from the +crossed position. + + +An application of an electric field in HG geometry led to a rather complex effect. The +colour of twisted domains slightly changed under the applied electric field and additional stripes +appeared across the twisted domain structure approximately parallel to the rubbing direction. +As the application of the field in the HG cell supplied only limited information and a detail +analysis is rather problematic, we studied HT cells under applied bias. In this geometry, the +applied electric field is approximately parallel to the molecular dipole moment and we can +observe a rearrangement of molecules. In Fig. 5, we demonstrate the HT texture for homologue +NF6 with and without applied electric field of about 5 V/m. In the upper part of Fig. 5, an area +without electrode is observed. When the field is switched on (Fig. 5(b)), the molecules reorient +along the field and the texture under the electrode area becomes black. After switching the +electric field off, the HT texture turns back to a lighter type, similar to the virgin texture (Fig. +5(a)), within several seconds. + +(a) +(b) +R +100 μm +c) +(d) +8 +We investigated the switching properties of the studied compounds in HT geometry. +Due to electrostatic interactions, the results are influenced both by the cell geometry and by the +character of the aligning layer. For n=1-4, the homologues NFn reveal strong vitrification and +an increase in viscosity when approaching the glass transition temperature Tg. This temperature +is relatively high and the samples feature a higher conductivity, which limited our studies for +these homologues. On the contrary, homologues NF5 and NF6 could be subjected to the applied +field for a longer time (several hours); the polarisation was measured repeatedly and the results +were reproducible. At the room temperature, these two homologues stay in LC phase for a long +time and they start to crystallise only after several hours. +For homologue NF5, the temperature dependence of the polarisation is presented in Fig. +6(a). The polarisation values are calculated by the time-integration of a switching current +profile. In Fig. 6(b), the switching current is plotted versus the applied electric field at a +frequency 10 Hz and at temperature 52 °C. For both homologues NF5 and NF6, we detected a +continuous increase in polarisation values on cooling process in the NF phase. A coexistence of +the Iso and NF phases was checked under the polarising microscope and it was observed only +in a narrow temperature interval of about 2 °C. The decrease in polarisation values shown in +Fig. 6(a) is connected with an increase in switching time. Such a slowing-down of molecular +dynamics is connected with an increase in the sample viscosity. + +Fig. 5. +Texture of NF6 in 5 m thick HT cell, (a) without electric field and (b) under +applied electric field of about 5 V/m perpendicular to the cell. The orientation of polarisers +and of the applied electric field are marked by black symbols for illustration. Upper part of the +figure shows an area without electrodes. + +To analyse the effect of the applied field in different cell geometries, we prepared a +home-made gap-cell with in-plane electrodes. Two glass slides were separated by copper 35 +m thick ribbons, with a gap distance of about 1 mm. In this cell, the domains disappeared + +(a) +(b) +No electrodes +Electrode-edge +E +X +P +100μm +9 +under the applied electric field as all the molecules were aligned along the applied electric field. +After the switching-off of the external electric field, the domain structure was partially +reconstructed in several seconds. Microphotographs can be found in Supplemental file (Fig. +S8). Unfortunately, the thickness of 35 m was rather large to reach homogeneous alignment +through the whole cell thickness. Additionally, we are aware that the applied electric field was +not homogeneously distributed. Technological tasks of the cell preparation and detailed +analysis of the defects in electric field are still under work. + +Fig. 6. +(a) The temperature dependence of the polarisation of NF5, which was +calculated from the polarisation current. (b) The current profile at a temperature T=52 °C is +demonstrated with a triangular profile of the applied electric field at a frequency 10 Hz. + + +We measured the dielectric spectra of all compounds in a temperature range from the +isotropic liquid to RT in order to study the molecular dynamics. The applied measuring field +was smaller than 0.01 V/m (higher probing fields could influence the dielectric measurements + +(a) +4 +(μC/cm² +3 +2 +P +0 +40 +45 +50 +55 +60 +65 +70 +T(°C) +(b) +10 +4 +current (arb. units) +5 +2 +E +(V/μm) +0 +0 +-5 +-2 +-10 +-4 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +10 +as the studied compounds are really sensitive to external fields). In the ferroelectric NF phase, +we found one distinct quite strong relaxation mode appearing at the Iso-NF phase transition on +cooling and remaining visible down to RT. On the other hand, when the sample is in the +crystalline state, this mode is not present and the permittivity is low (10). We demonstrate +three-dimensional plots of the real, ’, and imaginary, ’’, parts of permittivity versus frequency +and temperature, T, for compound NF5 in Fig. 7. For homologues NF2, NF3 and NF6, the 3D- +plots of permittivity are shown in Supplemental file, Figs. S9-S11. All the presented dielectric +data were obtained in 12 m thick cells with gold electrodes and no surfactant layers. +We encountered a disturbing effect of surfactant, similarly as it was mentioned in +previous works dealing with dielectric spectroscopy of the NF phase [16]. For such a type of +polar phase, it was reported that the polymer layers effectively influence the permittivity +measurements. Due to a non-conductive character of polymer layers on the cell surfaces, there +is a barrier which causes a spatial variation of the charge and influences the measured effective +permittivity values. We fitted the dielectric data to the Cole-Cole formula (see Supplemental +file for the details) to obtain information about the dielectric strength, , and the relaxation +frequency, fr. We detected large only slightly temperature dependent values of  up to 15103. +In contrast, the relaxation frequency decreases within the whole temperature range of the NF +phase on cooling and follows the Arrhenius law. Such behaviour is documented in Fig. 8 for +homologue NF6, which followed Arrhenius behaviour ideally and the activation energy, Ea, +was calculated to be 102 kJ/mol. For other compounds, the linearity of fr in logarithmic scale +(versus 1/T in absolute temperature scale) was confirmed only far from the Iso-NF phase +transition (see Fig. S12 in Supplemental file). Non-homogeneity of molecular alignment and/or +influence of electrodes should be taken into consideration to explain the deviation from +Arrhenius law. + + +11 + +Fig. 7. +3D-plot of (a) real, ’, and (b) imaginary, ’’, parts of the permittivity versus the +frequency and the temperature, T, for compound NF5. Dielectric measurements were performed +in 12 m cell with gold electrodes and no surfactant layer. + +16 +14 +12 +10 +8 +3 +6 +4 +30 +2 +40 +50 +0 +60 +101 +102 +70 +103 +frequency (Hz) +104 +80 +105 +106 +6 +(103) +2 +30 +40 +50 +101 +102 +60 +103 +70 +frequency (Hz) +104 +105 +80 +106 +12 + +Fig. 8. +Temperature dependences of (a) the dielectric strength, , and relaxation +frequency, fr, for NF6 in 12 m cell without surfactant layer. In the inset fr is presented in the +logarithmic scale versus reciprocal temperature, 1/T, in Kelvins and the activation energy, EA, +was established from the slope. + +Strong polar character of the NF phase was proved by SHG measurements. The SHG +experiments were carried out in transmission configuration according the scheme described in +Supplemental file. We utilised HG cells and SHG measurement results are presented for +compounds NF5 and NF6 in Fig. 9. On cooling the sample from the isotropic phase, the SHG +signal abruptly grows from zero value at the transition temperature to NF phase. With ongoing +temperature decrease, the SHG intensity slows down its increase, reaches the maximum and +slowly starts to decrease. All of this happens within the NF phase, where we would expect a +gradual increase in the SHG signal upon cooling. Moreover, even for the weakest applied +intensity of the fundamental laser beam, a small drop in SHG intensity was detected in +subsequent measuring runs at the same temperature. From this it follows that the decrease in +the SHG signal upon cooling may be explained by partial decomposition of our samples caused +by rather strong intensity of the pulse laser beam. +X-ray scattering experiments confirmed nematic character of the observed mesophase. +Nematic phase is characterised by the long-range orientational order and only broad diffuse +peaks of low intensity can be detected. For homologue NF5, the signal at small scattering angles +is rather wide and can be fitted with two signals, with maxima corresponding to 18.8 Å and +10.5 Å at T=75°C, 22.5 Å and 10.4 Å at T=30°C. As the length of molecules, l, can be +approximately established as l~20.9 Å, the peak at the small scattering angle matches perfectly +to the long dimension of the molecules. The peak at a wide-angle region has also a very broad + +口 +60 +12 +■ +4 +E,=102 kJ/mol +口 +(zH) +口 +40 +(10° +8 +口 +(zH) +2 +3V +口 +1.3 +1.4 +1.5 +1.6 +20 +4 +10" RT (Jmoll +0 +0 +40 +45 +50 +55 +60 +T(C) +13 +profile with the maximum corresponding to 4.4 Å for all measuring temperatures, and it +corresponds to an average distance between the molecules. + +Fig. 9. +SHG signals for NF5 and NF6 in HG cells + +4. +Conclusions +We proposed a new structural modification of highly polar molecules self-assembling +and forming the ferroelectric nematic phase NF. All the prepared compounds exhibit a direct +phase transition to the ferroelectric nematic phase on cooling from the isotropic phase. In the +presented homologue series, a prolongation of a side-chain resulted in the NF phase persistence +down to the room temperatures and stability for at least several hours. Ferroelectric character +of the nematic phase was proven by several experimental techniques. Characteristic textural +features for the ferroelectric nematics were observed in several sample geometries. The +ferroelectric switching process was detected and the polarisation was calculated from the +measured polarisation current. The values of polarisation were found to increase continuously +on cooling from the isotropic phase, reaching up to 4 C/cm2. For all the studied homologues, +the dielectric studies show a strong polar mode characteristic for the NF phase, disappearing in +the isotropic or crystalline phases. The dielectric strength of this mode exceeds values of about +15103, which is the maximum reached for the NF phase up to now. Nevertheless, the +characterisation of the defects in the NF phase and the role of the electrodes are not yet +completely solved and will need a deeper insight. The dipole moment of the molecules was +calculated and established to be about 14 D, which is larger than the value reported for DIO or +RM734. +The discovery of ferroelectricity for nematics opened new opportunities in the liquid +crystal research and generally in the field of condensed matter. The NF phase represents a highly +polar structure responsive to very small applied fields and it features a variety of new effects + +0.6 +NF5 +0.4 +NF6 +0.2 +0.0 +30 +40 +50 +60 +70 +80 +90 +T (C) +14 +induced by the confining surfaces. Generally, the application potential of the NF phase is +immense and not yet completely explored. Our particular room-temperature-stable soft phase +exhibits huge dielectric constant and can be important in future for the development of memory +devices, capacitors and actuators. + +Disclosure statement +No potential conflict of interest was reported by the authors. + +Acknowledgments +Authors acknowledge project MAGNELIQ, that received funding from the European Union’s +Horizon 2020 research and innovation programme under grant agreement No 899285; and +project 22-16499S from the Czech Science Foundation. V.N. is grateful to Damian Pociecha +and Ewa Gorecka from Warsaw University for their help with x-ray measurements. + +References +[1] +Handbook of Liquid Crystals, 2nd Edition, ed. J. W. Goodby, P. J. Collings, T. Kato, +C. Tschierske, H. Gleeson and P. Raynes, Wiley-VCH, 2014. +[2] +H. Takezoe and Y. Takanishi, Bent-core liquid crystals: Their mysterious and attractive +world. Japanese Journal of Applied Physics, 45 (2A), pp. 597-625, 2006. +[3] +M. 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How far can we push the rigid oligomers/polymers +toward ferroelectric nematic liquid crystals? J. Am. Chem. Soc. 143,17857−17861, +2021. + + + +1 + +Supplemental information + +Dimethylamino terminated ferroelectric nematogens revealing high +permittivity + +Martin Cigl, Natalia Podoliak, Tomáš Landovský, Dalibor Repček, Petr Kužel, +and Vladimíra Novotná* + +Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, Prague, Czech +Republic + +Contents +1. +Synthesis and compound characterisation +1.1. +General synthesis +1.2. +Synthetic procedures +1.3. +Equipment and apparatus +2. +Mesomorphic properties +2.2. +Textures +2.3. +Dielectric spectroscopy and electro-optical properties + + +1. +Syntheses and compound characterisation +1.1. +General synthesis +All starting materials and reagents were purchased from Sigma-Aldrich, Acros Organics or +Lach:Ner. All solvents used for the synthesis were “p.a.” grade. Tetrahydrofuran was further +distilled from calcium hydride to obtain sufficiently dry solvent. 1H NMR spectra were +recorded on Varian VNMRS300 instrument; deuteriochloroform (CDCl3) and +hexadeuteriodimethyl sulfoxide (DMSO-d6) were used as solvents and the signals of the +solvent served as an internal standard. Chemical shifts () are given in ppm and J values are +given in Hz. Elemental analyses were carried out on Elementar vario EL III instrument. The +purity of all final compounds was checked by HPLC analysis (high-pressure pump ECOM +Alpha; column WATREX Biospher Si 100, 250 × 4 mm, 5 m; detector WATREX UVD +250) and were found to be >99.8 %. Column chromatography was carried out using Merck +Kieselgel 60 (60−100 μm). + +2 + +Synthesis of materials started from commercial 4-aminosalicylic acid (1, see Scheme +1). Its amino group was protected by acetylation and the carboxylic group was protected by +alkylative esterification by methyl iodide, so as neither of the two groups interfere with the +alkylation of phenolic hydroxyl. Protected derivative 2 was then alkylated by 1- +bromoalkanes to get a series of alkyl homologues 3-n. In the next steps, the acetyl group +was cleaved by acidic hydrolysis under mild conditions and the liberated amino group was +alkylated by dimethyl sulphate yielding the key intermediate, acid 4-n. The lowest alkyl +homologue (4-1) was synthesised directly from acid 1 by alkylation with the excess of +dimethyl sulphate. The second part of the molecular core was synthesised from 4- +hydroxybenzoic acid (5), which was protected by the reaction with 3,4-dihydro-2H-pyrane +and reacted with 4-nitrophenol in a DCC-mediated esterification. The protected hydroxyl +group was then liberated by the treatment with p-toluenesulfonic acid. The final step of the +synthesis was esterification of acids 4-n with phenol 6 mediated by EDC. + + +Scheme 1. +Synthetic procedures for the preparation of target compounds NFn. + +1.2. +Synthetic procedures +Methyl 4-acetamido-2-hydroxybenzoate (2) +Acetic hydride (30 mL, 0.31 mol) was added dropwise to the suspension of powdered 4- +aminosalicylic acid (20.0 g, 0.13 mol) in acetonitrile (250 mL). The reaction mixture was +stirred for 2 h and the resulting suspension filtered. The filter cake was washed by the small +amount of acetonitrile to remove the residue of acetic acid and dried in a vacuum dryer at +40 °C. Yield 24.77 g (96 %). + +3 + +Dry 4-acetamidosalicylic acid (24.0 g, 0.12 mol) was dissolved in DMF. Powdered KHCO3 +was added with stirring, resulting in CO2 evolution. Then methyl iodide was added dropwise +and the reaction mixture stirred for 6 h under anhydrous conditions (CaCl2 tube). The +resulting suspension was poured into water and neutralised with concentrated HCl. White +precipitate was filtered off and crystallised from 50% aqueous methanol. Yield 24.43 g +(95 %). 1H NMR (DMSO-d6) : 10.23 (1 H, s), 7.71 (1 H, d, J=8.8 Hz), 7.37 (1 H, d, J=1.8 +Hz), 7.05 (1 H, dd, J=8.8, 2.3 Hz), 3.85 (3 H, s), 2.07 (3 H, s). + +General procedure for alkylation of benzoate 2 +Benzoate 2 was dissolved in dry DMF and powdered K2CO3 and KI (omitted if iodoalkane +was used) were added with stirring. Mixture was heated to 50 °C and 1-bromoalkane was +added. Reaction was stirred at 50 °C under anhydrous conditions (CaCl2 tube) for 10 h. The +cooled resulting mixture was poured into cold water, neutralised with concentrated HCl and +the precipitated product was filtered off and crystallised from ethanol. +Methyl 4-acetamido-2-ethoxybenzoate (3-1) +The reaction of benzoate 2 (5.0 g, 23.90 mmol) with ethyl iodide (5.45 g, 39.32 mmol) in +the presence of K2CO3 (5.0 g, 36.18 mmol) in dry DMF (50 mL) yielded 4.83 g (85 %) of +3a. 1H NMR (CDCl3) : 11.25 (1 H, br. s.), 8.36 (1 H, d, J=2.3 Hz), 7.94 (1 H, d, J=9.4 Hz), +6.58 (1 H, dd, J=9.1, 2.6 Hz), 4.11 (2 H, q, J=7.0 Hz), 3.89 (3 H, s), 2.23 (3 H, s), 1.42 (3 +H, t, J=7.0 Hz). +Methyl 4-acetamido-2-propoxybenzoate (3-2) +The reaction of benzoate 2 (5.0 g, 23.90 mmol) with 1-bromopropane (8.71 g, 70.82 mmol) +in the presence of K2CO3 (10.0 g, 72.36 mmol) in dry DMF (60 mL) yielded 3.53 g (58 %) +of 3b. 1H NMR (CDCl3) :10.95 (1 H, s), 7.79 (1 H, d, J=8.8 Hz), 7.60 (1 H, d, J=2.3 Hz), +6.81 (1 H, dd, J=8.8, 2.3 Hz), 4.09 (2 H, t, J=6.5 Hz), 1.81 - 2.05 (2 H, m), 1.10 (3 H, t, +J=7.3 Hz). +Methyl 4-acetamido-2-butoxybenzoate (3-c) +The reaction of benzoate 2 (5.0 g, 23.90 mmol) with 1-bromobutane (4.53 g, 32.40 mmol) +in the presence of K2CO3 (5.0 g, 36.18 mmol) in dry DMF (50 mL) yielded 5.42 g (86 %) +of 3c. 1H NMR (CHLOROFORM-d)  ppm 10.85 (1 H, d, J=8.8 Hz), 7.77 (1 H, s), 7.60 (1 +H, d, J=2.3 Hz 6.79 (1 H, dd, J=8.8, 2.3 Hz), 4.04 (2 H, t, J=6.5 Hz), 3.86 (2 H, s), 2.19 (3 +H, s), 1.72 - 1.93 (2 H, m), 1.42 - 1.61 (2 H, m), 0.97 (3 H, t, J=7.3 Hz). +Methyl 4-acetamido-2-(pentyloxy)benzoate (3-d) +The reaction of benzoate 2 (10.0 g, 47.80 mmol) with 1-iodopentane (19.88 g, 98.37 mmol) +in the presence of K2CO3 (15.0 g, 0.11 mol) in dry DMF (100 mL) yielded 9.81 g (75 %) of +3d. 1H NMR (CDCl3) :10.86 (1 H, s), 7.80 (1 H, d, J=8.8 Hz), 7.58 (1 H, d, J=2.3 Hz), 6.79 +(1 H, dd, J=8.8, 2.3 Hz), 4.01 (2 H, t, J=6.7 Hz), 1.77 - 2.03 (2 H, m), 1.31 - 1.57 (4 H, m), +0.91 (3 H, t, J=7.3 Hz). +Methyl 4-acetamido-2-(hexyloxy)benzoate (3-e) +The reaction of benzoate 2 (10.0 g, 47.80 mmol) with 1-bromohexane (15.78 g, +95.60 mmol) in the presence of K2CO3 (15.0 g, 0.11 mol) in dry DMF (100 mL) yielded + +4 + +9.73 g (69 %) of 3e. 1H NMR (CDCl3) : 10.83 (1 H, s), 7.79 (1 H, d, J=8.8 Hz), 7.59 (1 H, +d, J=2.3 Hz), 6.80 (1 H, dd, J=8.8, 2.3 Hz), 4.02 (2 H, t, J=6.7 Hz), 3.86 (3 H, s), 2.19 (3 H, +s), 1.71 - 1.91 (2 H, m), 1.42 - 1.56 (2 H, m), 1.20 - 1.41 (4 H, m), 0.91 (3 H, t, J=7.3 Hz). + +General procedure for deacetylation of amino group +Methyl 4-acetamido-2-(alkoxy)benzoate 3 was dissolved in methanol at 50 °C and the +concentrated H2SO4 was carefully added dropwise. The reaction mixture was stirred at 50 +°C for 30 min and then poured into cold water and neutralised with NaOH. The neutral +dispersion of the product in water was extracted with ethyl acetate, combined organic layers +were washed with water and brine. After drying with anhydrous MgSO4, the solvent was +removed on rotary evaporator and the residue was purified by column chromatography on +silica gel to yield methyl 2-(alkoxy)-4-aminobenzoate as an intermediate. +NOTE: The presence of water in the deacetylation reaction (e.g. use of diluted H2SO4) leads +to considerable amounts of decarboxylation byproduct. + +General procedure for methylation of amino group +Methyl 2-(alkoxy)-4-aminobenzoate was dissolved in DMSO and powdered K2CO3 was +added with stirring. A mixture was heated to 50 °C and dimethyl sulphate was added +dropwise. The reaction mixture was stirred at the same temperature and under anhydrous +conditions (CaCl2 tube) overnight. The progress of the reaction was monitored using TLC +(CH2Cl2-acetone 95 : 5). The resulting mixture was filtered and solid Na2S was added to the +filtrate. The mixture was stirred for 4 h at room temperature and then poured into water. +After 30 min of standing, the solution was neutralised with the concentrated acetic acid and +the precipitated product was collected by filtration. A crude product was purified by column +chromatography on silica gel and crystallised from methanol. +4-(Dimethylamino)-2-methoxybenzoic acid (4-1) +Compound 4a was synthesised by direct methylation of acid 1 using the general procedure +for methylation of amino group described above. Reaction of benzoic acid 1 (10.0 g, +65.30 mmol) with dimethyl sulphate (42.45 g, 0.33 mol) in the presence of K2CO3 (50.0 g, +0.36 mol) in DMSO (150 mL) and subsequent treatment with Na2S (16.0 g, 0.21 mol) +yielded 10.01 g (79 %) of 4-1. +4-(Dimethylamino)-2-ethoxybenzoic acid (4-2) +Following the general procedure, starting from methyl 4-acetamido-2-ethoxybenzoate 3-1 +(4.80 g, 20.23 mmol), which was deacylated using H2SO4 (5.0 mL, 96%) in methanol +(50 mL). The free amine was methylated using dimethyl sulphate (10.52 g, 80.90 mmol) and +K2CO3 (12.0 g, 86.82 mmol) in DMSO (50 mL) followed by the treatment with Na2S +(1.60 g, 20.50 mmol) yielded 2.29 g (54 %) of 4-2. 1H NMR (CDCl3) : 10.71 (1 H, br. s.), +7.99 (1 H, d, J=9.4 Hz), 6.37 (1 H, dd, J=8.8, 2.3 Hz), 6.10 (1 H, d, J=2.3 Hz), 4.29 (2 H, q, +J=7.0 Hz), 3.05 (6 H, s), 1.55 (3 H, t, J=7.0 Hz). +4-(Dimethylamino)-2-propoxybenzoic acid (4-3) +Using the described general procedure: 4-acetamido-2-propoxybenzoate 3-2 (3.50 g, +13.93 mmol) was deacylated using H2SO4 (3.5 mL, 96%) in methanol (40 mL). Liberated + +5 + +amine was methylated using dimethyl sulphate (7.24 g, 55.68 mmol) and K2CO3 (7.70 g, +55.71 mmol) in DMSO (50 mL) followed by the treatment with Na2S (1.10 g, 14.09 mmol) +yielded 1.91 g (61 %) of 4-3. 1H NMR (CDCl3) : 10.73 (1 H, br. s.), 8.01 (1 H, d, J=9.4 +Hz), 6.42 (1 H, dd, J=9.1, 2.1 Hz), 6.19 (1 H, d, J=1.8 Hz), 4.19 (2 H, t, J=6.5 Hz), 3.07 (6 +H, s), 1.95 (2 H, sext., 7.2 Hz), 1.10 (3 H, t, J=7.3 Hz) +2-Butoxy-4-(dimethylamino)benzoic acid (4-4) +Using the general deacetylation and alkylation protocol: 4-acetamido-2-butoxybenzoate +3-3 (5.30 g, 19.98 mmol) was deacylated using H2SO4 (5.0 mL, 96%) in methanol (50 mL). +Liberated amine was methylated using dimethyl sulphate (10.40 g, 79.98 mmol) and K2CO3 +(11.50 g, 83.21 mmol) in DMSO (50 mL) followed by the treatment with Na2S (1.60 g, +20.50 mmol) yielded 2.42 g (51 %) of 4-4. 1H NMR (CDCl3) : 10.71 (1 H, br. s.), 8.00 (1 +H, d, J=8.8 Hz), 6.40 (1 H, dd, J=8.8, 2.3 Hz), 6.17 (1 H, d, J=2.3 Hz), 4.21 (2 H, t, J=6.7 +Hz), 3.06 (6 H, s), 1.85 - 1.94 (2 H, m), 1.43 - 1.58 (2 H, m), 0.92 (3 H, , t, J=7.3 Hz). +4-(Dimethylamino)-2-(pentyloxy)benzoic acid (4-5) +Using the above mentioned general protocols: 4-acetamido-2-(pentyloxy)benzoate 3-4 +(9.50 g, 34.76 mmol) was deacylated using H2SO4 (5.0 mL, 96%) in methanol (150 mL). +Liberated amine was methylated using dimethyl sulphate (27.12 g, 0.21 mol) and K2CO3 +(29.0 g, 0.21 mol) in DMSO (150 mL) followed by the treatment with Na2S (10.90 g, +0.14 mol) yielded 5.41 g (62 %) of 4-3. 1H NMR (CDCl3) : 10.70 (1 H, br. s.), 8.00 (1 H, +d, J=8.8 Hz), 6.41 (1 H, dd, J=8.8, 2.3 Hz), 6.18 (1 H, d, J=2.3 Hz), 4.21 (2 H, t, J=6.7 Hz), +3.06 (6 H, s), 1.82 - 1.99 (2 H, m), 1.31 - 1.55 (4 H, m), 0.94 (3 H, , t, J=7.3 Hz). +4-(Dimethylamino)-2-(hexyloxy)benzoic acid (4-6) +Using the above mentioned general protocols: 4-acetamido-2-(hexyloxy)benzoate 3-3 +(9.50 g, 32.38 mmol) was deacylated using H2SO4 (5.0 mL, 96%) in methanol (150 mL). +Liberated amine was methylated using dimethyl sulphate (21.10 g, 0.16 mol) and K2CO3 +(23.0 g, 0.17 mol) in DMSO (150 mL) followed by the treatment with Na2S (7.20 g, +0.10 mol) yielded 4.82 g (56 %) of 4-6. 1H NMR (CDCl3) : 10.72 (1 H, br. s.), 7.99 (1 H, +d, J=8.8 Hz), 6.37 (1 H, dd, J=8.8, 2.3 Hz), 6.11 (1 H, d, J=2.3 Hz), 4.20 (2 H, t, J=6.7 Hz), +3.06 (6 H, s), 1.85 - 1.99 (2 H, m), 1.21 - 1.51 (6 H, m), 0.91 (3 H, , t, J=7.3 Hz). +4-Nitrophenyl 4-hydroxybenzoate (6) +3,4-Dihydro-2H-pyrane (14.30 g, 0.17 mol) was added dropwise to the suspension of 4- +hydroxybenzoic acid (13.80 g, 0.10 mol) in diethylether (200 ml). The reaction mixture was +stirred overnight under anhydrous conditions (CaCl2 tube) and then filtered. The filter cake +contained the majority of desired 4-((tetrahydro-2H-pyran-2-yl)oxy)benzoic acid. The +filtrate was vigorously stirred with aqueous NaOH (80 ml, 10%) for 30 min, and then the +aqueous layer was separated and neutralised by HCl. The pH was further adjusted to ca. 4 +using acetic acid. The precipitated solid was collected, washed with cold water and dried +under vacuum and finally combined with the dry portion obtained from the filter cake. + +4-((Tetrahydro-2H-pyran-2-yl)oxy)benzoic acid (31.35 g, 0.14 mol) and 4-nitrophenol +(19.60 g, 0.14 mol) were dissolved in dry THF (250 mL) and cooled to ca. 10 °C. Then N,N´- +dicyclohexylcarbodiimide (DCC, 30.60 g, 0.15 mol ) and 4-(dimethylamino)-pyridine +(DMAP, 5.60 g, 46.22 mmol) were added and the reaction mixture was stirred under + +6 + +anhydrous conditions for 12 h. The precipitated N,N´-dicyclohexylurea was filtered off and +the filtrate diluted with ethyl acetate (100 mL). The resulting solution was washed with +diluted HCl (100 mL, 1 : 15), then with water and the solvents were removed on rotary +evaporator. The solid residue was dissolved in CHCl3-methanol mixture (1 : 1) and +toluenesulfonic acid (4.0 g, 23.22 mmol) was added. The reaction mixture was stirred at +45 °C for 1 h and then evaporated to dryness on rotary evaporator. A crude product was +crystallised from acetone. Yield 28.71 g (66 %). 1H NMR (DMSO-d6)  8.33 (2 H, d, J=8.6 +Hz), 8.08 (2 H, d, J=8.8 Hz), 7.58 (2 H, d, J=8.6 Hz), 7.20 (2 H, d, J=8.8 Hz), 5.62 – 5.65 +(1 H, m), 3.63 - 3.75 (1 H, m), 3.48 - 3.60 (1 H, m), 1.21 - 1.97 (6 H, m). + +General procedure for EDC-mediated esterification +2-(Alkoxy)-4-(dimethylamino)benzoic acid 4-n and 4-nitrophenyl 4-hydroxybenzoate (6) + were suspended in dry dichloromethane (50 ml) and cooled to 2 – 8 °C in ice-water bath. +Then +N-(3-dimethylaminopropyl)-N′-ethylcarbodiimide +hydrochloride +(EDC) +and +4-(N,N-dimethylamino)pyridine (DMAP) (0.1 g, 0.82 mmol) were added. The reaction +mixture was stirred for 2 hours under anhydrous conditions and the temperature was let rise +as ice in the cooling bath melted. The resulting solution diluted with CH2Cl2 and washed +with water and brine. Organic layer was dried over anhydrous magnesium sulphate and +evaporated on the rotary evaporator. The residue was purified by column chromatography +on silica gel in CH2Cl2-acetone eluent and recrystallised from acetone. +4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-methoxybenzoate (NF1) +4-(Dimethylamino)-2-methoxybenzoic acid (4-1, 78.1 mg, 0.40 mmol) was esterified with +4-nitrophenyl 4-hydroxybenzoate (6, 104.5 mg, 0.40 mmol) using EDC (81 mg, 0.42 mmol) +and DMAP (51.0 mg, 0.42 mmol) in dichloromethane (2.0 mL) as described in general +procedure. Yield 92.5 mg (53 %). 1H NMR (CDCl3) : 8.33 (2 H, d, J=8.8 Hz), 8.24 (2 H, +d, J=8.8 Hz), 8.03 (1 H, d, J=9.4 Hz), 7.41 (4 H, dd, J=14.1, 8.8 Hz), 6.33 (1 H, dd, J=8.8, +2.3 Hz), 6.16 (1 H, d, J=2.3 Hz), 3.95 (3 H, s), 3.11 (6 H, s). Anal. calcd. for C23H20N2O7: +C 63.30, H 4.62, N 6.42; found C 63.86, H 4.68, N 6.47 %. +4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-ethoxybenzoate (NF2) +4-(Dimethylamino)-2-methoxybenzoic acid (4-1) +The reaction of 4-(dimethylamino)-2-ethoxybenzoic acid (4-2, 1.0 g, 4.78 mmol) was +esterified with 4-nitrophenyl 4-hydroxybenzoate (6, 1.24 g, 4.78 mmol) using EDC (1.0 g, +5.16 mmol) and DMAP (0.29 g, 2.39 mmol) in dichloromethane (30 mL) yielded 1.03 g +(48 %). 1H NMR (CDCl3) : 8.33 (2 H, d, J=9.4 Hz), 8.24 (2 H, d, J=8.2 Hz), 8.01 (1 H, d, +J=9.4 Hz), 7.33 - 7.48 (4 H, m), 6.32 (1 H, dd, J=8.8, 2.3 Hz), 6.16 (1 H, d, J=1.8 Hz), 4.15 +(2 H, d, J=7.0 Hz), 3.08 (6 H, s), 1.49 (3 H, t, J=7.0 Hz). 13C{H} NMR (CDCl3) : 163.72 +(s), 163.10 (s), 162.29 (s), 156.46 (s), 155.73 (s), 155.24 (s), 145.31 (s), 134.35 (s), 131.79 +(s), 125.24 (s), 125.03 (s), 122.67 (s), 122.62 (s), 104.59 (s), 103.90 (s), 95.65 (s), 64.41 (s), +40.11 (s), 14.77 (s). Anal. calcd. for C24H22N2O7: C 64.00, H 4.92, N 6.11; found C 63.87, +H 4.98, N 6.11 %. +4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-propoxybenzoate (NF3) +Starting from 4-(dimethylamino)-2-propoxybenzoic acid (4-3, 1.25 g, 5.78 mmol) and 4- +nitrophenyl 4-hydroxybenzoate (6, 1.50 g, 5.78 mmol) with EDC (1.18 g, 6.03 mmol) and + +7 + +DMAP (0.68 g, 5.61 mmol) in dichloromethane (50 mL) yielded 1.36 g (51 %). 1H NMR +(CDCl3) : 8.34 (2 H, d, J=8.8 Hz), 8.24 (2 H, d, J=8.8 Hz), 8.00 (1 H, d, J=9.4 Hz), 7.33 - +7.50 (4 H, m), 6.32 (1 H, dd, J=8.8, 2.3 Hz), 6.15 (1 H, d, J=2.3 Hz), 4.04 (2 H, t, J=6.5 Hz), +3.09 (6 H, s), 1.81 - 1.98 (2 H, m), 1.08 (3 H, t, J=7.3 Hz). 13C{H} NMR (CDCl3) : 163.73 +(s), 163.29 (s), 162.34 (s), 156.54 (s), 155.74 (s), 155.27 (s), 145.36 (s), 134.46 (s), 131.84 +(s), 125.26 (s), 125.04 (s), 122.65 (s), 122.60 (s), 104.66 (s), 103.84 (s), 95.45 (s), 70.18 (s), +40.11 (s), 22.64 (s), 10.68 (s). Anal. calcd. for C25H24N2O7: C 64.65, H 5.21, N 6.03; found +C 64.56, H 5.19, N 5.98 %. +4-[(4-Nitrophenoxy)carbonyl]phenyl 2-butoxy-4-(dimethylamino)benzoate (NF4) +Esterification of 2-butoxy-4-(dimethylamino)benzoic acid (4-4, 2.0 g, 8.43 mmol) with 4- +nitrophenyl 4-hydroxybenzoate (6, 2.50 g, 9.64 mmol) using EDC (2.0 g, 10.22 mmol) and +DMAP (0.58 g, 4.78 mmol) in dichloromethane (70 mL) yielded 2.31 g (51 %). 1H NMR +(CDCl3) : 8.34 (2 H, d, J=9.4 Hz), 8.24 (2 H, d, J=8.2 Hz), 8.00 (1 H, d, J=8.8 Hz), 7.31 - +7.50 (4 H, m), 6.32 (1 H, dd, J=8.8, 2.3 Hz), 6.16 (1 H, d, J=1.8 Hz), 4.08 (2 H, t, J=6.5 Hz), +3.09 (6 H, s), 1.74 - 1.96 (2 H, m), 1.42 - 1.66 (2 H, m), 0.95 (3 H, t, J=7.3 Hz). 13C{H} +NMR (CDCl3) : 163.73 (s), 163.33 (s), 162.30 (s), 156.52 (s), 155.74 (s), 155.25 (s), 145.34 +(s), 134.47 (s), 131.82 (s), 125.25 (s), 125.02 (s), 122.66 (s), 122.60 (s), 104.65 (s), 103.83 +(s), 95.43 (s), 68.35 (s), 40.15 (s), 31.30 (s), 19.25 (s), 13.85 (s). Anal. calcd. for C26H26N2O7: +C 65.26, H 5.48, N 5.85; found C 65.15, H 5.16, N 5.80 %. +4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-(pentyloxy)benzoate (NF5) +Following the general procedure above 4-(dimethylamino)-2-propoxybenzoic acid (4-3, +2.0 g, 7.96 mmol) and 4-nitrophenyl 4-hydroxybenzoate (6, 2.06 g, 7.94 mmol) were +reacted in the presence of EDC (1.68 g, 8.59 mmol) and DMAP (0.50 g, 4.13 mmol) in +dichloromethane (70 mL) yielded 1.76 g (45 %). 1H NMR (CDCl3) : 8.33 (2 H, d, J=9.2 +Hz), 8.24 (2 H, d, J=8.6 Hz), 7.99 (1 H, d, J=9.2 Hz), 7.32 - 7.49 (4 H, m), 6.32 (1 H, dd, +J=8.9, 2.3 Hz), 6.15 (1 H, d, J=2.0 Hz), 4.07 (2 H, t, J=6.6 Hz), 3.08 (6 H, s), 1.79 - 1.94 (2 +H, m), 1.26 - 1.55 (4 H, m), 0.83 - 0.94 (3 H, m). 13C{H} NMR (CDCl3) : 163.73 (s), 163.33 +(s), 162.30 (s), 156.52 (s), 155.74 (s), 155.25 (s), 145.34 (s), 134.47 (s), 131.82 (s), 125.25 +(s), 125.02 (s), 122.66 (s), 122.60 (s), 104.65 (s), 103.83 (s), 95.43 (s), 68.67 (s), 40.13 (s), +28.92 (s), 28.17 (s), 22.42 (s), 14.00 (s). Anal. calcd. for C27H28N2O7: C 65.84, H 5.73, N +5.69; found C 65.59, H 5.78, N 5.65 %. +4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-(hexyloxy)benzoate (NF6) +The reaction of 4-(dimethylamino)-2-ethoxybenzoic acid (4-2, 2.10 g, 7.91 mmol) was +esterified with 4-nitrophenyl 4-hydroxybenzoate (6, 2.10 g, 8.10 mmol) using EDC (1.70 g, +8.69 mmol) and DMAP (0.96 g, 7.92 mmol) in dichloromethane (70 mL) yielded 1.63 g +(41 %). 1H NMR (CDCl3) : 8.32 (2 H, d, J=9.4 Hz), 8.23 (2 H, d, J=8.8 Hz), 7.99 (1 H, d, +J=9.4 Hz), 7.31 - 7.52 (4 H, m), 6.32 (1 H, dd, J=8.8, 2.3 Hz), 6.15 (1 H, d, J=1.8 Hz), 4.07 +(2 H, t, J=6.7 Hz), 1.77 - 1.93 (2 H, m), 1.41 - 1.57 (2 H, m), 1.18 - 1.39 (4 H, m), 0.76 - +0.94 (3 H, m). 13C{H} NMR (CDCl3) : 163.70 (s), 163.32 (s), 162.27 (s), 156.54 (s), 155.74 +(s), 155.25 (s), 145.35 (s), 134.46 (s), 131.77 (s), 125.23 (s), 125.01 (s), 122.64 (s), 122.56 +(s), 104.69 (s), 103.85 (s), 95.48 (s), 68.70 (s), 40.07 (s), 31.52 (s), 29.20 (s), 25.69 (s), 22.53 +(s), 13.99 (s). Anal. calcd. for C28H30N2O7: C 66.39, H 5.97, N 5.53; found C 66.21, H 5.90, +N 5.49 %. + +8 + +1.3. +Equipment and apparatus +The compounds were studied by differential scanning calorimetry (DSC). Perkin- +Elmer 7 Pyris calorimeter (Perkin Elmer, Shelton, CT, USA) was utilised and the +measurements were conducted on cooling/heating runs at a rate of 10 K/min. The +calorimeter was calibrated to the extrapolated onsets for the melting points of water, indium +and zinc. A small amount of the studied compound (2-5 mg) was sealed into an aluminium +pan and put into the calorimeter chamber. A nitrogen medium was utilised during the +calorimetric measurements. The phase transition temperatures and the corresponding +enthalpies were established from the second heating and the subsequent cooling runs. +Textures were observed under the polarising microscope Eclipse E600Pol (Nikon, +Tokyo, Japan). We analysed the samples in various geometries. Two kinds of commercial +cells were purchased with the thickness of 5 m: HG cells with homogeneous anchoring +(orienting molecules parallel to the cell surface) and HT cells with surfactant adjusting +homeotropic arrangement of molecules (perpendicular to the surface). These cells consist of +glasses with ITO transparent electrodes and materials were filled in the isotropic phase by +capillary action. The Linkam E350 heating/cooling stage with TMS 93 temperature +programmer (Linkam, Tadworth, UK) was utilised, with the temperature stabilisation within +±0.1 K. +The switching current profile versus time was detected by a digital oscilloscope +Tektronix DPO4034 (Tektronix, Beaverton, OR, USA). Polarisation, P, was determined by +the integration of the current profile when the electric field of triangular modulation at a +frequency of 10 Hz was applied with the magnitude of 10 V/m. +We measured the dielectric spectroscopy by Schlumberger 1260 impedance analyser +(Schlumberger, Houston, TX, USA) and stabilised the temperature within ±0.1 K during the +frequency sweeps in a range of 1 Hz ÷ 1 MHz. The permittivity, (f) =−i which is +frequency dependent, was analysed with support of a modified version of the Cole-Cole +formula: + +) +2 +( +) +( +1 +0 +) +1 +( +* +m +n +r +Af +f +i +f +if ++ +− ++ + += +− +− + + + + + + + + (1), +where fr is the relaxation frequency,  is the dielectric strength,  is the distribution +parameter of relaxation,  is the permittivity of vacuum,  is the high frequency +permittivity, n, m, and A are the parameters of fitting. In formula (1) an ionic conductivity +and ITO electrode effects were taken into consideration. The measured values of the real +part of the permittivity,  and the imaginary part,  were simultaneously fitted to obtain +the parameters fr and . +The polarisation current profile of electric field was detected by Tektronix DPO4034 +digital oscilloscope (Tektronix, Oregon, US). The driving voltage from a generator (Agilent, +California, US) was amplified by a linear amplifier providing the amplitude up to ±120 V. +The temperature-dependent second harmonic generation (SHG) measurements were +conducted using an optical setup based on Ti:sapphire femtosecond laser (Spitfire ACE), + +9 + +which was amplified to produce 40 fs long pulses with 5 kHz repetition rate and central +wavelength of 800 nm. For SHG we utilised HG cells and placed them into a Linkam stage, +the temperature was stabilised with an accuracy ±0.1 K. The samples were illuminated by a +collimated beam with pulses fluence of approximately 0.01 mJ/cm2. The SHG signal +generated in transmission configuration was appropriately filtered, then detected by an +avalanche photodiode and amplified using a lock-in amplifier. The scheme of SHG +measurements is shown in Figure S1. +For the x-ray studies, the Bruker D8 GADDS system was utilised: parallel CuK +beam formed by Goebel mirror monochromator, 0.5 mm collimator, modified Linkam +heating stage, Vantec 2000 area detector. The samples for the diffraction experiments were +prepared in a form of droplets on heated surface. + +Figure S1. +SHG measurement scheme. + +2. +Mesomorphic properties + +Clanek o kapalnych krystalech – Vladka Novotna +Chopper +Mirror +800 nm +Filters +ND, High pass +Mirror +400 nm +Lens +Avalanche +photodiode +Sample in HG cell +Linkam stage +Boxcar/Lock-in +Lock-in +reference +Temperature controller + +10 + +Figure S2. +Vitrification process and creation of a fibre after melting of NF5. + + +Figure S3. +The microphotograph of the texture for homologue NF2 detected in a 5 m +HG cell. The width of the photo corresponds to about 200m. + +Figure S4. +The microphotograph of NF4 homologue in 5 m HG-A cell. The Polariser +orientation (white) and the rubbing direction (red) are marked. + +R +50μm11 + + +Figure S5 +The texture of NF5 in 5 m HG-A cell under a microscope with (a) crossed +polarisers, (b) and (c) with uncrossed position of polarisers (analyser rotated by an angle +about 20 degrees). All figures show the same part of the sample; red arrow represents the +rubbing direction and white arrows indicate the orientation of polarisers. + +Figure S6. +The microphotograph of NF6 homologue in 5 m HG-P cell. The rubbing +direction, R, is marked with a red line. + + +(a) +(b) +C +50μm +R50um12 + + +Figure S7. +The photo of NF6 homologue in 5 m HG-A cell after the application of the +external electric field of about 2 V/m. The rubbing direction, R, is marked with a red line. + +Figure S8. +The texture of NF6 in a special home-made gap-cell with a thickness of about +35 m, defined by two copper electrodes. One electrode is located at the right upper corner +out of the figure; the orientation of the applied electric field, E, is marked with the black +arrow. For (a) no electric field was applied and for (b) the electric field of about 0.2 V/m +was applied. + +R +P +50μm(a) +(b) +E13 + + +Figure S9. +3D-plot of (a) real, ’, and (b) imaginary, ’’, parts of the permittivity versus +frequency and temperature, T, for compound NF2. Dielectric measurements were performed +in 12 m cell with gold electrodes and no surfactant layer. + + +(a) +25000 +20000 +15000 +3 +10000 +5000 +102 +40 +frequency (Hz) +103 +60 +104 +80 +100 +105 +120 +(°C) +T +140 +106 +160 +(b) +8000 +6000 +3 +4000 +2000 +Q +102× +40 +frequency (Hz) +103× +60 +80 +104 +100 +105 +120 +140 +T (C) +106 +16014 + + +Figure S10. +3D-plot of (a) real, ’, and (b) imaginary, ’’, parts of the permittivity versus +frequency and temperature, T, for compound NF3. Dielectric measurements were performed +in 12 m cell with gold electrodes and no surfactant layer. + + + +(a) +20000- +15000 +000013 +5000 +60 +80 +101 +100 +102 +120 +(0d +103 +frequency (Hz) +104 +105 +140 +106 +(b) +6000 +5000 +4000 +3 +3000 +2000 +1000 +60 +80 +101 +102 +100 +103 +frequency (Hz) +104 +120 +105 +140 +10615 + + +Figure S11. +3D-plot of (a) real, ’, and (b) imaginary, ’’, parts of the permittivity versus +frequency and temperature, T, for compound NF6. Dielectric measurements were performed +in 12 m cell with gold electrodes and no surfactant layer. + + + +(a) +18000 +16000 +14000 +12000 +10000 +32 +8000 +6000 +4000 +2000 +30 +0 +40 +101 +50 +102 +°℃) +103 +60 +frequency (Hz) +104 +70 +105 +80 +106 +(b) +8000 +6000 +4000 +2000 +30 +40 +101 +50 +102 +103 +60 +frequency (Hz) +104 +(0。 +70 +105 +106 +*8016 + + +Figure S12. +Temperature dependences of the dielectric strength, , and the relaxation +frequency, fr, for NF5 in 12 m cell without surfactant layer. In the inset fr is presented in +the logarithmic scale versus reciprocal temperature, 1/T, in Kelvins and the activation energy +EA was established from the slope. + +400 +14 +350 +12 +6 +300 +10 +5 +250 +.0) +4 +8 +m +3 +200 +(zH) +6 +2 +=94 kJ/mol +150 +0.0030 +0.0032 +4 +100 +10" RT (Jmol') +2 +50 +0 +0 +30 +40 +50 +60 +70 +80 +T(C)17 + + +Figure S13. +For NF5 at the temperature T=30° C (a) the x-ray intensity versus the +scattering angle, . (b) 2D pattern of the intensity at the same temperature. Scattering angles +are in the logarithmic scale. + +Figure S14. +A model of NF1 molecule with the orientation of the dipole moment (blue +arrow). + +(a) +Intensity(arb.units) +4.4 A +10.4 A +22.5 A +0.1 +5 +10 +15 +20 +25 +30 +35 +(b) +20(deg.) +30 +28 +26 +24 +22 +20 +18 +16 \ No newline at end of file diff --git a/89E4T4oBgHgl3EQfDAsR/content/tmp_files/load_file.txt b/89E4T4oBgHgl3EQfDAsR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb8c67c05cbf9ec3af04d68a7c9270c2f6859959 --- /dev/null +++ b/89E4T4oBgHgl3EQfDAsR/content/tmp_files/load_file.txt @@ -0,0 +1,1411 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf,len=1410 +page_content='1 Dimethylamino terminated ferroelectric nematogens revealing high permittivity Martin Cigl, Natalia Podoliak, Tomáš Landovský, Dalibor Repček, Petr Kužel, and Vladimíra Novotná* Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, Prague, Czech Republic Abstract Since the recent discoveries, ferroelectric nematics became of upmost interest due to their outstanding ferroelectric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In this work, we prepared a series of polar molecules revealing a ferroelectric nematic phase (NF) with a very high dielectric constant (>104).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' A new motif, which differs from previously reported molecular structures, was optimized to support the NF phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For all homologues the NF phase was observed directly on the cooling from the isotropic phase and ferroelectric behaviour was investigated by dielectric spectroscopy, second harmonic generation, polarization current measurements and by analysis of textures in the polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The presented materials combine ferroelectricity with giant permittivity in a fluid media at room temperatures, so they appear to be extremely attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Polarity of molecules with the strong susceptibility to the electric field represent high potential for various applications in energy-efficient memory devices or capacitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Introduction In thermotropic liquid crystals (LCs) molecules can self-assemble and create intermediate phases (mesophases) in a certain temperature range between liquid and crystalline phases [1], combining the fluidity of liquids with the anisotropy characteristic for crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Anisotropic properties of LC medium manifest itself as a result of the anisotropic shape of (partially) ordered constituent molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' A large variety of phases and structures can be observed in LCs, which are susceptible to external field and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Many LC phases reveal a large electro-optical response, which became a background of mass-production technological applications (monitors, sensors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' First, the ferroelectricity in LCs was associated with chirality of the constituent molecules and only a tilted smectic phase formed by chiral rod-like molecules [1] was considered to feature ferroelectricity (FE) and/or antiferroelectricity (AF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' With the discovery of bent-core materials [2], it was found that non-chiral mesogens may also form FE and AF phases as the close packing and hindered rotation can lead to the structural chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Nevertheless, due to a higher viscosity, smectic phases never reached such broad application range as nematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Recent discoveries stimulated renewed intensive progress in the field of nematic liquid crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For a conventional nematic phase, the director orientations n and –n are indistinguishable due to the thermal fluctuations, so they form only non-polar phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' However, 2 as far back as in 1918, Max Born [3] predicted a possibility of a ferroelectric fluid, in which all the dipoles point in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In such a nematic ferroelectric state (NF), the dipole moments μ should be strong enough such that the dipole-dipole interactions overwhelm the thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In 2017, a real breakthrough was announced in the development of LCs, as the first two ferroelectric nematics (denoted RM734 and DIO) were reported simultaneously by two research teams [4-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Both materials reveal extremely high longitudinal dipole moments (about 10 D), anomalously huge dielectric anisotropy Δε, and a spontaneous polarisation of about 4 μC/cm2, which is an order of magnitude higher than the previously reported values in other ferroelectric LC phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Recently, these materials have been intensively studied [7-17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Mandle at al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' [9] synthesised a homologue series relevant to the molecular structure of RM734 and analysed the mesomorphic properties and tendencies leading to the NF phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The compounds have been intensively studied by Ljubljana researchers [10-12] and by the Boulder group [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The existence of ferroelectric domains with a different macroscopic orientation of the dipoles in the absence of electric field was reported [10-14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Details of polar nature of self-assembly, evolution of topological objects and analysis of their character [12,17,18] are under intensive research progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Currently, the research is focused on the preparation and characterisation of new compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Machine learning procedures were applied to predict ideal conditions for the NF phase presence, including a dipole moment value, aspect ratio, length of the molecule as well as the dipolar angle [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In spite of the fact that these conditions are rather restrictive, development in the designing of prosperous molecular structures was promoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' At the moment, microscopic organisation of the polar molecules and the mechanism of the phase transition to the ferroelectric nematic phase undergo intensive research and stimulating debates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' A theoretical description of the ferroelectric nematic phase has been proposed [20,21], and chiral analogues of highly polar molecules were developed recently [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Additionally, a possibility of oligomer synthesis was shown [23] and new phases and effects introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In any case, the ferroelectric properties combined with the giant permittivity in a fluid media represent an attractive rapidly developing subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Since the discovery of NF phase, the ongoing research is mostly concentrated on the design of new molecular structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Up to now, the library of NF materials is strictly limited to a couple of general structures possessing a suitable aspect ratio and a large enough dipole moment, which develops due to the effective electron donating and withdrawing groups within the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In this contribution, we demonstrate newly designed structural motif (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In contrast to the previously reported molecular designs [3-5,8-19], which utilise an oxygen-based electron donating group, we synthesised a series possessing a more efficient nitrogen electron donating group in the terminal part of the aromatic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Such a design yields higher dipole moment along the long molecular axis compared to other published materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' To modify the lateral interactions, which are strong in highly polar systems, we introduced a lateral alkyl chain with varied number of carbon atoms from 1 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Based on these considerations, we synthesised a series of compounds (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1) which exhibit the NF phase directly below the isotropic liquid on cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' By tuning the lateral substitution, we shifted the temperature interval of NF down to the room temperature (RT), at which it may eventually relax to a stable glassy state preserving the ferroelectric behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Chemical formula of compounds NFn with n = 1 - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Materials and methods Chemical formula of the studied compounds is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Synthesis of materials started from commercial 4-aminosalicylic acid (1, see Scheme 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Its amino group was protected by acetylation and the carboxylic group was protected by alkylative esterification by methyl iodide, so as neither of the two groups interfere with the alkylation of phenolic hydroxyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Protected derivative 2 was then alkylated by 1-bromoalkanes to get a series of alkyl homologues 3-n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In the next steps, the acetyl group was cleaved by acidic hydrolysis under mild conditions and the liberated amino group was alkylated by dimethyl sulphate yielding the key intermediate, acid 4-n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The lowest alkyl homologue (4-1) was synthesised directly from acid 1 by alkylation with the excess of dimethyl sulphate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The second part of the molecular core was synthesised from 4-hydroxybenzoic acid (5), which was protected by the reaction with 3,4-dihydro-2H- pyrane and reacted with 4-nitrophenol in a DCC-mediated esterification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The protected hydroxyl group was then liberated by the treatment with p-toluenesulfonic acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The final step of the synthesis was esterification of acids 4-n with phenol 6 mediated by EDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Differential scanning calorimetry (DSC) measurements were performed to acquire thermal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For electro-optical studies, a polarising optical microscope was used, equipped with a heating/cooling stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Details about the compound characterisation and experimental apparatus are in Supplemental file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Scheme 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Synthesis of the studied polar nematogens denoted NFn with n varying from 1 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Results We studied all newly synthesised homologues by DSC and observed textures and their changes in polarising microscope to assess the phase behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' We performed DSC measurements in a broad temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' We established the melting point (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=') from the first heating run, during which we observed a direct transformation from the crystalline to the isotropic (Iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=') phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' After the first heating of the fresh sample, we followed with a cooling run from the Iso phase down to -25°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' On the cooling run, the compounds transformed to the liquid crystalline state at a significantly lower temperature, Tiso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Under the polarising microscope, we observed characteristic textures in the LC state, which were previously ascribed to the ferroelectric nematic phase, NF [12-18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In the following description, the properties of NF phase are systematically uncovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The analysed DSC data are summarised in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Compounds NF1, NF2, and NF3 did not crystallise during the cooling run, however, they crystallised during the subsequent heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' These homologues revealed the ferroelectric nematic phase only during the cooling of the sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' temperature stabilisation or heating of the sample caused the crystallisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The homologue NF4 revealed the shortest temperature range of the NF phase and crystallised at about 74°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' On the other hand, the longest homologues NF5 and NF6 did not crystallise during the DSC measurements at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For these homologues, the NF phase persisted during the second and third cooling-heating DSC cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The stability of the NF phase for these two homologues was confirmed during electro-optical measurements: the NF phase was stable for several hours at RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The DSC thermograph for the homologue NF6 is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For the first heating of the sample, the melting point (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=') was established;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' for the second heating run, the NF phase melted at a temperature corresponding to Tiso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' A glassy transition was clearly distinguishable and its temperature, Tg, was determined from the onset calculated at a half heat capacity, cp, see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Glassy properties and ability to form fibres from the melted compound NF5 is demonstrated in Supplemental file (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' DSC thermograph detected for NF6 during the first and second heating and cooling runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 8 the first heating the second heating 6 the cooling Heat flow ( mW) glassy state N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 20 0 20 40 60 80 100 120 T(C) 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Calorimetric data taken from DSC measurements: melting point, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=', detected at the first heating run, the NF-Iso phase transition temperature, Tiso, and the glassy transition temperature, Tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' All temperatures are presented in °C, and enthalpy changes, \uf044H, in J/g, are in square brackets at the corresponding temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' \uf044H (J/g) Tiso (\uf06fC) \uf044H (J/g) Tg \uf044H (J/g) NF1 188 [+98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='6] 170 [-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='73] 24 [+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='47] NF2 150 [+71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 136 [-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='59] 30 [+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='42] NF3 156 [+73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8] 116 [-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='13] 15 [+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='27] NF4 144 [+80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='2] 96 [-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='11] NF5 120 [+55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='1] 82 [-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='86] 9 [+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='44] NF6 104 [+51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4] 65 [-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='33] 4 [+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='28] In the polarising microscope, we observed various textural features in different commercial or home-made cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' There are two basic geometries for rod-shaped liquid crystalline molecules: in HG cells, the molecules are oriented along the cell surface, and in the HT cell, a homeotropic anchoring ensured molecular orientation perpendicular to this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In the HG cell, the alignment is provided by rubbed polyimide layers with a small pretilt to arrange defect-free textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The pretilt results in nonzero polar surface energy as was pointed out by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Two kinds of HG cells were available, with parallel (HG-P) or antiparallel (HG-A) rubbing directions on opposite glass surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Let us start with HG-A cells and compare the results for various cell thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In this geometry, we observed two kinds of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The texture in 5\uf06dm HG-A cell for the studied homologue NF6 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The dominating type of domains are twisted domains, which were described for the NF phase in literature [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The twisted domains are recognisable when slightly uncrossing the analyser from the crossed position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Another type of domains can be observed in less extensive areas of the HG-A samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In the upper right part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 3, we found “red-colour” domains with characteristic borderline approximately parallel to the rubbing direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The red colour was typical for these domains in 5 \uf06dm HG-A cell, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' S2-S5 in Supplemental for other homologues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Extinction position in these domains are not easy to be established and the colour of these domains changes when rotating the sample with respect to the polariser position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' We did not find twisted domains in HG-P cells with parallel alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In this geometry, we observed homogeneously aligned area as well as “red” domains, as is demonstrated for NF6 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' S6 in Supplemental file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' We concentrate on twisted domains, which are very frequent in the HG-A geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' We found that for very thin HG-A sample, the twisted domain can be extended for a large area by quick cooling from the isotropic phase (with a rate \uf03e20 K/min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Rather big twisted domains separated by a zig-zag borderline are demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4(a) for NF6 in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='6 \uf06dm HG-A cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' One can see that the borderline between the twisted domains is oriented approximately perpendicularly to the rubbing direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' When we turn the analyser from the crossed position by an angle of ~ 20 degrees, we clearly observe two kinds of domains (see insets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' the sense of twist is opposite for two neighbouring domains and they are separated by 2\uf070 6 disclination line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In the paper by Sebastian [12], similar domains were observed for another type of ferroelectric nematogen and designated “sierra-domains”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In our particular case, these twisted domains reveal sharper contour and can be renamed as “shark-domains”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For the homologue NF5, the twisted domains are demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' S4 in Supplemental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Schematic picture of molecular twist between surfaces with antiparallel alignment is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Microphotograph of NF6 homologue in 5 \uf06dm HG-A cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The red arrow (R) marks the rubbing direction, the white arrows show the polariser (P) / analyser (A) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' R 50um 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Textures of NF6 in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='6 \uf06dm HG-A cell under a polarizing microscope (a) between crossed polarisers, the red arrow marks the rubbing direction, R, the orientation of the analyser (A) and the polariser (P) is schematically shown by white arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In the figure (b) there is a schematic arrangement of molecules in neighbouring twisted domains between glass surfaces with antiparallel rubbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The part of the figure (a) marked by white lines is shown in (c) and (d) when A is rotated by an angle of about 20 degrees counterclockwise or clockwise from the crossed position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' An application of an electric field in HG geometry led to a rather complex effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The colour of twisted domains slightly changed under the applied electric field and additional stripes appeared across the twisted domain structure approximately parallel to the rubbing direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' As the application of the field in the HG cell supplied only limited information and a detail analysis is rather problematic, we studied HT cells under applied bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In this geometry, the applied electric field is approximately parallel to the molecular dipole moment and we can observe a rearrangement of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 5, we demonstrate the HT texture for homologue NF6 with and without applied electric field of about 5 V/\uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In the upper part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 5, an area without electrode is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' When the field is switched on (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 5(b)), the molecules reorient along the field and the texture under the electrode area becomes black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' After switching the electric field off, the HT texture turns back to a lighter type, similar to the virgin texture (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 5(a)), within several seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' (a) (b) R 100 μm c) (d) 8 We investigated the switching properties of the studied compounds in HT geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Due to electrostatic interactions, the results are influenced both by the cell geometry and by the character of the aligning layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For n=1-4, the homologues NFn reveal strong vitrification and an increase in viscosity when approaching the glass transition temperature Tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' This temperature is relatively high and the samples feature a higher conductivity, which limited our studies for these homologues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' On the contrary, homologues NF5 and NF6 could be subjected to the applied field for a longer time (several hours);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' the polarisation was measured repeatedly and the results were reproducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' At the room temperature, these two homologues stay in LC phase for a long time and they start to crystallise only after several hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For homologue NF5, the temperature dependence of the polarisation is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The polarisation values are calculated by the time-integration of a switching current profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 6(b), the switching current is plotted versus the applied electric field at a frequency 10 Hz and at temperature 52 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For both homologues NF5 and NF6, we detected a continuous increase in polarisation values on cooling process in the NF phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' A coexistence of the Iso and NF phases was checked under the polarising microscope and it was observed only in a narrow temperature interval of about 2 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The decrease in polarisation values shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 6(a) is connected with an increase in switching time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Such a slowing-down of molecular dynamics is connected with an increase in the sample viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Texture of NF6 in 5 \uf06dm thick HT cell, (a) without electric field and (b) under applied electric field of about 5 V/\uf06dm perpendicular to the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The orientation of polarisers and of the applied electric field are marked by black symbols for illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Upper part of the figure shows an area without electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' To analyse the effect of the applied field in different cell geometries, we prepared a home-made gap-cell with in-plane electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Two glass slides were separated by copper 35 \uf06dm thick ribbons, with a gap distance of about 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In this cell, the domains disappeared (a) (b) No electrodes Electrode-edge E X P 100μm 9 under the applied electric field as all the molecules were aligned along the applied electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' After the switching-off of the external electric field, the domain structure was partially reconstructed in several seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Microphotographs can be found in Supplemental file (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Unfortunately, the thickness of 35 \uf06dm was rather large to reach homogeneous alignment through the whole cell thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Additionally, we are aware that the applied electric field was not homogeneously distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Technological tasks of the cell preparation and detailed analysis of the defects in electric field are still under work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' (a) The temperature dependence of the polarisation of NF5, which was calculated from the polarisation current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' (b) The current profile at a temperature T=52 °C is demonstrated with a triangular profile of the applied electric field at a frequency 10 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' We measured the dielectric spectra of all compounds in a temperature range from the isotropic liquid to RT in order to study the molecular dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The applied measuring field was smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='01 V/\uf06dm (higher probing fields could influence the dielectric measurements (a) 4 (μC/cm² 3 2 P 0 40 45 50 55 60 65 70 T(°C) (b) 10 4 current (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' units) 5 2 E (V/μm) 0 0 5 2 10 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='10 10 as the studied compounds are really sensitive to external fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In the ferroelectric NF phase, we found one distinct quite strong relaxation mode appearing at the Iso-NF phase transition on cooling and remaining visible down to RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' On the other hand, when the sample is in the crystalline state, this mode is not present and the permittivity is low (\uf03c10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' We demonstrate three-dimensional plots of the real, \uf065’, and imaginary, \uf065’’, parts of permittivity versus frequency and temperature, T, for compound NF5 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For homologues NF2, NF3 and NF6, the 3D- plots of permittivity are shown in Supplemental file, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' S9-S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' All the presented dielectric data were obtained in 12 \uf06dm thick cells with gold electrodes and no surfactant layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' We encountered a disturbing effect of surfactant, similarly as it was mentioned in previous works dealing with dielectric spectroscopy of the NF phase [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For such a type of polar phase, it was reported that the polymer layers effectively influence the permittivity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Due to a non-conductive character of polymer layers on the cell surfaces, there is a barrier which causes a spatial variation of the charge and influences the measured effective permittivity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' We fitted the dielectric data to the Cole-Cole formula (see Supplemental file for the details) to obtain information about the dielectric strength, \uf044\uf065, and the relaxation frequency, fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' We detected large only slightly temperature dependent values of \uf044\uf065 up to 15\uf0b4103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In contrast, the relaxation frequency decreases within the whole temperature range of the NF phase on cooling and follows the Arrhenius law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Such behaviour is documented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 8 for homologue NF6, which followed Arrhenius behaviour ideally and the activation energy, Ea, was calculated to be 102 kJ/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For other compounds, the linearity of fr in logarithmic scale (versus 1/T in absolute temperature scale) was confirmed only far from the Iso-NF phase transition (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' S12 in Supplemental file).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Non-homogeneity of molecular alignment and/or influence of electrodes should be taken into consideration to explain the deviation from Arrhenius law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 3D-plot of (a) real, \uf065’, and (b) imaginary, \uf065’’, parts of the permittivity versus the frequency and the temperature, T, for compound NF5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Dielectric measurements were performed in 12 \uf06dm cell with gold electrodes and no surfactant layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 16 14 12 10 8 3 6 4 30 2 40 50 0 60 101 102 70 103 frequency (Hz) 104 80 105 106 6 (103) 2 30 40 50 101 102 60 103 70 frequency (Hz) 104 105 80 106 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Temperature dependences of (a) the dielectric strength, \uf044\uf065, and relaxation frequency, fr, for NF6 in 12 \uf06dm cell without surfactant layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In the inset fr is presented in the logarithmic scale versus reciprocal temperature, 1/T, in Kelvins and the activation energy, EA, was established from the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Strong polar character of the NF phase was proved by SHG measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The SHG experiments were carried out in transmission configuration according the scheme described in Supplemental file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' We utilised HG cells and SHG measurement results are presented for compounds NF5 and NF6 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' On cooling the sample from the isotropic phase, the SHG signal abruptly grows from zero value at the transition temperature to NF phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' With ongoing temperature decrease, the SHG intensity slows down its increase, reaches the maximum and slowly starts to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' All of this happens within the NF phase, where we would expect a gradual increase in the SHG signal upon cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Moreover, even for the weakest applied intensity of the fundamental laser beam, a small drop in SHG intensity was detected in subsequent measuring runs at the same temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' From this it follows that the decrease in the SHG signal upon cooling may be explained by partial decomposition of our samples caused by rather strong intensity of the pulse laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' X-ray scattering experiments confirmed nematic character of the observed mesophase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Nematic phase is characterised by the long-range orientational order and only broad diffuse peaks of low intensity can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For homologue NF5, the signal at small scattering angles is rather wide and can be fitted with two signals, with maxima corresponding to 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Å and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='5 Å at T=75°C, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='5 Å and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 Å at T=30°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' As the length of molecules, l, can be approximately established as l~20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='9 Å, the peak at the small scattering angle matches perfectly to the long dimension of the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The peak at a wide-angle region has also a very broad 口 60 12 ■ 4 E,=102 kJ/mol 口 (zH) 口 40 (10° 8 口 (zH) 2 3V 口 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='6 20 4 10" RT (Jmoll 0 0 40 45 50 55 60 T(C) 13 profile with the maximum corresponding to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 Å for all measuring temperatures, and it corresponds to an average distance between the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' SHG signals for NF5 and NF6 in HG cells 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Conclusions We proposed a new structural modification of highly polar molecules self-assembling and forming the ferroelectric nematic phase NF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' All the prepared compounds exhibit a direct phase transition to the ferroelectric nematic phase on cooling from the isotropic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In the presented homologue series, a prolongation of a side-chain resulted in the NF phase persistence down to the room temperatures and stability for at least several hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Ferroelectric character of the nematic phase was proven by several experimental techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Characteristic textural features for the ferroelectric nematics were observed in several sample geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The ferroelectric switching process was detected and the polarisation was calculated from the measured polarisation current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The values of polarisation were found to increase continuously on cooling from the isotropic phase, reaching up to 4 \uf06dC/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For all the studied homologues, the dielectric studies show a strong polar mode characteristic for the NF phase, disappearing in the isotropic or crystalline phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The dielectric strength of this mode exceeds values of about 15\uf0b4103, which is the maximum reached for the NF phase up to now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Nevertheless, the characterisation of the defects in the NF phase and the role of the electrodes are not yet completely solved and will need a deeper insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The dipole moment of the molecules was calculated and established to be about 14 D, which is larger than the value reported for DIO or RM734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The discovery of ferroelectricity for nematics opened new opportunities in the liquid crystal research and generally in the field of condensed matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The NF phase represents a highly polar structure responsive to very small applied fields and it features a variety of new effects 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='6 NF5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 NF6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 30 40 50 60 70 80 90 T (C) 14 induced by the confining surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Generally, the application potential of the NF phase is immense and not yet completely explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Our particular room-temperature-stable soft phase exhibits huge dielectric constant and can be important in future for the development of memory devices, capacitors and actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Disclosure statement No potential conflict of interest was reported by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Acknowledgments Authors acknowledge project MAGNELIQ, that received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 899285;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' and project 22-16499S from the Czech Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' is grateful to Damian Pociecha and Ewa Gorecka from Warsaw University for their help with x-ray measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' References [1] Handbook of Liquid Crystals, 2nd Edition, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' [14] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Chen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Korblova, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Glaser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Maclennan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Walba, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Clark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Polar in-plane surface orientation of a ferroelectric nematic liquid crystal: Polar monodomains and twisted state electro-optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' PNAS 118 (22) 2104092118, 2021.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' [20] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Madhusudana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Simple molecular model for ferroelectric nematic liquid crystals exhibited by small rodlike mesogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Physical Review E, 104, 014704, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' [21] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Kats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Stability of the uniform ferroelectric nematic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Physical Review E, 103, 012704, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' [22] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Kougo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Wan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Aya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Spontaneous helielectric nematic liquid crystals: Electric analog to helimagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' PNAS, 118 (42) e2111101118, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Xia, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Kougo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Lei, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Dai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Cen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Jiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Aya, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' How far can we push the rigid oligomers/polymers toward ferroelectric nematic liquid crystals?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 143,17857−17861, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1 Supplemental information Dimethylamino terminated ferroelectric nematogens revealing high permittivity Martin Cigl, Natalia Podoliak, Tomáš Landovský, Dalibor Repček, Petr Kužel, and Vladimíra Novotná* Institute of Physics of the Czech Academy of Sciences, Na Slovance 2, Prague, Czech Republic Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Synthesis and compound characterisation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' General synthesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Synthetic procedures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Equipment and apparatus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Mesomorphic properties 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Textures 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Dielectric spectroscopy and electro-optical properties 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Syntheses and compound characterisation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' General synthesis All starting materials and reagents were purchased from Sigma-Aldrich, Acros Organics or Lach:Ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' All solvents used for the synthesis were “p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='a.” grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Tetrahydrofuran was further distilled from calcium hydride to obtain sufficiently dry solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR spectra were recorded on Varian VNMRS300 instrument;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' deuteriochloroform (CDCl3) and hexadeuteriodimethyl sulfoxide (DMSO-d6) were used as solvents and the signals of the solvent served as an internal standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Chemical shifts (\uf064) are given in ppm and J values are given in Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Elemental analyses were carried out on Elementar vario EL III instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The purity of all final compounds was checked by HPLC analysis (high-pressure pump ECOM Alpha;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' column WATREX Biospher Si 100, 250 × 4 mm, 5 \uf06dm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' detector WATREX UVD 250) and were found to be >99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Column chromatography was carried out using Merck Kieselgel 60 (60−100 μm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 2 Synthesis of materials started from commercial 4-aminosalicylic acid (1, see Scheme 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Its amino group was protected by acetylation and the carboxylic group was protected by alkylative esterification by methyl iodide, so as neither of the two groups interfere with the alkylation of phenolic hydroxyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Protected derivative 2 was then alkylated by 1- bromoalkanes to get a series of alkyl homologues 3-n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In the next steps, the acetyl group was cleaved by acidic hydrolysis under mild conditions and the liberated amino group was alkylated by dimethyl sulphate yielding the key intermediate, acid 4-n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The lowest alkyl homologue (4-1) was synthesised directly from acid 1 by alkylation with the excess of dimethyl sulphate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The second part of the molecular core was synthesised from 4- hydroxybenzoic acid (5), which was protected by the reaction with 3,4-dihydro-2H-pyrane and reacted with 4-nitrophenol in a DCC-mediated esterification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The protected hydroxyl group was then liberated by the treatment with p-toluenesulfonic acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The final step of the synthesis was esterification of acids 4-n with phenol 6 mediated by EDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Scheme 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Synthetic procedures for the preparation of target compounds NFn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Synthetic procedures Methyl 4-acetamido-2-hydroxybenzoate (2) Acetic hydride (30 mL, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='31 mol) was added dropwise to the suspension of powdered 4- aminosalicylic acid (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='13 mol) in acetonitrile (250 mL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The reaction mixture was stirred for 2 h and the resulting suspension filtered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The filter cake was washed by the small amount of acetonitrile to remove the residue of acetic acid and dried in a vacuum dryer at 40 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Yield 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='77 g (96 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 3 Dry 4-acetamidosalicylic acid (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='12 mol) was dissolved in DMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Powdered KHCO3 was added with stirring, resulting in CO2 evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Then methyl iodide was added dropwise and the reaction mixture stirred for 6 h under anhydrous conditions (CaCl2 tube).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The resulting suspension was poured into water and neutralised with concentrated HCl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' White precipitate was filtered off and crystallised from 50% aqueous methanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Yield 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='43 g (95 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (DMSO-d6) \uf064: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='23 (1 H, s), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='71 (1 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='37 (1 H, d, J=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='05 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='85 (3 H, s), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='07 (3 H, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' General procedure for alkylation of benzoate 2 Benzoate 2 was dissolved in dry DMF and powdered K2CO3 and KI (omitted if iodoalkane was used) were added with stirring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Mixture was heated to 50 °C and 1-bromoalkane was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Reaction was stirred at 50 °C under anhydrous conditions (CaCl2 tube) for 10 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The cooled resulting mixture was poured into cold water, neutralised with concentrated HCl and the precipitated product was filtered off and crystallised from ethanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Methyl 4-acetamido-2-ethoxybenzoate (3-1) The reaction of benzoate 2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='90 mmol) with ethyl iodide (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='45 g, 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='32 mmol) in the presence of K2CO3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='18 mmol) in dry DMF (50 mL) yielded 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='83 g (85 %) of 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='25 (1 H, br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='36 (1 H, d, J=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='94 (1 H, d, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='58 (1 H, dd, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='6 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='11 (2 H, q, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='89 (3 H, s), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='23 (3 H, s), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='42 (3 H, t, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Methyl 4-acetamido-2-propoxybenzoate (3-2) The reaction of benzoate 2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='90 mmol) with 1-bromopropane (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='71 g, 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='82 mmol) in the presence of K2CO3 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='36 mmol) in dry DMF (60 mL) yielded 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='53 g (58 %) of 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='95 (1 H, s), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='79 (1 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='60 (1 H, d, J=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='81 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='09 (2 H, t, J=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='5 Hz), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='81 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='05 (2 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='10 (3 H, t, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Methyl 4-acetamido-2-butoxybenzoate (3-c) The reaction of benzoate 2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='90 mmol) with 1-bromobutane (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='53 g, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='40 mmol) in the presence of K2CO3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='18 mmol) in dry DMF (50 mL) yielded 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='42 g (86 %) of 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CHLOROFORM-d) \uf064 ppm 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='85 (1 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='77 (1 H, s), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='60 (1 H, d, J=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='79 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='04 (2 H, t, J=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='5 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='86 (2 H, s), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='19 (3 H, s), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='72 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='93 (2 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='42 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='61 (2 H, m), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='97 (3 H, t, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Methyl 4-acetamido-2-(pentyloxy)benzoate (3-d) The reaction of benzoate 2 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='80 mmol) with 1-iodopentane (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='88 g, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='37 mmol) in the presence of K2CO3 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='11 mol) in dry DMF (100 mL) yielded 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='81 g (75 %) of 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='86 (1 H, s), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='80 (1 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='58 (1 H, d, J=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='79 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='01 (2 H, t, J=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='7 Hz), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='77 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='03 (2 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='31 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='57 (4 H, m), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='91 (3 H, t, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Methyl 4-acetamido-2-(hexyloxy)benzoate (3-e) The reaction of benzoate 2 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='80 mmol) with 1-bromohexane (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='78 g, 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='60 mmol) in the presence of K2CO3 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='11 mol) in dry DMF (100 mL) yielded 4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='73 g (69 %) of 3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='83 (1 H, s), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='79 (1 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='59 (1 H, d, J=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='80 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='02 (2 H, t, J=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='7 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='86 (3 H, s), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='19 (3 H, s), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='71 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='91 (2 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='42 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='56 (2 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='20 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='41 (4 H, m), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='91 (3 H, t, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' General procedure for deacetylation of amino group Methyl 4-acetamido-2-(alkoxy)benzoate 3 was dissolved in methanol at 50 °C and the concentrated H2SO4 was carefully added dropwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The reaction mixture was stirred at 50 °C for 30 min and then poured into cold water and neutralised with NaOH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The neutral dispersion of the product in water was extracted with ethyl acetate, combined organic layers were washed with water and brine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' After drying with anhydrous MgSO4, the solvent was removed on rotary evaporator and the residue was purified by column chromatography on silica gel to yield methyl 2-(alkoxy)-4-aminobenzoate as an intermediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' NOTE: The presence of water in the deacetylation reaction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' use of diluted H2SO4) leads to considerable amounts of decarboxylation byproduct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' General procedure for methylation of amino group Methyl 2-(alkoxy)-4-aminobenzoate was dissolved in DMSO and powdered K2CO3 was added with stirring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' A mixture was heated to 50 °C and dimethyl sulphate was added dropwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The reaction mixture was stirred at the same temperature and under anhydrous conditions (CaCl2 tube) overnight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The progress of the reaction was monitored using TLC (CH2Cl2-acetone 95 : 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The resulting mixture was filtered and solid Na2S was added to the filtrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The mixture was stirred for 4 h at room temperature and then poured into water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' After 30 min of standing, the solution was neutralised with the concentrated acetic acid and the precipitated product was collected by filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' A crude product was purified by column chromatography on silica gel and crystallised from methanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4-(Dimethylamino)-2-methoxybenzoic acid (4-1) Compound 4a was synthesised by direct methylation of acid 1 using the general procedure for methylation of amino group described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Reaction of benzoic acid 1 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='30 mmol) with dimethyl sulphate (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='45 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='33 mol) in the presence of K2CO3 (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='36 mol) in DMSO (150 mL) and subsequent treatment with Na2S (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='21 mol) yielded 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='01 g (79 %) of 4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4-(Dimethylamino)-2-ethoxybenzoic acid (4-2) Following the general procedure, starting from methyl 4-acetamido-2-ethoxybenzoate 3-1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='80 g, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='23 mmol), which was deacylated using H2SO4 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 mL, 96%) in methanol (50 mL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The free amine was methylated using dimethyl sulphate (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='52 g, 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='90 mmol) and K2CO3 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='82 mmol) in DMSO (50 mL) followed by the treatment with Na2S (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='60 g, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='50 mmol) yielded 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='29 g (54 %) of 4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='71 (1 H, br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='99 (1 H, d, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='37 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='10 (1 H, d, J=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='29 (2 H, q, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='05 (6 H, s), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='55 (3 H, t, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4-(Dimethylamino)-2-propoxybenzoic acid (4-3) Using the described general procedure: 4-acetamido-2-propoxybenzoate 3-2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='50 g, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='93 mmol) was deacylated using H2SO4 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='5 mL, 96%) in methanol (40 mL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Liberated 5 amine was methylated using dimethyl sulphate (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='24 g, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='68 mmol) and K2CO3 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='70 g, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='71 mmol) in DMSO (50 mL) followed by the treatment with Na2S (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='10 g, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='09 mmol) yielded 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='91 g (61 %) of 4-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='73 (1 H, br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='01 (1 H, d, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='42 (1 H, dd, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='1 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='19 (1 H, d, J=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='19 (2 H, t, J=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='5 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='07 (6 H, s), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='95 (2 H, sext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=', 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='2 Hz), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='10 (3 H, t, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz) 2-Butoxy-4-(dimethylamino)benzoic acid (4-4) Using the general deacetylation and alkylation protocol: 4-acetamido-2-butoxybenzoate 3-3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='30 g, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='98 mmol) was deacylated using H2SO4 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 mL, 96%) in methanol (50 mL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Liberated amine was methylated using dimethyl sulphate (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='40 g, 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='98 mmol) and K2CO3 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='50 g, 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='21 mmol) in DMSO (50 mL) followed by the treatment with Na2S (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='60 g, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='50 mmol) yielded 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='42 g (51 %) of 4-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='71 (1 H, br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='00 (1 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='40 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='17 (1 H, d, J=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='21 (2 H, t, J=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='7 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='06 (6 H, s), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='85 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='94 (2 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='43 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='58 (2 H, m), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='92 (3 H, , t, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4-(Dimethylamino)-2-(pentyloxy)benzoic acid (4-5) Using the above mentioned general protocols: 4-acetamido-2-(pentyloxy)benzoate 3-4 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='50 g, 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='76 mmol) was deacylated using H2SO4 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 mL, 96%) in methanol (150 mL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Liberated amine was methylated using dimethyl sulphate (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='12 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='21 mol) and K2CO3 (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='21 mol) in DMSO (150 mL) followed by the treatment with Na2S (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='90 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='14 mol) yielded 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='41 g (62 %) of 4-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='70 (1 H, br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='00 (1 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='41 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='18 (1 H, d, J=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='21 (2 H, t, J=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='7 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='06 (6 H, s), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='82 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='99 (2 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='31 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='55 (4 H, m), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='94 (3 H, , t, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4-(Dimethylamino)-2-(hexyloxy)benzoic acid (4-6) Using the above mentioned general protocols: 4-acetamido-2-(hexyloxy)benzoate 3-3 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='50 g, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='38 mmol) was deacylated using H2SO4 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 mL, 96%) in methanol (150 mL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Liberated amine was methylated using dimethyl sulphate (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='10 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='16 mol) and K2CO3 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='17 mol) in DMSO (150 mL) followed by the treatment with Na2S (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='20 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='10 mol) yielded 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='82 g (56 %) of 4-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='72 (1 H, br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='99 (1 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='37 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='11 (1 H, d, J=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='20 (2 H, t, J=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='7 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='06 (6 H, s), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='85 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='99 (2 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='21 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='51 (6 H, m), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='91 (3 H, , t, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4-Nitrophenyl 4-hydroxybenzoate (6) 3,4-Dihydro-2H-pyrane (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='30 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='17 mol) was added dropwise to the suspension of 4- hydroxybenzoic acid (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='80 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='10 mol) in diethylether (200 ml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The reaction mixture was stirred overnight under anhydrous conditions (CaCl2 tube) and then filtered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The filter cake contained the majority of desired 4-((tetrahydro-2H-pyran-2-yl)oxy)benzoic acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The filtrate was vigorously stirred with aqueous NaOH (80 ml, 10%) for 30 min, and then the aqueous layer was separated and neutralised by HCl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The pH was further adjusted to ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4 using acetic acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The precipitated solid was collected, washed with cold water and dried under vacuum and finally combined with the dry portion obtained from the filter cake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4-((Tetrahydro-2H-pyran-2-yl)oxy)benzoic acid (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='35 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='14 mol) and 4-nitrophenol (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='60 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='14 mol) were dissolved in dry THF (250 mL) and cooled to ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 10 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Then N,N´- dicyclohexylcarbodiimide (DCC, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='60 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='15 mol ) and 4-(dimethylamino)-pyridine (DMAP, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='60 g, 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='22 mmol) were added and the reaction mixture was stirred under 6 anhydrous conditions for 12 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The precipitated N,N´-dicyclohexylurea was filtered off and the filtrate diluted with ethyl acetate (100 mL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The resulting solution was washed with diluted HCl (100 mL, 1 : 15), then with water and the solvents were removed on rotary evaporator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The solid residue was dissolved in CHCl3-methanol mixture (1 : 1) and toluenesulfonic acid (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='22 mmol) was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The reaction mixture was stirred at 45 °C for 1 h and then evaporated to dryness on rotary evaporator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' A crude product was crystallised from acetone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Yield 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='71 g (66 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (DMSO-d6) \uf064\uf03a 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='33 (2 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='6 Hz), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='08 (2 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='58 (2 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='6 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='20 (2 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='62 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='65 (1 H, m), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='63 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='75 (1 H, m), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='48 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='60 (1 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='21 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='97 (6 H, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' General procedure for EDC-mediated esterification 2-(Alkoxy)-4-(dimethylamino)benzoic acid 4-n and 4-nitrophenyl 4-hydroxybenzoate (6) were suspended in dry dichloromethane (50 ml) and cooled to 2 – 8 °C in ice-water bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Then N-(3-dimethylaminopropyl)-N′-ethylcarbodiimide hydrochloride (EDC) and 4-(N,N-dimethylamino)pyridine (DMAP) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='1 g, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='82 mmol) were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The reaction mixture was stirred for 2 hours under anhydrous conditions and the temperature was let rise as ice in the cooling bath melted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The resulting solution diluted with CH2Cl2 and washed with water and brine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Organic layer was dried over anhydrous magnesium sulphate and evaporated on the rotary evaporator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The residue was purified by column chromatography on silica gel in CH2Cl2-acetone eluent and recrystallised from acetone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-methoxybenzoate (NF1) 4-(Dimethylamino)-2-methoxybenzoic acid (4-1, 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='1 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='40 mmol) was esterified with 4-nitrophenyl 4-hydroxybenzoate (6, 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='5 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='40 mmol) using EDC (81 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='42 mmol) and DMAP (51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 mg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='42 mmol) in dichloromethane (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 mL) as described in general procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Yield 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='5 mg (53 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='33 (2 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='24 (2 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='03 (1 H, d, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='41 (4 H, dd, J=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='1, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='33 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='16 (1 H, d, J=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='95 (3 H, s), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='11 (6 H, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' calcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' for C23H20N2O7: C 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='30, H 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='62, N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='42;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' found C 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='86, H 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='68, N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='47 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-ethoxybenzoate (NF2) 4-(Dimethylamino)-2-methoxybenzoic acid (4-1) The reaction of 4-(dimethylamino)-2-ethoxybenzoic acid (4-2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='78 mmol) was esterified with 4-nitrophenyl 4-hydroxybenzoate (6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='24 g, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='78 mmol) using EDC (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='16 mmol) and DMAP (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='29 g, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='39 mmol) in dichloromethane (30 mL) yielded 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='03 g (48 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='33 (2 H, d, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 Hz), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='24 (2 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='2 Hz), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='01 (1 H, d, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='33 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='48 (4 H, m), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='32 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='16 (1 H, d, J=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='15 (2 H, d, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='08 (6 H, s), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='49 (3 H, t, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 13C{H} NMR (CDCl3) \uf064: 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='72 (s), 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='10 (s), 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='29 (s), 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='46 (s), 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='73 (s), 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='24 (s), 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='31 (s), 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='35 (s), 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='79 (s), 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='24 (s), 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='03 (s), 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='67 (s), 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='62 (s), 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='59 (s), 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='90 (s), 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='65 (s), 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='41 (s), 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='11 (s), 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='77 (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' calcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' for C24H22N2O7: C 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='00, H 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='92, N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' found C 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='87, H 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='98, N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='11 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-propoxybenzoate (NF3) Starting from 4-(dimethylamino)-2-propoxybenzoic acid (4-3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='25 g, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='78 mmol) and 4- nitrophenyl 4-hydroxybenzoate (6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='50 g, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='78 mmol) with EDC (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='18 g, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='03 mmol) and 7 DMAP (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='68 g, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='61 mmol) in dichloromethane (50 mL) yielded 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='36 g (51 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='34 (2 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='24 (2 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='00 (1 H, d, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='33 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='50 (4 H, m), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='32 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='15 (1 H, d, J=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='04 (2 H, t, J=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='5 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='09 (6 H, s), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='81 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='98 (2 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='08 (3 H, t, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 13C{H} NMR (CDCl3) \uf064: 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='73 (s), 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='29 (s), 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='34 (s), 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='54 (s), 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='74 (s), 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='27 (s), 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='36 (s), 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='46 (s), 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='84 (s), 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='26 (s), 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='04 (s), 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='65 (s), 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='60 (s), 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='66 (s), 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='84 (s), 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='45 (s), 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='18 (s), 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='11 (s), 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='64 (s), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='68 (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' calcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' for C25H24N2O7: C 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='65, H 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='21, N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='03;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' found C 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='56, H 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='19, N 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='98 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4-[(4-Nitrophenoxy)carbonyl]phenyl 2-butoxy-4-(dimethylamino)benzoate (NF4) Esterification of 2-butoxy-4-(dimethylamino)benzoic acid (4-4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='43 mmol) with 4- nitrophenyl 4-hydroxybenzoate (6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='50 g, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='64 mmol) using EDC (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='22 mmol) and DMAP (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='58 g, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='78 mmol) in dichloromethane (70 mL) yielded 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='31 g (51 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='34 (2 H, d, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 Hz), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='24 (2 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='2 Hz), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='00 (1 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='31 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='50 (4 H, m), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='32 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='16 (1 H, d, J=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='08 (2 H, t, J=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='5 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='09 (6 H, s), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='74 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='96 (2 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='42 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='66 (2 H, m), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='95 (3 H, t, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 13C{H} NMR (CDCl3) \uf064: 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='73 (s), 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='33 (s), 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='30 (s), 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='52 (s), 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='74 (s), 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='25 (s), 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='34 (s), 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='47 (s), 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='82 (s), 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='25 (s), 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='02 (s), 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='66 (s), 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='60 (s), 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='65 (s), 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='83 (s), 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='43 (s), 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='35 (s), 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='15 (s), 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='30 (s), 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='25 (s), 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='85 (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' calcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' for C26H26N2O7: C 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='26, H 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='48, N 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='85;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' found C 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='15, H 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='16, N 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='80 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-(pentyloxy)benzoate (NF5) Following the general procedure above 4-(dimethylamino)-2-propoxybenzoic acid (4-3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 g, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='96 mmol) and 4-nitrophenyl 4-hydroxybenzoate (6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='06 g, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='94 mmol) were reacted in the presence of EDC (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='68 g, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='59 mmol) and DMAP (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='50 g, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='13 mmol) in dichloromethane (70 mL) yielded 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='76 g (45 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='33 (2 H, d, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='2 Hz), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='24 (2 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='6 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='99 (1 H, d, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='2 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='32 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='49 (4 H, m), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='32 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='9, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='15 (1 H, d, J=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='07 (2 H, t, J=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='6 Hz), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='08 (6 H, s), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='79 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='94 (2 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='26 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='55 (4 H, m), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='83 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='94 (3 H, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 13C{H} NMR (CDCl3) \uf064: 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='73 (s), 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='33 (s), 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='30 (s), 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='52 (s), 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='74 (s), 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='25 (s), 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='34 (s), 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='47 (s), 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='82 (s), 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='25 (s), 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='02 (s), 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='66 (s), 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='60 (s), 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='65 (s), 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='83 (s), 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='43 (s), 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='67 (s), 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='13 (s), 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='92 (s), 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='17 (s), 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='42 (s), 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='00 (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' calcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' for C27H28N2O7: C 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='84, H 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='73, N 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='69;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' found C 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='59, H 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='78, N 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='65 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 4-[(4-Nitrophenoxy)carbonyl]phenyl 4-(dimethylamino)-2-(hexyloxy)benzoate (NF6) The reaction of 4-(dimethylamino)-2-ethoxybenzoic acid (4-2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='10 g, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='91 mmol) was esterified with 4-nitrophenyl 4-hydroxybenzoate (6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='10 g, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='10 mmol) using EDC (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='70 g, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='69 mmol) and DMAP (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='96 g, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='92 mmol) in dichloromethane (70 mL) yielded 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='63 g (41 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 1H NMR (CDCl3) \uf064: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='32 (2 H, d, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 Hz), 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='23 (2 H, d, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='99 (1 H, d, J=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 Hz), 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='31 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='52 (4 H, m), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='32 (1 H, dd, J=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3 Hz), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='15 (1 H, d, J=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='8 Hz), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='07 (2 H, t, J=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='7 Hz), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='77 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='93 (2 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='41 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='57 (2 H, m), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='18 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='39 (4 H, m), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='76 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='94 (3 H, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 13C{H} NMR (CDCl3) \uf064: 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='70 (s), 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='32 (s), 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='27 (s), 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='54 (s), 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='74 (s), 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='25 (s), 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='35 (s), 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='46 (s), 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='77 (s), 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='23 (s), 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='01 (s), 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='64 (s), 122.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='20 (s), 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='69 (s), 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='53 (s), 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='99 (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' calcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' for C28H30N2O7: C 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='39, H 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='97, N 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='53;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' found C 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='21, H 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='90, N 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='49 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Equipment and apparatus The compounds were studied by differential scanning calorimetry (DSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Perkin- Elmer 7 Pyris calorimeter (Perkin Elmer, Shelton, CT, USA) was utilised and the measurements were conducted on cooling/heating runs at a rate of 10 K/min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The calorimeter was calibrated to the extrapolated onsets for the melting points of water, indium and zinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' A small amount of the studied compound (2-5 mg) was sealed into an aluminium pan and put into the calorimeter chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' A nitrogen medium was utilised during the calorimetric measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The phase transition temperatures and the corresponding enthalpies were established from the second heating and the subsequent cooling runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Textures were observed under the polarising microscope Eclipse E600Pol (Nikon, Tokyo, Japan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' We analysed the samples in various geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Two kinds of commercial cells were purchased with the thickness of 5 \uf06dm: HG cells with homogeneous anchoring (orienting molecules parallel to the cell surface) and HT cells with surfactant adjusting homeotropic arrangement of molecules (perpendicular to the surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' These cells consist of glasses with ITO transparent electrodes and materials were filled in the isotropic phase by capillary action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The Linkam E350 heating/cooling stage with TMS 93 temperature programmer (Linkam, Tadworth, UK) was utilised, with the temperature stabilisation within ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The switching current profile versus time was detected by a digital oscilloscope Tektronix DPO4034 (Tektronix, Beaverton, OR, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Polarisation, P, was determined by the integration of the current profile when the electric field of triangular modulation at a frequency of 10 Hz was applied with the magnitude of 10 V/\uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' We measured the dielectric spectroscopy by Schlumberger 1260 impedance analyser (Schlumberger, Houston, TX, USA) and stabilised the temperature within ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='1 K during the frequency sweeps in a range of 1 Hz ÷ 1 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The permittivity, \uf065\uf02a(f) =\uf065\uf0a2−i\uf065\uf0a2\uf0a2\uf02c which is frequency dependent, was analysed with support of a modified version of the Cole-Cole formula: ) 2 ( ) ( 1 0 ) 1 ( m n r Af f i f if + − + \uf044 = − − \uf0a5 \uf070\uf065 \uf073 \uf065 \uf065 \uf065 \uf061 (1), where fr is the relaxation frequency, \uf044\uf065 is the dielectric strength, \uf061 is the distribution parameter of relaxation, \uf065\uf030 is the permittivity of vacuum, \uf065\uf0a5 is the high frequency permittivity, n, m, and A are the parameters of fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In formula (1) an ionic conductivity and ITO electrode effects were taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The measured values of the real part of the permittivity, \uf065\uf0a2\uf02c and the imaginary part, \uf065\uf0a2\uf0a2\uf02c were simultaneously fitted to obtain the parameters fr and \uf044\uf065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The polarisation current profile of electric field was detected by Tektronix DPO4034 digital oscilloscope (Tektronix, Oregon, US).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The driving voltage from a generator (Agilent, California, US) was amplified by a linear amplifier providing the amplitude up to ±120 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The temperature-dependent second harmonic generation (SHG) measurements were conducted using an optical setup based on Ti:sapphire femtosecond laser (Spitfire ACE), 9 which was amplified to produce 40 fs long pulses with 5 kHz repetition rate and central wavelength of 800 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For SHG we utilised HG cells and placed them into a Linkam stage, the temperature was stabilised with an accuracy ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The samples were illuminated by a collimated beam with pulses fluence of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='01 mJ/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The SHG signal generated in transmission configuration was appropriately filtered, then detected by an avalanche photodiode and amplified using a lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The scheme of SHG measurements is shown in Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For the x-ray studies, the Bruker D8 GADDS system was utilised: parallel CuK\uf061 beam formed by Goebel mirror monochromator, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='5 mm collimator, modified Linkam heating stage, Vantec 2000 area detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The samples for the diffraction experiments were prepared in a form of droplets on heated surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' SHG measurement scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Mesomorphic properties Clanek o kapalnych krystalech – Vladka Novotna Chopper Mirror 800 nm Filters ND, High pass Mirror 400 nm Lens Avalanche photodiode Sample in HG cell Linkam stage Boxcar/Lock-in Lock-in reference Temperature controller 10 Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Vitrification process and creation of a fibre after melting of NF5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The microphotograph of the texture for homologue NF2 detected in a 5 \uf06dm HG cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The width of the photo corresponds to about 200\uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The microphotograph of NF4 homologue in 5 \uf06dm HG-A cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The Polariser orientation (white) and the rubbing direction (red) are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' R 50μm11 Figure S5 The texture of NF5 in 5 \uf06dm HG-A cell under a microscope with (a) crossed polarisers, (b) and (c) with uncrossed position of polarisers (analyser rotated by an angle about 20 degrees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' All figures show the same part of the sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' red arrow represents the rubbing direction and white arrows indicate the orientation of polarisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The microphotograph of NF6 homologue in 5 \uf06dm HG-P cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The rubbing direction, R, is marked with a red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' (a) (b) C 50μm R50um12 Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The photo of NF6 homologue in 5 \uf06dm HG-A cell after the application of the external electric field of about 2 V/\uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The rubbing direction, R, is marked with a red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' The texture of NF6 in a special home-made gap-cell with a thickness of about 35 \uf06dm, defined by two copper electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' One electrode is located at the right upper corner out of the figure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' the orientation of the applied electric field, E, is marked with the black arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For (a) no electric field was applied and for (b) the electric field of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='2 V/\uf06dm was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' R P 50μm(a) (b) E13 Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 3D-plot of (a) real, \uf065’, and (b) imaginary, \uf065’’, parts of the permittivity versus frequency and temperature, T, for compound NF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Dielectric measurements were performed in 12 \uf06dm cell with gold electrodes and no surfactant layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' (a) 25000 20000 15000 3 10000 5000 102 40 frequency (Hz) 103 60 104 80 100 105 120 (°C) T 140 106 160 (b) 8000 6000 3 4000 2000 Q 102× 40 frequency (Hz) 103× 60 80 104 100 105 120 140 T (C) 106 16014 Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 3D-plot of (a) real, \uf065’, and (b) imaginary, \uf065’’, parts of the permittivity versus frequency and temperature, T, for compound NF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Dielectric measurements were performed in 12 \uf06dm cell with gold electrodes and no surfactant layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' (a) 20000- 15000 000013 5000 60 80 101 100 102 120 (0d 103 frequency (Hz) 104 105 140 106 (b) 6000 5000 4000 3 3000 2000 1000 60 80 101 102 100 103 frequency (Hz) 104 120 105 140 10615 Figure S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 3D-plot of (a) real, \uf065’, and (b) imaginary, \uf065’’, parts of the permittivity versus frequency and temperature, T, for compound NF6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Dielectric measurements were performed in 12 \uf06dm cell with gold electrodes and no surfactant layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' (a) 18000 16000 14000 12000 10000 32 8000 6000 4000 2000 30 0 40 101 50 102 °℃) 103 60 frequency (Hz) 104 70 105 80 106 (b) 8000 6000 4000 2000 30 40 101 50 102 103 60 frequency (Hz) 104 (0。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 70 105 106 8016 Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Temperature dependences of the dielectric strength, \uf044\uf065, and the relaxation frequency, fr, for NF5 in 12 \uf06dm cell without surfactant layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' In the inset fr is presented in the logarithmic scale versus reciprocal temperature, 1/T, in Kelvins and the activation energy EA was established from the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' 400 14 350 12 6 300 10 5 250 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0) 4 8 m 3 200 (zH) 6 2 =94 kJ/mol 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='0032 4 100 10" RT (Jmol\') 2 50 0 0 30 40 50 60 70 80 T(C)17 Figure S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' For NF5 at the temperature T=30° C (a) the x-ray intensity versus the scattering angle, \uf051\uf02c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' (b) 2D pattern of the intensity at the same temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Scattering angles are in the logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' Figure S14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' A model of NF1 molecule with the orientation of the dipole moment (blue arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content=' (a) Intensity(arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='units) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 A 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E4T4oBgHgl3EQfDAsR/content/2301.04865v1.pdf'} +page_content='4 A 22.' metadata={'source': 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2 Jan 2023 +The Hamilton-Jacobi analysis for higher-order modified gravity +Alberto Escalante∗ and Aldair Pantoja† +Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla. +Apartado Postal J-48 72570, Puebla Pue., M´exico, +(Dated: January 3, 2023) +The Hamilton-Jacobi [HJ] study for the Chern-Simons [CS] modification of general relativity +[GR] is performed. The complete structure of the Hamiltonians and the generalized brackets are +reported, from these results the HJ fundamental differential is constructed and the symmetries of +the theory are found. By using the Hamiltonians we remove an apparent Ostrogradsky’s instability +and the new structure of the hamiltonian is reported. In addition, the counting of physical degrees +of freedom is developed and some remarks are discussed. +PACS numbers: 98.80.-k,98.80.Qc +I. +INTRODUCTION +It is well-known that GR is a successful framework for describing the classical behavior of the grav- +itational field and its relation with the geometry of space-time [1–6]. From the canonical point of +view, GR is a background independent gauge theory with diffeomorphisms invariance; the extended +Hamiltonian is a linear combination of first class constraints and propagates two physical degrees +of freedom [7]. From the quantum point of view, the quantization program of gravity is a difficult +task to perform. In fact, from the nonperturbative scheme, the non-linearity of the gravitational +field, manifested in the constraints, obscures the quantization making the complete description of +a nonperturbative quantum theory of gravity still an open problem [8, 9]. On the other hand, the +perturbative point of view of the path-integral method leads to the non-renormalizability problem +[10, 11] with all the tools that have been developed in quantum field theory have not worked suc- +cessfully. In this respect, it is common to study modified theories of gravity in order to obtain +insights in the classical or quantum regime; with the expectation that these theories will provide +new ideas or allow the development of new tools to carry out the quantization program, with an +example of this being the so-called higher order theories [12–15]. In fact, higher-order theories are +good candidates for fixing the infinities that appear in the renormalization problem of quantum +gravity. It is claimed that adding higher order terms quadratic in the curvature to gravity could +help avoid this problem; since these terms have a dimensionless coupling constant, which ensures +∗Electronic address: aescalan@ifuap.buap.mx +†Electronic address: jpantoja@ifuap.buap.mx + +2 +that the final theory is divergence-free [16, 17]. The study of higher-order theories is a modern topic +in physics, these theories are relevant in dark energy physics [18, 19], generalized electrodynamics +[20–22] and string theories [23, 24]. Furthermore, an interesting model in four dimensions can be +found in the literature, in which the Einstein-Hilbert [EH] action is extended by the addition of +a Chern-Simons four-current coupled with an auxiliary field, thus, under a particular choice of the +auxiliary field the resulting action will be a close model to GR [25]. In fact, at Lagrangian level +the theory describes the propagation of two degrees of freedom corresponding to gravitational waves +traveling with velocity c, but these propagate with different polarization intensities violating spatial +reflection symmetry. Moreover, the Schwarzchild metric is a solution of the equations of motion, +thus, the modified theory and the EH action share the same classical tests. On the other hand, +at hamiltonian level the theory is a higher-order gauge theory [26] whose Hamiltonian analysis is +known not to be easy to perform. In this respect, the analysis of constrained higher-order systems is +usually developed by using the Ostrogradsky-Dirac [OD] [27–30] or the Gitman-Lyakhovich-Tyutin +[GLT ] [31, 32] methods. OD scheme is based on the extension of the phase space by considering +to the fields and their velocities as canonical coordinates and then introducing an extensi´on to their +canonical momenta. However, the identification of the constraints is not easy to develop; in some +cases, the constraints are fixed by hand in order to obtain a consistent algebra [33] and this yields +the opportunity to work with alternative methods. On the other hand, the GLT framework is based +on the introduction of extra variables which transforms a problem with higher time derivatives to +one with only first-order ones then, by using the Dirac brackets the second class constraints and the +extra variables can be removed [34]. +Nevertheless, there is an alternative scheme for analyzing higher-order theories: +the so-called +Hamilton-Jacobi method. The HJ scheme for regular field theories was developed by G¨uler [35, 36] +and later extended for singular systems in [37, 38]. It is based on the identification of the constraints, +called Hamiltonians. These Hamiltonians can be either involutive or non-involutive and they are used +for constructing a generalized differential, where the characteristic equations, the gauge symmetries, +and the generalized HJ brackets of the theory can be identified. It is important to remark that the +identification of the Hamiltonians is performed by means of the null vectors, thus, the Hamiltonians +will have the correct structure without fix them by hand as is done in other approaches, then the +identification of the symmetries will be, in general, more economical than other schemes [39–43]. +With all of above the aims of this paper is to develop a detailed HJ analysis of the theory reported +in [25]. +In fact, we shall analyze this model beyond the Lagrangian approach reported in [25]; +we shall see that the Jackiw-Yi [JY ] model is a higher-order theory and it is mandatory to study +this theory due to its closeness with GR. However, it is well-known that in higher-order theories +could be present ghost degrees of freedom associated to Ostrogradsky’s instabilities [44], namely, +the hamiltonian function is unbounded and this is reflected with the presence of linear terms of the +canonical momenta in the hamiltonian. In this respect, it is important to comment that if there are +constraints, then it is possible to heal those instabilities [45, 46]; in our case the JY model will show +an apparent Ostrogradsky’s instability since linear terms in the momenta will appear, however, we + +3 +will heal the theory by using the complete set of Hamiltonians, thereby exorcising the associated +ghosts. +The paper is organized as follows. In Sect. II, we start with the CS modification of GR, we will work +in the perturbative context, say, we will expand the metric around the Minkowski background. We +shall observe that the modified theory is of higher-order in the temporal derivatives, then we shall +introduce a change of variables in order to express the action in terms of only first-order temporal +derivatives. The change of variable will allows us to develop the HJ analysis in an easy way; the +identification of the Hamiltonians, the construction of the generalized differential and the symmetries +will be identified directly. In Sect. III we present the conclusions and some remarks. +II. +THE HAMILTON-JACOBI ANALYSIS +The modified EH action is given by [25] +S[gµν] = +� +M +� +R√−g + 1 +4θ∗Rσ +τ +µνRτ +σµν +� +d4x, +(1) +where M is the space-time manifold, gµν the metric tensor, R the scalar curvature, g the determinant +of the metric, Rαβµν the Riemman tensor and θ is a coupling field. In general, θ can be viewed as +an external quantity or as a local dynamical variable, however, in order to obtain an action close to +GR we are going to choose θ = t +Ω. Along the paper we will use grek letters for labeling space-time +indices µ = 0, 1, 2, 3 and latin letters for space indices i = 1, 2, 3. In addition, we will work within +the perturbative context expanding the metric around the Minkowski background +gµν = ηµν + hµν, +(2) +where hµν is the perturbation. By substituting the expression for θ and by taking into account eq. +(2) in (1) we obtain the following linearized action +S[hµν] = −1 +2 +� +M +hµν � +Glin +µν + Clin +µν +� +d4x, +(3) +where Glin +µν is the linearized version of the Einstein tensor and Clin +µν is a linearized Cotton-type tensor +Clin +µν = − 1 +4Ω[ǫ0µλγ∂λ(□hγν − ∂ν∂αhαγ)+ ǫ0νλγ∂λ(□hγµ − ∂µ∂αhαγ)] [25] defined in four-dimensions. +Now we shall suppose that the space-time has a topology M ∼= R × Σ, where R is an evolution +parameter and Σ is a Cauchy hypersurface. Hence, by performing the 3 + 1 decomposition of the +action (3) we write down the corresponding Lagrangian density +L = +� �1 +2 +˙hij ˙hij − ∂jh0i∂jh0i − 1 +2∂khij∂khij − 1 +2 +˙hii ˙hjj + ∂jh00∂jhii + 1 +2∂khii∂khjj − 2∂ih0i ˙hjj +−∂ih00∂jhij − ∂ihij∂jhkk + 2∂jh0i ˙hij + ∂ihi0∂jh0j + ∂khki∂jhij + 1 +µǫijk(−¨hli∂jhlk ++2˙hli∂j∂lh0k + ∂lhmi∂m∂jhlk + ∇2h0i∂jh0k + ∇2hmi∂jhmk) +� +d3x, +(4) +where we have defined µ ≡ 2Ω and ǫijk ≡ ǫ0ijk. As it was commented above, we will reduce the +order of the time derivatives of the Lagrangian (4) by extending the configuration space, this is done + +4 +by introducing the following change of variable +Kij = 1 +2(˙hij − ∂ih0j − ∂jh0i), +(5) +here Kij is related with the so-called extrinsic curvature [47, 48]. Thus, by substituting (5) into (4) +we rewrite the Lagrangian in the following new fashion +L = +� � +2KijKij − 2KiiKjj − h00Rijij − hijRij + 1 +2hiiRijij + 1 +µǫijk(4Kil∂jKkl + ∂mhim∂j∂lhkl ++∇2him∂jhkm) + ψij(˙hij − ∂ih0j − ∂jh0i − 2Kij) +� +d3x, +(6) +where we have added the Lagrange multipliers ψij enforcing the the relation (5), and the expressions +Rijij and Rij are defined in the following way +Rijij ≡ ∂i∂jhij − ∇2hii, +(7) +Rij ≡ 1 +2(∂i∂khjk + ∂j∂khik − ∂i∂jhkk − ∇2hij). +(8) +Now, we calculate the canonical momenta associated with the dynamical variables +π00 = +∂L +∂ ˙h00 += 0, +(9) +π0i = +∂L +∂ ˙h0i += 0, +(10) +πij = +∂L +∂ ˙hij += ψij, +(11) +P ij = +∂L +∂ ˙Kij += 0, +(12) +Λij = +∂L +∂ ˙ψij += 0. +(13) +Thus, from the equations (9)-(13) we identify the following HJ Hamiltonians of the theory +H′ ≡ H0 + Π = 0, +(14) +H00 +1 +≡ π00 = 0, +(15) +H0i +2 +≡ π0i = 0, +(16) +Hij +3 +≡ πij − ψij = 0, +(17) +Hij +4 +≡ P ij = 0, +(18) +Hij +5 +≡ Λij = 0, +(19) +where H0 is the canonical hamiltonian defined as usual H0 = ˙hµνπµν + ˙KijP ij + ˙ψijΛij − L and +Π = ∂0S [39–43]. Moreover, the fundamental Poisson brackets [PB] between the canonical variables + +5 +are given by +{hµν, παβ} = 1 +2(δα +µδβ +ν + δα +ν δβ +µ)δ3(x − y), +(20) +{Kij, πkl} = 1 +2(δk +i δl +j + δk +j δl +i)δ3(x − y), +(21) +{ψij, Λkl} = 1 +2(δi +kδj +l + δj +kδi +l)δ3(x − y). +(22) +Furthermore, in the HJ scheme, the dynamics of the system is governed by the fundamental differ- +ential defined as +dF = {F, HI}dωI, +(23) +where F is any function defined on the phase space, HI is the set of all Hamiltonians (14)-(19) +and ωI are the parameters related to them. It is important to remark, that in the HJ method the +Hamiltonians are classified as involutive and non-involutive. Involutive ones are those whose PB +with all Hamiltonians, including themselves, vanish; otherwise, they are called non-involutive. Be- +cause of integrability conditions, the non-involutive Hamiltonians are removed from the fundamental +differential (23) by introducing the so-called generalized brackets, these new brackets are given by +{f, g}∗ = {f, g} − {f, Ha′}C−1 +a′b′{Hb′, g}, +(24) +where Ca′b′ is the matrix formed with the PB between all non-involutive Hamiltonians. +From +(14)-(19) the non-involutive Hamiltonians are Hij +3 and Hij +5 , whose PB is +{Hij +3 , Hij +5 } = −1 +2(ηikηjl + ηilηkj)δ3(x − y), +(25) +therefore, the matrix Ca′b′ given by +Ca′b′ = + + +0 +− 1 +2(ηikηjl + ηilηkj) +1 +2(ηikηjl + ηilηkj) +0 + +δ3(x − y), +(26) +and its inverse C−1 +a′b′ takes the form +C−1 +a′b′ = + + +0 +1 +2(ηikηjl + ηilηkj) +− 1 +2(ηikηjl + ηilηkj) +0 + + δ3(x − y). +(27) +In this manner, the following non-vanishing generalized brackets between the fields arise +{hµν, παβ}∗ = 1 +2(δα +µδβ +ν + δβ +µδα +ν )δ3(x − y), +(28) +{Kij, P kl}∗ = 1 +2(δk +i δl +j + δl +iδk +j )δ3(x − y), +(29) +{hµν, ψαβ}∗ = 1 +2(δα +µδβ +ν + δβ +µδα +ν )δ3(x − y), +(30) +{ψij, Λkl}∗ = 0, +(31) +we observe from (31) that the canonical variables (ψij, Λkl) can be removed which implies that we +can perform the substitution of πij = ψij and Λij = 0, hence, the canonical hamiltonian takes the + +6 +form +H0 = +� +[2KiiKjj − 2KijKij + h00Rijij + hijRij − 1 +2hiiRijij − 1 +µǫijk(4Kil∂jKkl + ∂mhim∂j∂lhkl ++∇2him∂jhkm) − 2h0j∂iπij + 2Kijπij]d3x. +(32) +It is worth to comment, that the canonical hamiltonian has linear terms in the momenta πij and +this fact could be related to Ostrogradsky’s instabilities. Nevertheless, it is well-known that those +instabilities could be healed by means the correct identification of the constraints [45, 46]. In this +respect, an advantage of the HJ scheme is that the constraints are identified directly and it is +not necessary to fix them by hand, then with the generalized brackets and the identification of the +Hamiltonians we can remove the linear canonical momenta terms. In fact, by using the Hamiltonians +(14)-(19) the canonical hamiltonian takes the following form +H′ +0 = +� +[1 +2πijπij − 1 +4πiiπjj + hijRij − 1 +µǫijk(4Kil∂jKkl + ∂mhim∂j∂lhkl + ∇2hil∂jhkl) +− 4 +µ2 (2∂iKij∂jKkk + 2∂iKjk∂iKjk − 2∂jKik∂iKjk − ∂jKik∂kKij − ∂kKii∂kKjj]d3x. +hence, the Ostrogradsky instability has been healed and the associated ghost was exorcised. +On the other hand, with all these results we rewrite the fundamental differential in terms of either +involutive Hamiltonians or generalized brackets, this is +dF = +� +[{F, H′}∗dt + {F, H00 +1 }∗dω1 +00 + {F, H0i +2 }∗dω2 +0i + {F, Hij +4 }∗dω4 +ij]d3y. +(33) +thus, we will search if there are more Hamiltonians in the theory. For this aim, we shall take into +account either the generalized differential (33) or the Frobenius integrability conditions which, ensure +that system is integrable, this is +dHa = 0, +(34) +where Ha ≡ (H00 +1 , H0i +2 , Hij +4 ) are all involutive Hamiltonians. From integrability conditions (34) the +following 10 new Hamiltonians arise +H00 +6 +≡ ∇2hii − ∂i∂jhij = 0, +(35) +H0i +7 +≡ ∂jπij = 0, +(36) +Hij +8 +≡ πij − 2Kij + 2ηijKkk − 2 +µ(ǫiklηjm + ǫjklηim)∂kKlm = 0, +(37) +Now, we observe that the Hamiltonians Hij +4 , H00 +6 +and H8 are non-involutive, therefore they will be +removed by introducing a new set of generalized brackets. In this respect, if we calculate the matrix +whose entries will be all generalized brackets, say (28)-(31), between the non-involutive Hamiltonians, +we will find null vectors, say vi = ( 1 +2∂i∂jζ, δikζ, 0), where ζ is an arbitrary function. Hence, from the +contraction of the null vectors with the Hamiltonians [42, 43], we will find the following involutive +Hamiltonian +H9 = ∇2hii − ∂i∂jhij + 1 +2∂i∂jP ij, +(38) + +7 +thus, there are only 12 non-involutive Hamiltonians (Hij +4 , Hij +8 ) whose generalized brackets are given +by +{Hij +4 , Hij +8 }∗ = 2[ 1 +2µ(ǫikmηjl + ǫjkmηil + ǫilmηjk + ǫilmηik)∂m + 1 +2(ηikηjl ++ηjkηil) − ηijηkl]δ3(x − y). +(39) +In this manner, we proceed to construct the new set of HJ generalized brackets, namely { , }∗∗, in +the same way as we did before with the brackets (28)-(31). The non-trivial new generalized brackets +are given by +{hij, πkl}∗∗ = 1 +2(δk +i δl +j + δl +iδk +j )δ3(x − y), +(40) +{Kij, P kl}∗∗ = 0, +(41) +{hij, Kkl}∗∗ = 1 +4(ηikηjl + ηilηjk − ηijηkl)δ3(x − y) + µ2 +4Ξ[[(ηikηjl + ηilηjk − ηijηkl)∇2 + (ηij∂k∂l ++ηkl∂i∂j)](∇2 + µ2) − 3∂i∂j∂k∂l − 3µ2 +4 (ηik∂j∂l + ηil∂j∂k + ηjk∂i∂l + ηjl∂i∂k) ++µ +4 [(ǫikmηjl + ǫjkmηil + ǫilmηjk + ǫjlmηik)(∇2 + µ2) + 3(ǫikm∂j∂l + ǫjkm∂i∂l ++ǫilm∂j∂k + ǫjlm∂i∂k)]∂m]δ3(x − y), +(42) +where Ξ ≡ −µ2(∇2 + µ2)(∇2 + µ2 +4 ). It is worth commenting, that some brackets were reported in +[26], however, there are some differences. In fact, in this paper we have used an alternative analy- +sis and new variables were introduced; the introduction of the variables allowed us to identify the +brackets (42) directly and they have a more compact form than those reported in [26]. Moreover, +the tedious classification of the constrains into first class and second class as usually is done, in the +HJ scheme it is not necessary. Thus, we can observe that the HJ is more economical. +With the new set of either involutives Hamiltonians or generalized brackets, the fundamental differ- +ential takes the following new form +dF = +� +[{F, H′(y)}∗∗dt + {F, H00 +1 (y)}∗∗dω1 +00 + {F, H0i +2 (y)}∗∗dω2 +0i + {F, H0i +7 (y)}∗∗dω7 +0i ++ {F, H9(y)}∗∗dω9]d3y, +(43) +where +H00 +1 += π00, +(44) +H0i +2 += π0i, +(45) +H0i +7 += ∂jπij, +(46) +H9 = ∇2hii − ∂i∂jhij. +(47) +From integrability conditions of H0i +7 and H9 we find +dH0i +7 += 0, +(48) +dH9 = −∂i∂jπij = −∂iH0i +7 = 0, +(49) + +8 +therefore, there are not further Hamiltonians. It is worth to comment, that the Hamiltonians given +in (47) are related to those reported in [49] where only linearized gravity was studied. However, there +are differences: from on side, the PB reported in [49] and the generalized brackets found in (40)-(42) +are different. On the other hand, the contribution of the modification is present in the generalized +brackets, and this fact will be relevant in the study of quantization because the generalized brackets +will be changed to commutators and the contribution could provide differences with respect standard +linearized gravity. +Now, we will calculate the HJ characteristic equations, they are given by +dh00 = dθ1 +00, +(50) +dh0i = 1 +2dθ2 +0i, +(51) +dhij = [2Kij + ∂ih0j + ∂jh0i]dt − 1 +2(δk +i ∂j + δk +j ∂i)dθ7 +0k, +(52) +dπ00 = −Rij +ijdt, +(53) +dπ0i = 1 +2∂jπijdt, +(54) +dπij = [ηij∇2h00 − ∂i∂jh00 − ηijRkl +kl − 2Rij − 1 +µ[(ǫikl∂j + ǫjkl∂i)∂k∂mhlm +−(ǫiklηjm + ǫjklηim)∂k∇2hlm]]dt + (∂i∂j − ηij∇2)dθ9, +(55) +dKij = [−1 +2∂i∂jh00 − Rij + 1 +4ηijRkl +kl]dt + 1 +2∂i∂jdθ9, +(56) +dP ij = [0]dt, +(57) +from the characteristic equations we can identify the following facts: from equations (50)-(51) we +observe that the variables h00 and h0i are identified as Lagrange multipliers. Moreover, from (41) +and (57) we discard to P ij as degree of freedom because its time evolution vanishes. Furthermore, +we identify the equations of motion for hij and its momentum πij. In fact, by taking dθ7 +0k = 0 and +dθ9 = 0, we obtain +˙hij = 2Kij + ∂ih0j + ∂jh0i, +(58) +˙πij = ηij∇2h00 − ∂i∂jh00 − ηijRkl +kl − 2Rij − 1 +µ[(ǫikl∂j + ǫjkl∂i)∂k∂mhlm +−(ǫiklηjm + ǫjklηim)∂k∇2hlm], +(59) +˙Kij = −1 +2∂i∂jh00 − Rij + 1 +4ηijRklkl. +(60) +We observe that (58) corresponds to the definition of Kij, thus, if we use (58) and +˙Kij we will +obtain a second order time equation for hij as expected, then there are six degrees of freedom +associated with the perturbation. In this manner, we calculate the number of physical degrees of +freedom as follows: there are 12 canonical variables (hij, πij) and eight involutive Hamiltonians +(H00 +1 , H0i +2 , H0i +7 , H9), thus +DOF = 1 +2[12 − 8] = 2, +and thus, the theory has two physical degrees of freedom just like GR [25, 26]. +On the other hand, if in the characteristics equations we take dt = 0, then we identify the following + +9 +canonical transformations +δh00 = δω1 +00, +(61) +δh0i = 1 +2δω2 +0i, +(62) +δhij = −1 +2(δk +i ∂j + δk +j ∂i)δω7 +0k, +(63) +moreover, we can then identify the corresponding gauge transformations of the theory by considering +that the Lagrangian (6) will be invariant under (61)-(63) if the variation δS = 0 [50], this is +δS = +� ∂S +∂hµν +δhµν + +∂S +∂(∂αhµν)δ(∂αhµν) + +∂S +∂(∂α∂βhµν)δ(∂α∂βhµν) +� +(64) += +� �� +−□hµν + □hλληµν − ∂α∂λhαληµν − ∂µ∂νhλλ + 2∂µ∂λhνλ + 1 +µǫ0µλγ(∂ν∂α∂λhαγ +−∂λ□hν +γ)) δhµν] d4x = 0, +(65) +thus, by taking account (61)-(63) into the variation, we obtain the following +δS = +� +[Rij +ijδω1 +00 + 1 +2[2∇2h0 +i + 2∂i ˙hj +j − 2∂i∂jh0j − 2∂j ˙hij + 1 +µǫ0ijk(∂j∇2h0k − ∂j∂l ˙hkl)]δω2 +0i +−1 +2[¨hij − ¨hk +kηij + 2∂k ˙h0kηij − 2∂i ˙h0 +j + ∂i∂jh00 − ∇2h00ηij + 2Rij − Rkl +klηij ++ 1 +µǫ0ikl(∂k¨hjl − ∂j∂k ˙h0l + ∂j∂k∂mhlm − ∂k∇2hjl)]δ(∂iω7 +0j + ∂jω7 +0i)]d4x = 0. +(66) +Now, we define ∂0ξ ≡ δω1 +00, so after long algebraic work we find that the variation takes the form +δS = +� +[−∂j ˙hij + ∂ihjj + ∇2h0i − ∂i∂jh0j + 1 +2µǫ0ijk(∂j∇2h0k − ∂j∂l ˙hkl)](−∂iξ + δω2 +0i + ∂0δω7 +0i)d4x, += 0, +(67) +hence, the action will be invariant under (61)-(63) if the the parameters ω′s obey +δω2 +0i = −∂0δω7 +0i + ∂iξ. +(68) +Now, we will write (68) in a new fashion. +In fact, we introduce the following 4-vector ξµ ≡ +( 1 +2ξ, − 1 +2δω7 +0i) ≡ (ξ0, ξi); then ξ = 2ξ0 and δω7 +0i = −2ξi. Hence, the relation (68) takes the form +1 +2δω2 +0i = ∂0ξi + ∂iξ0, +(69) +finally, from the equations (61)-(63) and (69) the following gauge transformations are identified +δhµν = ∂µξν + ∂νξµ. +(70) +all these results are in agreement with those reported in [26], thus, our study complete and extends +those reported in the literature. +III. +CONCLUSIONS AND REMARKS +In this paper a detailed HJ analysis for the higher-order modified gravity has been performed. +We introduced a new set of variables in a different way than other approaches and reported in + +10 +the literature, then the full set of involutive and non-involutive Hamiltonians were identified. +The correct identification of the Hamiltonians allow us to avoid the Ostrogradsky instability by +removing the terms with linear momenta, healing the canonical Hamiltonian. Furthermore, the HJ +generalized brackets and the fundamental differential were obtained from which the characteristic +equations and the gauge symmetries were identified. The complete identification of the Hamiltoni- +ans allowed us to carry out the counting of the physical degrees of freedom, concluding that the +modified theory and GR shares the same number of physical degrees of freedom. In this manner, we +have all elements to analize the theory in the quantum context. In fact, with our perturbative HJ +study either constraints or the generalized brackets are under control, thus, we could use the tools +developed in the canonical quantization of field theories in order to make progress in this program +[51]. Furthermore, our analysis will be relevant for the study of the theory in the non-perturbative +scenario. 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Ganz and K. Noui, Reconsidering the Ostrogradsky theorem: higher-derivatives Lagrangians, ghost +and degeneracy, Class. Quantum Grav. 38, 075005 (2021). +[46] Tai-jun Chen, M. Fasiello, Eugene A. Lim, Andrew J. Tolley, Higher derivative theories with constraints: +Exorcising Ostrogradski’s Ghost, JCAP 130, 042, (2013). +[47] T. Frankel, The Geometry of Physics 3rd, Cambridge University Press, (2012). +[48] H. Fuhri, S. Hortner, Phys. Rev. D 103, 105014, (2021). +[49] M. Bertin, B. Pimentel, C. Valcarcel and G. Zambrano, Hamilton-Jacobi formalism for linearized gravity, +Class. Quantum Grav. 28, 175015 (2011). +[50] M.C. Bertin, B.M. Pimentel, C.E. Valc´arcel, G.E.R. Zambrano, Involutive constrained systems and +Hamilton-Jacobi formalism, J. Math. Phys. 55, 112901 (2014). +[51] R. Amorim and J. Barcelos, Functional versus canonical quantization of nonlocal massive vector-gauge +theory, J. Math. Phys. 40, 585 (1999). +[52] A. Escalante and A. Pantoja, The perturbative and non-perturbative canonical analysis of the Chern- +Simons modification of General Relativity, in progress. + diff --git a/CtAyT4oBgHgl3EQf4fpz/content/tmp_files/load_file.txt b/CtAyT4oBgHgl3EQf4fpz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2821201daa9fb08c3727f7e1923c112b754678fe --- /dev/null +++ b/CtAyT4oBgHgl3EQf4fpz/content/tmp_files/load_file.txt @@ -0,0 +1,397 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf,len=396 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content='00787v1 [gr-qc] 2 Jan 2023 The Hamilton-Jacobi analysis for higher-order modified gravity Alberto Escalante∗ and Aldair Pantoja† Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Apartado Postal J-48 72570, Puebla Pue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=', M´exico, (Dated: January 3, 2023) The Hamilton-Jacobi [HJ] study for the Chern-Simons [CS] modification of general relativity [GR] is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' The complete structure of the Hamiltonians and the generalized brackets are reported, from these results the HJ fundamental differential is constructed and the symmetries of the theory are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' By using the Hamiltonians we remove an apparent Ostrogradsky’s instability and the new structure of the hamiltonian is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In addition, the counting of physical degrees of freedom is developed and some remarks are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' PACS numbers: 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content='-k,98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content='Qc I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' INTRODUCTION It is well-known that GR is a successful framework for describing the classical behavior of the grav- itational field and its relation with the geometry of space-time [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' From the canonical point of view, GR is a background independent gauge theory with diffeomorphisms invariance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' the extended Hamiltonian is a linear combination of first class constraints and propagates two physical degrees of freedom [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' From the quantum point of view, the quantization program of gravity is a difficult task to perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In fact, from the nonperturbative scheme, the non-linearity of the gravitational field, manifested in the constraints, obscures the quantization making the complete description of a nonperturbative quantum theory of gravity still an open problem [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' On the other hand, the perturbative point of view of the path-integral method leads to the non-renormalizability problem [10, 11] with all the tools that have been developed in quantum field theory have not worked suc- cessfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In this respect, it is common to study modified theories of gravity in order to obtain insights in the classical or quantum regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' with the expectation that these theories will provide new ideas or allow the development of new tools to carry out the quantization program, with an example of this being the so-called higher order theories [12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In fact, higher-order theories are good candidates for fixing the infinities that appear in the renormalization problem of quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' It is claimed that adding higher order terms quadratic in the curvature to gravity could help avoid this problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' since these terms have a dimensionless coupling constant, which ensures ∗Electronic address: aescalan@ifuap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content='buap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content='mx †Electronic address: jpantoja@ifuap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content='buap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content='mx 2 that the final theory is divergence-free [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' The study of higher-order theories is a modern topic in physics, these theories are relevant in dark energy physics [18, 19], generalized electrodynamics [20–22] and string theories [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Furthermore, an interesting model in four dimensions can be found in the literature, in which the Einstein-Hilbert [EH] action is extended by the addition of a Chern-Simons four-current coupled with an auxiliary field, thus, under a particular choice of the auxiliary field the resulting action will be a close model to GR [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In fact, at Lagrangian level the theory describes the propagation of two degrees of freedom corresponding to gravitational waves traveling with velocity c, but these propagate with different polarization intensities violating spatial reflection symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Moreover, the Schwarzchild metric is a solution of the equations of motion, thus, the modified theory and the EH action share the same classical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' On the other hand, at hamiltonian level the theory is a higher-order gauge theory [26] whose Hamiltonian analysis is known not to be easy to perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In this respect, the analysis of constrained higher-order systems is usually developed by using the Ostrogradsky-Dirac [OD] [27–30] or the Gitman-Lyakhovich-Tyutin [GLT ] [31, 32] methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' OD scheme is based on the extension of the phase space by considering to the fields and their velocities as canonical coordinates and then introducing an extensi´on to their canonical momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' However, the identification of the constraints is not easy to develop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' in some cases, the constraints are fixed by hand in order to obtain a consistent algebra [33] and this yields the opportunity to work with alternative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' On the other hand, the GLT framework is based on the introduction of extra variables which transforms a problem with higher time derivatives to one with only first-order ones then, by using the Dirac brackets the second class constraints and the extra variables can be removed [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Nevertheless, there is an alternative scheme for analyzing higher-order theories: the so-called Hamilton-Jacobi method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' The HJ scheme for regular field theories was developed by G¨uler [35, 36] and later extended for singular systems in [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' It is based on the identification of the constraints, called Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' These Hamiltonians can be either involutive or non-involutive and they are used for constructing a generalized differential, where the characteristic equations, the gauge symmetries, and the generalized HJ brackets of the theory can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' It is important to remark that the identification of the Hamiltonians is performed by means of the null vectors, thus, the Hamiltonians will have the correct structure without fix them by hand as is done in other approaches, then the identification of the symmetries will be, in general, more economical than other schemes [39–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' With all of above the aims of this paper is to develop a detailed HJ analysis of the theory reported in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In fact, we shall analyze this model beyond the Lagrangian approach reported in [25];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' we shall see that the Jackiw-Yi [JY ] model is a higher-order theory and it is mandatory to study this theory due to its closeness with GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' However, it is well-known that in higher-order theories could be present ghost degrees of freedom associated to Ostrogradsky’s instabilities [44], namely, the hamiltonian function is unbounded and this is reflected with the presence of linear terms of the canonical momenta in the hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In this respect, it is important to comment that if there are constraints, then it is possible to heal those instabilities [45, 46];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' in our case the JY model will show an apparent Ostrogradsky’s instability since linear terms in the momenta will appear, however, we 3 will heal the theory by using the complete set of Hamiltonians, thereby exorcising the associated ghosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' II, we start with the CS modification of GR, we will work in the perturbative context, say, we will expand the metric around the Minkowski background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' We shall observe that the modified theory is of higher-order in the temporal derivatives, then we shall introduce a change of variables in order to express the action in terms of only first-order temporal derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' The change of variable will allows us to develop the HJ analysis in an easy way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' the identification of the Hamiltonians, the construction of the generalized differential and the symmetries will be identified directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' III we present the conclusions and some remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' THE HAMILTON-JACOBI ANALYSIS The modified EH action is given by [25] S[gµν] = � M � R√−g + 1 4θ∗Rσ τ µνRτ σµν � d4x, (1) where M is the space-time manifold, gµν the metric tensor, R the scalar curvature, g the determinant of the metric, Rαβµν the Riemman tensor and θ is a coupling field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In general, θ can be viewed as an external quantity or as a local dynamical variable, however, in order to obtain an action close to GR we are going to choose θ = t Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Along the paper we will use grek letters for labeling space-time indices µ = 0, 1, 2, 3 and latin letters for space indices i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In addition, we will work within the perturbative context expanding the metric around the Minkowski background gµν = ηµν + hµν, (2) where hµν is the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' By substituting the expression for θ and by taking into account eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (2) in (1) we obtain the following linearized action S[hµν] = −1 2 � M hµν � Glin µν + Clin µν � d4x, (3) where Glin µν is the linearized version of the Einstein tensor and Clin µν is a linearized Cotton-type tensor Clin µν = − 1 4Ω[ǫ0µλγ∂λ(□hγν − ∂ν∂αhαγ)+ ǫ0νλγ∂λ(□hγµ − ∂µ∂αhαγ)] [25] defined in four-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Now we shall suppose that the space-time has a topology M ∼= R × Σ, where R is an evolution parameter and Σ is a Cauchy hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Hence, by performing the 3 + 1 decomposition of the action (3) we write down the corresponding Lagrangian density L = � �1 2 ˙hij ˙hij − ∂jh0i∂jh0i − 1 2∂khij∂khij − 1 2 ˙hii ˙hjj + ∂jh00∂jhii + 1 2∂khii∂khjj − 2∂ih0i ˙hjj −∂ih00∂jhij − ∂ihij∂jhkk + 2∂jh0i ˙hij + ∂ihi0∂jh0j + ∂khki∂jhij + 1 µǫijk(−¨hli∂jhlk +2˙hli∂j∂lh0k + ∂lhmi∂m∂jhlk + ∇2h0i∂jh0k + ∇2hmi∂jhmk) � d3x, (4) where we have defined µ ≡ 2Ω and ǫijk ≡ ǫ0ijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' As it was commented above, we will reduce the order of the time derivatives of the Lagrangian (4) by extending the configuration space, this is done 4 by introducing the following change of variable Kij = 1 2(˙hij − ∂ih0j − ∂jh0i), (5) here Kij is related with the so-called extrinsic curvature [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Thus, by substituting (5) into (4) we rewrite the Lagrangian in the following new fashion L = � � 2KijKij − 2KiiKjj − h00Rijij − hijRij + 1 2hiiRijij + 1 µǫijk(4Kil∂jKkl + ∂mhim∂j∂lhkl +∇2him∂jhkm) + ψij(˙hij − ∂ih0j − ∂jh0i − 2Kij) � d3x, (6) where we have added the Lagrange multipliers ψij enforcing the the relation (5), and the expressions Rijij and Rij are defined in the following way Rijij ≡ ∂i∂jhij − ∇2hii, (7) Rij ≡ 1 2(∂i∂khjk + ∂j∂khik − ∂i∂jhkk − ∇2hij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (8) Now, we calculate the canonical momenta associated with the dynamical variables π00 = ∂L ∂ ˙h00 = 0, (9) π0i = ∂L ∂ ˙h0i = 0, (10) πij = ∂L ∂ ˙hij = ψij, (11) P ij = ∂L ∂ ˙Kij = 0, (12) Λij = ∂L ∂ ˙ψij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (13) Thus, from the equations (9)-(13) we identify the following HJ Hamiltonians of the theory H′ ≡ H0 + Π = 0, (14) H00 1 ≡ π00 = 0, (15) H0i 2 ≡ π0i = 0, (16) Hij 3 ≡ πij − ψij = 0, (17) Hij 4 ≡ P ij = 0, (18) Hij 5 ≡ Λij = 0, (19) where H0 is the canonical hamiltonian defined as usual H0 = ˙hµνπµν + ˙KijP ij + ˙ψijΛij − L and Π = ∂0S [39–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Moreover, the fundamental Poisson brackets [PB] between the canonical variables 5 are given by {hµν, παβ} = 1 2(δα µδβ ν + δα ν δβ µ)δ3(x − y), (20) {Kij, πkl} = 1 2(δk i δl j + δk j δl i)δ3(x − y), (21) {ψij, Λkl} = 1 2(δi kδj l + δj kδi l)δ3(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (22) Furthermore, in the HJ scheme, the dynamics of the system is governed by the fundamental differ- ential defined as dF = {F, HI}dωI, (23) where F is any function defined on the phase space, HI is the set of all Hamiltonians (14)-(19) and ωI are the parameters related to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' It is important to remark, that in the HJ method the Hamiltonians are classified as involutive and non-involutive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Involutive ones are those whose PB with all Hamiltonians, including themselves, vanish;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' otherwise, they are called non-involutive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Be- cause of integrability conditions, the non-involutive Hamiltonians are removed from the fundamental differential (23) by introducing the so-called generalized brackets, these new brackets are given by {f, g}∗ = {f, g} − {f, Ha′}C−1 a′b′{Hb′, g}, (24) where Ca′b′ is the matrix formed with the PB between all non-involutive Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' From (14)-(19) the non-involutive Hamiltonians are Hij 3 and Hij 5 , whose PB is {Hij 3 , Hij 5 } = −1 2(ηikηjl + ηilηkj)δ3(x − y), (25) therefore, the matrix Ca′b′ given by Ca′b′ = \uf8eb \uf8ed 0 − 1 2(ηikηjl + ηilηkj) 1 2(ηikηjl + ηilηkj) 0 \uf8f6 \uf8f8δ3(x − y), (26) and its inverse C−1 a′b′ takes the form C−1 a′b′ = \uf8eb \uf8ed 0 1 2(ηikηjl + ηilηkj) − 1 2(ηikηjl + ηilηkj) 0 \uf8f6 \uf8f8 δ3(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (27) In this manner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' the following non-vanishing generalized brackets between the fields arise {hµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' παβ}∗ = 1 2(δα µδβ ν + δβ µδα ν )δ3(x − y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (28) {Kij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' P kl}∗ = 1 2(δk i δl j + δl iδk j )δ3(x − y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (29) {hµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' ψαβ}∗ = 1 2(δα µδβ ν + δβ µδα ν )δ3(x − y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (30) {ψij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Λkl}∗ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (31) we observe from (31) that the canonical variables (ψij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Λkl) can be removed which implies that we can perform the substitution of πij = ψij and Λij = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' the canonical hamiltonian takes the 6 form H0 = � [2KiiKjj − 2KijKij + h00Rijij + hijRij − 1 2hiiRijij − 1 µǫijk(4Kil∂jKkl + ∂mhim∂j∂lhkl +∇2him∂jhkm) − 2h0j∂iπij + 2Kijπij]d3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (32) It is worth to comment, that the canonical hamiltonian has linear terms in the momenta πij and this fact could be related to Ostrogradsky’s instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Nevertheless, it is well-known that those instabilities could be healed by means the correct identification of the constraints [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In this respect, an advantage of the HJ scheme is that the constraints are identified directly and it is not necessary to fix them by hand, then with the generalized brackets and the identification of the Hamiltonians we can remove the linear canonical momenta terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In fact, by using the Hamiltonians (14)-(19) the canonical hamiltonian takes the following form H′ 0 = � [1 2πijπij − 1 4πiiπjj + hijRij − 1 µǫijk(4Kil∂jKkl + ∂mhim∂j∂lhkl + ∇2hil∂jhkl) − 4 µ2 (2∂iKij∂jKkk + 2∂iKjk∂iKjk − 2∂jKik∂iKjk − ∂jKik∂kKij − ∂kKii∂kKjj]d3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' hence, the Ostrogradsky instability has been healed and the associated ghost was exorcised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' On the other hand, with all these results we rewrite the fundamental differential in terms of either involutive Hamiltonians or generalized brackets, this is dF = � [{F, H′}∗dt + {F, H00 1 }∗dω1 00 + {F, H0i 2 }∗dω2 0i + {F, Hij 4 }∗dω4 ij]d3y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (33) thus, we will search if there are more Hamiltonians in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' For this aim, we shall take into account either the generalized differential (33) or the Frobenius integrability conditions which, ensure that system is integrable, this is dHa = 0, (34) where Ha ≡ (H00 1 , H0i 2 , Hij 4 ) are all involutive Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' From integrability conditions (34) the following 10 new Hamiltonians arise H00 6 ≡ ∇2hii − ∂i∂jhij = 0, (35) H0i 7 ≡ ∂jπij = 0, (36) Hij 8 ≡ πij − 2Kij + 2ηijKkk − 2 µ(ǫiklηjm + ǫjklηim)∂kKlm = 0, (37) Now, we observe that the Hamiltonians Hij 4 , H00 6 and H8 are non-involutive, therefore they will be removed by introducing a new set of generalized brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In this respect, if we calculate the matrix whose entries will be all generalized brackets, say (28)-(31), between the non-involutive Hamiltonians, we will find null vectors, say vi = ( 1 2∂i∂jζ, δikζ, 0), where ζ is an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Hence, from the contraction of the null vectors with the Hamiltonians [42, 43], we will find the following involutive Hamiltonian H9 = ∇2hii − ∂i∂jhij + 1 2∂i∂jP ij, (38) 7 thus, there are only 12 non-involutive Hamiltonians (Hij 4 , Hij 8 ) whose generalized brackets are given by {Hij 4 , Hij 8 }∗ = 2[ 1 2µ(ǫikmηjl + ǫjkmηil + ǫilmηjk + ǫilmηik)∂m + 1 2(ηikηjl +ηjkηil) − ηijηkl]δ3(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (39) In this manner, we proceed to construct the new set of HJ generalized brackets, namely { , }∗∗, in the same way as we did before with the brackets (28)-(31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' The non-trivial new generalized brackets are given by {hij, πkl}∗∗ = 1 2(δk i δl j + δl iδk j )δ3(x − y), (40) {Kij, P kl}∗∗ = 0, (41) {hij, Kkl}∗∗ = 1 4(ηikηjl + ηilηjk − ηijηkl)δ3(x − y) + µ2 4Ξ[[(ηikηjl + ηilηjk − ηijηkl)∇2 + (ηij∂k∂l +ηkl∂i∂j)](∇2 + µ2) − 3∂i∂j∂k∂l − 3µ2 4 (ηik∂j∂l + ηil∂j∂k + ηjk∂i∂l + ηjl∂i∂k) +µ 4 [(ǫikmηjl + ǫjkmηil + ǫilmηjk + ǫjlmηik)(∇2 + µ2) + 3(ǫikm∂j∂l + ǫjkm∂i∂l +ǫilm∂j∂k + ǫjlm∂i∂k)]∂m]δ3(x − y), (42) where Ξ ≡ −µ2(∇2 + µ2)(∇2 + µ2 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' It is worth commenting, that some brackets were reported in [26], however, there are some differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In fact, in this paper we have used an alternative analy- sis and new variables were introduced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' the introduction of the variables allowed us to identify the brackets (42) directly and they have a more compact form than those reported in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Moreover, the tedious classification of the constrains into first class and second class as usually is done, in the HJ scheme it is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Thus, we can observe that the HJ is more economical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' With the new set of either involutives Hamiltonians or generalized brackets, the fundamental differ- ential takes the following new form dF = � [{F, H′(y)}∗∗dt + {F, H00 1 (y)}∗∗dω1 00 + {F, H0i 2 (y)}∗∗dω2 0i + {F, H0i 7 (y)}∗∗dω7 0i + {F, H9(y)}∗∗dω9]d3y, (43) where H00 1 = π00, (44) H0i 2 = π0i, (45) H0i 7 = ∂jπij, (46) H9 = ∇2hii − ∂i∂jhij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (47) From integrability conditions of H0i 7 and H9 we find dH0i 7 = 0, (48) dH9 = −∂i∂jπij = −∂iH0i 7 = 0, (49) 8 therefore, there are not further Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' It is worth to comment, that the Hamiltonians given in (47) are related to those reported in [49] where only linearized gravity was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' However, there are differences: from on side, the PB reported in [49] and the generalized brackets found in (40)-(42) are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' On the other hand, the contribution of the modification is present in the generalized brackets, and this fact will be relevant in the study of quantization because the generalized brackets will be changed to commutators and the contribution could provide differences with respect standard linearized gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' we will calculate the HJ characteristic equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' they are given by dh00 = dθ1 00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (50) dh0i = 1 2dθ2 0i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (51) dhij = [2Kij + ∂ih0j + ∂jh0i]dt − 1 2(δk i ∂j + δk j ∂i)dθ7 0k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (52) dπ00 = −Rij ijdt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (53) dπ0i = 1 2∂jπijdt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (54) dπij = [ηij∇2h00 − ∂i∂jh00 − ηijRkl kl − 2Rij − 1 µ[(ǫikl∂j + ǫjkl∂i)∂k∂mhlm −(ǫiklηjm + ǫjklηim)∂k∇2hlm]]dt + (∂i∂j − ηij∇2)dθ9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (55) dKij = [−1 2∂i∂jh00 − Rij + 1 4ηijRkl kl]dt + 1 2∂i∂jdθ9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (56) dP ij = [0]dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (57) from the characteristic equations we can identify the following facts: from equations (50)-(51) we observe that the variables h00 and h0i are identified as Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Moreover, from (41) and (57) we discard to P ij as degree of freedom because its time evolution vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Furthermore, we identify the equations of motion for hij and its momentum πij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In fact, by taking dθ7 0k = 0 and dθ9 = 0, we obtain ˙hij = 2Kij + ∂ih0j + ∂jh0i, (58) ˙πij = ηij∇2h00 − ∂i∂jh00 − ηijRkl kl − 2Rij − 1 µ[(ǫikl∂j + ǫjkl∂i)∂k∂mhlm −(ǫiklηjm + ǫjklηim)∂k∇2hlm], (59) ˙Kij = −1 2∂i∂jh00 − Rij + 1 4ηijRklkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (60) We observe that (58) corresponds to the definition of Kij, thus, if we use (58) and ˙Kij we will obtain a second order time equation for hij as expected, then there are six degrees of freedom associated with the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In this manner, we calculate the number of physical degrees of freedom as follows: there are 12 canonical variables (hij, πij) and eight involutive Hamiltonians (H00 1 , H0i 2 , H0i 7 , H9), thus DOF = 1 2[12 − 8] = 2, and thus, the theory has two physical degrees of freedom just like GR [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' if in the characteristics equations we take dt = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' then we identify the following 9 canonical transformations δh00 = δω1 00,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (61) δh0i = 1 2δω2 0i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (62) δhij = −1 2(δk i ∂j + δk j ∂i)δω7 0k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (63) moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' we can then identify the corresponding gauge transformations of the theory by considering that the Lagrangian (6) will be invariant under (61)-(63) if the variation δS = 0 [50],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' this is δS = � ∂S ∂hµν δhµν + ∂S ∂(∂αhµν)δ(∂αhµν) + ∂S ∂(∂α∂βhµν)δ(∂α∂βhµν) � (64) = � �� −□hµν + □hλληµν − ∂α∂λhαληµν − ∂µ∂νhλλ + 2∂µ∂λhνλ + 1 µǫ0µλγ(∂ν∂α∂λhαγ −∂λ□hν γ)) δhµν] d4x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (65) thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' by taking account (61)-(63) into the variation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' we obtain the following δS = � [Rij ijδω1 00 + 1 2[2∇2h0 i + 2∂i ˙hj j − 2∂i∂jh0j − 2∂j ˙hij + 1 µǫ0ijk(∂j∇2h0k − ∂j∂l ˙hkl)]δω2 0i −1 2[¨hij − ¨hk kηij + 2∂k ˙h0kηij − 2∂i ˙h0 j + ∂i∂jh00 − ∇2h00ηij + 2Rij − Rkl klηij + 1 µǫ0ikl(∂k¨hjl − ∂j∂k ˙h0l + ∂j∂k∂mhlm − ∂k∇2hjl)]δ(∂iω7 0j + ∂jω7 0i)]d4x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (66) Now, we define ∂0ξ ≡ δω1 00, so after long algebraic work we find that the variation takes the form δS = � [−∂j ˙hij + ∂ihjj + ∇2h0i − ∂i∂jh0j + 1 2µǫ0ijk(∂j∇2h0k − ∂j∂l ˙hkl)](−∂iξ + δω2 0i + ∂0δω7 0i)d4x, = 0, (67) hence, the action will be invariant under (61)-(63) if the the parameters ω′s obey δω2 0i = −∂0δω7 0i + ∂iξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (68) Now, we will write (68) in a new fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In fact, we introduce the following 4-vector ξµ ≡ ( 1 2ξ, − 1 2δω7 0i) ≡ (ξ0, ξi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' then ξ = 2ξ0 and δω7 0i = −2ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Hence, the relation (68) takes the form 1 2δω2 0i = ∂0ξi + ∂iξ0, (69) finally, from the equations (61)-(63) and (69) the following gauge transformations are identified δhµν = ∂µξν + ∂νξµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' (70) all these results are in agreement with those reported in [26], thus, our study complete and extends those reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' CONCLUSIONS AND REMARKS In this paper a detailed HJ analysis for the higher-order modified gravity has been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' We introduced a new set of variables in a different way than other approaches and reported in 10 the literature, then the full set of involutive and non-involutive Hamiltonians were identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' The correct identification of the Hamiltonians allow us to avoid the Ostrogradsky instability by removing the terms with linear momenta, healing the canonical Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Furthermore, the HJ generalized brackets and the fundamental differential were obtained from which the characteristic equations and the gauge symmetries were identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' The complete identification of the Hamiltoni- ans allowed us to carry out the counting of the physical degrees of freedom, concluding that the modified theory and GR shares the same number of physical degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In this manner, we have all elements to analize the theory in the quantum context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In fact, with our perturbative HJ study either constraints or the generalized brackets are under control, thus, we could use the tools developed in the canonical quantization of field theories in order to make progress in this program [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Furthermore, our analysis will be relevant for the study of the theory in the non-perturbative scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' In fact, now the modified theory will be full background independent then we will compare the differences between the canonical structure of GR reported in the literature [8, 9] and that for the modified theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' However, all those ideas are still in progress and will be reported soon [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Data Availability Statement: No Data associated in the manuscript [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Einstein, The Foundation of the General Theory of Relativity, Annalen Phys 49, 769-822 (1916).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Einstein, The Field Equations of Gravitation, Sitzungsberichte, Royal Pruss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=', Berlin, 844-847 (1915).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' [3] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Dyson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Eddington and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Davison, A 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Escalante and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} +page_content=' Pantoja, The perturbative and non-perturbative canonical analysis of the Chern- Simons modification of General Relativity, in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtAyT4oBgHgl3EQf4fpz/content/2301.00787v1.pdf'} diff --git a/D9FJT4oBgHgl3EQfBywq/content/tmp_files/load_file.txt b/D9FJT4oBgHgl3EQfBywq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a67561ffc97c4691d67159e98761361f8b2db3e --- /dev/null +++ b/D9FJT4oBgHgl3EQfBywq/content/tmp_files/load_file.txt @@ -0,0 +1,1570 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf,len=1569 +page_content='Model-based Offline Reinforcement Learning with Local Misspecification Kefan Dong*, Yannis Flet-Berliac*, Allen Nie*, Emma Brunskill Stanford University {kefandong,yfletberliac,anie,ebrun}@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='edu Abstract We present a model-based offline reinforcement learning pol- icy performance lower bound that explicitly captures dynam- ics model misspecification and distribution mismatch and we propose an empirical algorithm for optimal offline policy se- lection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Theoretically, we prove a novel safe policy improve- ment theorem by establishing pessimism approximations to the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Our key insight is to jointly consider se- lecting over dynamics models and policies: as long as a dy- namics model can accurately represent the dynamics of the state-action pairs visited by a given policy, it is possible to approximate the value of that particular policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We analyze our lower bound in the LQR setting and also show compet- itive performance to previous lower bounds on policy selec- tion across a set of D4RL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Introduction Offline reinforcement learning (RL) could leverage histor- ical decisions made and their outcomes to improve data- driven decision-making in areas like marketing (Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2017), robotics (Quillen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Swazinna, Udluft, and Runkler 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020), recommendation systems (Swaminathan and Joachims 2015), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Offline RL is particularly useful when it is possible to deploy context-specific decision policies, but it is costly or infeasible to do online reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Prior work on offline RL for large state and/or action spaces has primarily focused on one of two extreme settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' One line of work makes minimal assumptions on the under- lying stochastic process, requiring only no confounding, and leverages importance-sampling estimators of potential poli- cies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=', Thomas, Theocharous, and Ghavamzadeh (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Unfortunately, such estimators have a variance that scales exponentially with the horizon (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2018b) and are often ill-suited to long horizon problems1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' An alternative, which is the majority of work in offline RL, is to make a number of assumptions on the domain, These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Copyright © 2022, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 1Marginalized importance sampling (MIS) methods (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Xie, Ma, and Wang 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Yin and Wang 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Liu, Bacon, and Brunskill 2020) help address this but rely on the system being Markov in the underlying state space behavior data generation process and the expressiveness of the function classes employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The work in this space typi- cally assumes the domain satisfies the Markov assumption, which has been recently shown in the off-policy evaluation setting to enable provably more efficient policy value esti- mation (Kallus and Uehara 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Historically, most work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=', Munos (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Farahmand, Munos, and Szepesv´ari (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Xie and Jiang (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Chen and Jiang (2019)) as- sumes the batch data set has coverage on any state-action pairs that could be visited under any possible policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' More recent work relaxes this strong requirement using a pes- simism under uncertainty approach that is model-based (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Kidambi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020), model-free (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020) or uses policy search (Curi, Berkenkamp, and Krause 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' van Hasselt, Hessel, and Aslanides 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Such work still relies on realizability/lack of misspecification assump- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For model-free approaches, a common assumption is that the value function class can represent all policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2020) assume that the value function class is closed under (modified) Bellman backups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' A recent exception is Xie and Jiang (2020), which only requires the optimal Q- function to be representable by the value function class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' However, their sample complexity scales non-optimally (Xie and Jiang 2020, Theorem 2), and they also make strong assumptions on the data coverage – essentially the dataset must visit all states with sufficient probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Model-based approaches such as Malik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2020) as- sume the dynamics class has no misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' These two lines of work hint at possibilities in the mid- dle: can we leverage the sample-efficient benefits of Markov structure and allow for minimal assumptions on the data- gathering process and potential model misspecification?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' This can be viewed as one step towards more best-in-class results for offline RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Such results are relatively rare in RL, which tends to focus on obtaining optimal or near-optimal policies for the underlying domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Yet in many important applications, it may be much more practical to hope to iden- tify a strong policy within a particular policy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Our insight is that the algorithm may be able to lever- age misspecified models and still leverage the Markov as- sumption for increased data efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In particular, we take a model-based offline RL approach to leverage dynamics models that can accurately fit the space of state-action pairs visited under a particular policy (local small misspecifica- tion), rather than being a good model of the entire possi- ble state-action space (global small misspecification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Our work is most closely related to the recently proposed Min- imax Model Learning (MML) algorithm (Voloshin, Jiang, and Yue 2021): MML optimizes for the model that mini- mizes a value-aware error which upper bounds the differ- ence of policy value in learned and real models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' If the con- sidered model class includes the true model, this can work very well, but when the models are misspecified, this can be- come overly conservative since it optimizes with respect to a worst-case potential state-action distribution shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The key feature of our algorithm is to jointly optimize pol- icy and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Prior model-based offline RL algorithms typically estimate dynamics first, and then optimize a policy w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' the learned dynamics (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Voloshin, Jiang, and Yue 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' But when the dynamics model class is misspecified, there may not exist a unique “good dynamics” that can approximate the value of every policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' As a result, the learned policy may have a good estimated value under the learned dynamics, but a poor performance in the real en- vironment, or the learned policy may be overly conservative due to the misestimated dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Our paper makes the following contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' First, we provide a finite sample bound that assumes a Markov model, leverages the pessimism principle to work with many data- gathering distributions, accounts for estimation error in the behavior policy and, most importantly, directly accounts for dynamics and value function model misspecification (see Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We prove the misspecification error of our method is much tighter than other approaches because we only look at the models’ ability to represent visited state- action pairs for a particular policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In that sense, we say our algorithm relies on small local model dynamics mis- specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In Theorem 6, we show that when the dynam- ics model class does not satisfy realizability, decoupling the learning of policy and dynamics is suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' This moti- vates our algorithm which jointly optimizes the policy and model dynamics across a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Because of the tighter pessimistic estimation, we can prove a novel safe policy im- provement theorem (see Theorem 4) for offline policy opti- mization (OPO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' While our primary contribution is theoreti- cal, our proposed method for policy selection improves over the state-of-the-art MML Voloshin, Jiang, and Yue (2021) in a simple linear Gaussian setting, and has solid performance on policy selection on a set of D4RL benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Related Works There is an extensive and growing body of research on of- fline RL and we focus here on methods that also assume a Markov domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Many papers focus on model-free meth- ods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=', Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2019, 2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Nachum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2019) and their follow-ups (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Zhang, Liu, and Whiteson 2020) learn a distribution correction term, on top of which they perform evaluation or policy optimization tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Uehara, Huang, and Jiang (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Jiang and Huang (2020) study the duality between learn- ing Q-functions and learning importance weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2020) explicitly consider the distribution shift in offline RL and propose conservative Bellman equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Another line of research uses model-based methods (Ki- dambi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Matsushima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Swazinna, Udluft, and Runkler 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Fu and Levine 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Farahmand, Barreto, and Nikovski 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Gelada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Delgrange, Nowe, and P´erez (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Voloshin, Jiang, and Yue (2021) learn the dynamics using different loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2020) build an uncertainty quan- tification on top of the learned dynamics and select a policy that optimizes the lower confidence bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (Argenson and Dulac-Arnold 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Zhan, Zhu, and Xu 2021) focus on pol- icy optimization instead of model learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In Table 1, we compare our error bounds with existing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Our statistical error (introduced by finite dataset) is comparable with VAML (Farahmand, Barreto, and Nikovski 2017), MBS-PI (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020) and MML (Voloshin, Jiang, and Yue 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In addition, we consider misspecification er- rors and safe policy improvement (SPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Algorithm Statistical Error Misspecification SPI VAML � O � p √n � 2 \x13(global) \x17 MBS-PI � O � Vmaxζ (1−γ)2√n � \x13(global) \x13 MML Rn3 \x13(global) \x17 Ours � O � Vmax 1−γ � ζ n � \x13(local) \x13 Table 1: Comparison of error bounds with prior works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Problem Setup A Markov Decision Process (MDP) is defined by a tuple ⟨T, r, S, A, γ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' S and A denote the state and action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' T : S × A → ∆(S) is the transition and r : S × A → R+ is the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' γ ∈ [0, 1) is the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For a policy π : S → ∆(A), the value function is defined as V π T (s) = Es0=s,at∼π(st),st+1∼T (st,at)[�∞ t=0 γtr(st, at)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let Rmax ≜ maxs,a r(s, a) be the maximal reward and Vmax ≜ Rmax/(1 − γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Without loss of generality, we as- sume that the initial state is fixed as s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We use η(T, π) ≜ V π T (s0) to denote the expected value of policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let ρπ T (s, a) ≜ (1 − γ) �∞ t=0 γt Prπ T (st = s, at = a | s0) be the normalized state-action distribution when we execute policy π in a domain with dynamics model T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For simplicity in this paper we assume the reward function is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' An offline RL algorithm takes a dataset D = {(si, ai, s′ i)}n i=1 as input, where n is the size of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Each (si, ai, s′ i) tuple is drawn independently from a behav- ior distribution µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We assume that µ is consistent with the MDP in the sense that µ(· | s, a) = T(s, a) for all (s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For simplicity, we use ˆE to denote the empirical distribu- tion over the dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In this paper, we assume that the 2VAML only considers linear function approximation and p is the dimension of the feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 3The Rademacher complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For the finite hypothesis, the best-known upper bound is in the same order of ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' algorithm has access to an estimated behavior distribution ˆµ such that TV(µ, ˆµ) is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' This estimation can be easily obtained using a separate dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=', Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The algorithm can access three (finite) function classes G, T , Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' G is a class of value functions, T a class of dy- namics and Π a class of policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We assume that g(s, a) ∈ [0, Vmax] for all g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We use T ⋆ to denote the ground- truth dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note that T ⋆ may not be in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Our goal is to return a policy π ∈ Π that maximizes η(T ⋆, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Main Results A standard model-based RL algorithm learns the dynamics models first, and then uses the learned dynamics to estimate the value of a policy, or optimize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In this approach, it is crucial to link the estimation error of the dynamics to the estimation error of the value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Therefore, as a starting point, we invoke the simulation lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Lemma 1 (Simulation Lemma (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Kakade and Langford 2002)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Consider two MDPs with dynamics T, T ⋆, and the same reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Then, η(T, π) − η(T ⋆, π) = γ 1 − γ E(s,a)∼ρπ T [ Es′∼T (s,a)[V π T ⋆(s′)] − Es′∼T ⋆(s,a)[V π T ⋆(s′)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (1) For a fixed ground-truth dynamics T ⋆, we define Gπ T (s, a) = Es′∼T (s,a)[V π T ⋆(s′)] − Es′∼T ⋆(s,a)[V π T ⋆(s′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The simulation lemma states that the dynamics will provide an accurate estimate of the policy value if Es′∼T (s,a)[V π T ⋆(s′)] matches Es′∼T ⋆(s,a)[V π T ⋆(s′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In other words, to obtain a good estimate of a policy value, it is suf- ficient to minimize the model error Gπ T (s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Since the value function V π T ⋆ is unknown, Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2020) upper bound the model error by introducing a class of test functions G : S → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' When V π T ⋆ ∈ G, we have |Gπ T (s,a)|≤supg∈G ��Es′∼T (s,a)g(s′)−Es′∼T ⋆(s,a)g(s′)] ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In an offline dataset D, typically we can only observe one sample from T ⋆(s, a) per state-action pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Hence the al- gorithm cannot compute this upper bound exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In ad- dition, the distribution of the dataset D is also different from the one required by the simulation lemma ρπ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' To ad- dress these issues, we explicitly introduce a density ratio w : S × A → R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For a test function g ∈ G and a dynam- ics model T, let f g T (s, a) ≜ Es′∼T (s,a)[g(s′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Recall that ˆE denotes the empirical expectation over dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Then our model loss is defined as ℓw(T, g) = |ˆE[w(s, a)(f g T (s, a) − g(s′))]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2) Distribution mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We aim to upper bound policy eval- uation error by the loss function even if there are state ac- tion pairs with small probability mass under behavior dis- tribution µ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=', the offline dataset does not have a perfect coverage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Following Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2020), we treat the un- known state-action pairs pessimistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let ζ be a fixed cutoff threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Recall that ˆµ is an estimation of the behav- ior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For a policy π and dynamics T, we define wπ,T (s, a) ≜ I � ρπ T (s,a) ˆµ(s,a) ≤ ζ � ρπ T (s,a) ˆµ(s,a) as the truncated den- sity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For a fixed policy π, when w = wπ,T , ���E(s,a)∼ρπ T � Gπ T (s, a) ���� ≤ ����E(s,a)∼ρπ T � I �ρπ T (s, a) ˆµ(s, a) ≤ ζ � Gπ T (s, a) ����� + ���E(s,a)∼ρπ T � I �ρπ T (s, a) ˆµ(s, a) > ζ � Gπ T (s, a) ���� ≤ |E(s,a)∼ˆµ � w(s, a)Gπ T (s, a) � | + Vmax ���E(s,a)∼ρπ T � I �ρπ T (s, a) ˆµ(s, a) > ζ ����� ≤ |E(s,a)∼µ � w(s, a)Gπ T (s, a) � | + ζVmaxTV (ˆµ, µ) + Vmax ����E(s,a)∼ρπ T � I �ρπ T (s, a) ˆµ(s, a) > ζ ������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Hence, ignoring statistical error due to finite dataset, we can upper bound the estimation error |η(T ⋆, π) − η(T, π)| by γ 1 − γ � sup g∈G ���ℓwπ,T (g, T) ��� + ζVmaxTV (ˆµ, µ) + VmaxE(s,a)∼ρπ T � I �ρπ T (s, a) ˆµ(s, a) > ζ ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (3) Intuitively, the first term measures the error caused by im- perfect dynamics T, the second term captures the estimation error of the behavior distribution, and the last term comes from truncating the density ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Pessimistic Policy Optimization with Model Misspecification In this section, we explicitly consider misspecifications of the function classes used for representing the value func- tion and dynamics models (G and T , respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Most prior theoretical work on model-based RL make strong as- sumptions on the realizability of the dynamics model class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For example, in the offline setting, Voloshin, Jiang, and Yue (2021) focus on exact realizability of the dynamics model (that is, T ⋆ ∈ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In the online setting, Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2020) pro- vide bounds where there is a linear regret term due to global model misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Their bounds require a T ∈ T such that TV (T(s, a), T ⋆(s, a)) ≤ ϵ for all (s, a), even if the state-action pair (s, a) is only visited under some poorly per- forming policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We now show that offline RL tasks can need much weaker realizability assumptions on the dynam- ics model class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Our key observation is that for a given dynamics T and policy π, computing the density ratio wπ,T is statistically efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note that to compute wπ,T we do not need any samples from the true dynamics: instead, we only need to be able to estimate the state-action density under a dynamics model T for policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' This allows us to explicitly utilize the density ratio to get a relaxed realizability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The local value function error for a particular dynamics model T and policy π is defined as ϵV (T, π) ≜ inf g∈G |E(s,a)∼µ[wπ,T (s, a)(Es′∼T (s,a)[(g − V π T ⋆)(s′)] + Es′∼T ⋆(s,a)[(g − V π T ⋆)(s′)])]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The term ϵV measures the local misspecification of the value function class – that is, the error between the true value of the policy V π T ⋆ and the best fitting value function in the class G – only on the state-action pairs that policy π visits under a particular potential dynamics model T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In con- trast, previous results (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Nachum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Voloshin, Jiang, and Yue 2021) take the global maximum error over all (reachable) (s, a), which can be much larger than the local misspecification error ϵV (T, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' With this local misspecification error, we can establish a pessimistic estimation of the true reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let E be a high probability event under which the loss function ℓwπ,T (T, g) is close to its expectation (randomness comes from the dataset D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In the Appendix, we define this event formally and prove that Pr(E) ≥ 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The following lemma gives a lower bound on the true reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Proofs, when omitted, are in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let ι = log(2|G||T ||Π|/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For any dynamics model T and policy π, we define lb(T, π) = η(T, π) − 1 1 − γ � sup g∈G ℓwπ,T (g, T) + VmaxE(s,a)∼ρπ T � I �ρπ T (s, a) ˆµ(s, a) > ζ ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (4) Then, under the event E, we have η(T ⋆, π) ≥ lb(T, π) − 1 1 − γ � ϵV (T, π) − 2Vmax � ζι/n − ζVmaxTV (ˆµ, µ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (5) We use this to define our offline policy selection Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Algorithm 1: Model-based Offline RL with Local Misspecification Error Require: estimated behavior distribution ˆµ, truncation threshold ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' for π ∈ Π, T ∈ T do Compute wπ,T (s, a) = I � ρπ T (s,a) ˆµ(s,a) ≤ ζ � ρπ T (s,a) ˆµ(s,a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Compute lb(T, π) by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' end π ← argmaxπ∈Π maxT ∈T lb(T, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In contrast to existing offline model-based algorithms (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Voloshin, Jiang, and Yue 2021), our algorithm optimizes the dynamics and policy jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For a given dy- namics model, policy pair, our Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 1 computes the trun- cated density ratio wπ,T which does not require collecting new samples and then uses this to compute a lower bound lb(T, π) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Finally, it outputs a policy that maximizes the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We will shortly show this joint optimiza- tion can lead to better offline learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Parameter ζ controls the truncation of the stationary im- portance weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Increasing ζ decreases the last term in the lower bound objective lb(T, π), but it may also increase the variance given the finite dataset size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note that by setting ζ = log(n) and letting n → ∞ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=', with infinite data), the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (4) and the statistical error converge to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Safe Policy Improvement We now derive a novel safe policy improvement result, up to the error terms given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Intuitively this guarantees that the policy returned by Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 1 will improve over the be- havior policy when possible, which is an attractive property in many applied settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note that recent work (Voloshin, Jiang, and Yue 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020) on model-based of- fline RL does not provide this guarantee when the dynamics model class is misspecified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For a fixed policy π, define ϵρ(π) ≜ infT ∈T E(s,a)∼ρπ T ⋆ [TV (T(s, a), T ⋆(s, a))], (6) ϵµ(π) ≜ E(s,a)∼ρπ T ⋆ � I �ρπ T ⋆(s, a) ˆµ(s, a) > ζ/2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (7) The term ϵρ measures the local misspecification error of the dynamics model class in being able to represent the dynam- ics for state-action pairs encountered for policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' ϵµ rep- resents that overlap of the dataset for an alternate policy π: such a quantity is common in much of offline RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In the fol- lowing theorem, we prove that the true value of the policy computed by Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 1 is lower bounded by that of the optimal policy in the function class with some error terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Consider a fixed parameter ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let ˆπ be the pol- icy computed by Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 1 and ˆT = argmaxT lb(T, ˆπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let ι = log(2|G||T ||Π|/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Then, with probability at least 1−δ, we have η(T ⋆, ˆπ) ≥ sup π � η(T ⋆, π) − 6Vmaxϵρ(π) (1 − γ)2 − Vmaxϵµ(π) 1 − γ � − ϵV ( ˆT, ˆπ) 1 − γ − 4Vmax 1 − γ � ζι n − 2ζVmaxTV (ˆµ, µ) 1 − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (8) To prove Theorem 4, we prove the tightness of lb(T, π) — the lower bound maxT lb(T, π) is at least as high as the true value of the policy with some errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Consequently, maxi- mizing the lower bound also maximizes the true value of the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Formally speaking, we have the following Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For any policy π ∈ Π, under the event E we have max T ∈T lb(T, π) ≥ η(T ⋆, π) − 6Vmaxϵρ(π)/(1 − γ)2 − 1 1 − γ � Vmaxϵµ(π) − 2Vmax � ζι/n − ζVmaxTV (ˆµ, µ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In the sequel, we present a proof sketch for Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In this proof sketch, we hide 1/(1 − γ) factors in the big- O notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For a fixed policy π, let ˆT be the minimizer of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We prove Lemma 5 by analyzing the terms in the definition of lb( ˆT, π) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (4)) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Following the definition of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (6), we can show that ∥ρπ ˆT − ρπ T ⋆∥1 ≤ O(ϵρ(π)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Consequently we get η( ˆT, π) ≥ η(T ⋆, π) − O(ϵρ(π)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Recall that 0 ≤ g(s, a) ≤ Vmax for all g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Then for any (s, a) we have supg∈G |Es′∼ ˆT (s,a)g(s′) − Es′∼T ⋆(s,a)g(s′)]| ≤ VmaxTV( ˆT(s, a), T ⋆(s, a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Com- bining the definition of ℓw(g, T), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (6) and statistical error we get supg∈G ℓwπ,T (g, T) ≤ � O(ϵρ(π) + 1/√n + VmaxTV (ˆµ, µ)) under event E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For the last term regarding distribution mismatch, we combine Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (7) and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We can upper bound this term by O(ϵρ(π) + ϵµ(π)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The final term arises due to the potential estimation error in the behavior policy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Theorem 4 follows directly from combining Lemma 3 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note that Theorem 4 accounts for estimation er- ror in the behavior policy, misspecification in the dynamics model class, and misspecification in the value function class, the latter two in a more local, tighter form than prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Illustrative Example To build intuition of where our approach may yield benefits, we provide an illustrative example where Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 1 has better performance than existing approaches: an MDP whose state space is partitioned into several parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The model class is re- stricted so that every model can only be accurate on one part of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' When each deterministic policy only vis- its one part of the state space, the local misspecification error is small — for each policy, there exists a dynamics model in the set which can accurately estimate the distribution of states and actions visited under that policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In contrast, if the dynamics are learned to fit the whole state space, the estima- tion error will be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' More precisely, for a fixed parameter d, consider a MDP where S = {s0, · · · , sd} ∪ {sg, sb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' There are d actions denoted by a1, · · · , ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The true dynamics are deterministic and given by T ⋆(s0, ai) = si, T ⋆(si, aj) = �sg, if I [i = j] , sb, if I [i ̸= j] , (9) T ⋆(sg, ai) = sg, T ⋆(sb, ai) = sb, ∀i ∈ [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (10) And the reward is r(s, ai) = I [s = sg] , ∀i ∈ [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The transition function class T is parameterized by θ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For a fixed θ, the transition for states s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' , sd is Tθ(si, aj) = �sg, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 1 2 � 1 + e⊤ j θ � , sb, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 1 2 � 1 − e⊤ j θ � , (11) where ej is the j-th standard basis of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The transitions for states s0, sg, sb is identical to the true dynamics T ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' But the transition model Tθ in the function class must use the same parameter θ to approximate the dynamics in states s1, · · · , sd, which makes it misspecified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Decoupling learning the dynamics model and policy is suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Most prior algorithms first learn a dynamics model and then do planning with that model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' However, note here that the optimal action induced by MDP planning given a particular Tθ is suboptimal (assuming a uniformly random tie-breaking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' This is because, for any given θ, that dynam- ics model will estimate the dynamics of states s1, · · · , sd as being identical, with identical resulting value functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note this is suboptimality will occur in this example even if the dataset is large and covers the state–action pairs visited by any possible policy (ϵµ(π) = 0), the value function class is tabular and can represent any value function ϵV = 0, the behavior policy is known or the resulting estimation error is small (TV (ˆµ, µ) = 0, and ζ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In such a case, Theo- rem 4 guarantees that with high probability, our algorithm will learn the optimal policy because there exist couplings of the dynamics models and optimal policies such that the local misspecification error ϵρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' This demonstrates that prior algorithms (including MML (Voloshin, Jiang, and Yue 2021)) that decouple the learning of dynamics and policy can be suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We now state this more formally: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Consider any (possibly stochastic) algorithm that outputs an estimated dynamics Tθ ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let πθ be the greedy policy w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Tθ (with ties breaking uniformly at ran- dom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Then max π η(T ⋆, π) − η(T ⋆, πθ) ≥ (A − 1)γ2 A(1 − γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (12) As a side point, we also show that the off-policy estima- tion error in Voloshin, Jiang, and Yue (2021) is large when the dynamics model class is misspecified in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We defer this result to the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Experiments While our primary contribution is theoretical, we now inves- tigate how our method can be used for offline model-based policy selection with dynamics model misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We first empirically evaluate our method on Linear-Quadratic Regulator (LQR), a commonly used environment in optimal control theory (Bertsekas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2000), in order to assess: Can Algorithm 1 return the optimal policy when we have both model and distribution mismatch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We also evaluate our ap- proach using D4RL (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020), a standard offline RL benchmark for continuous control tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Here we consider: Given policies and dynamics pairs obtained using state-of- the-art offline model-based RL methods with ensemble dy- namics, does Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 1 allow picking the best policy, outper- forming previous methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Linear-Quadratic Regulator (LQR) LQR is defined by a linear transition dynamics st+1 = Ast + Bat + η, where st ∈ Rn and at ∈ Rm are state and action at time step t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' η ∼ N(0, σ2I) is ran- dom noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' LQR has a quadratic reward function R(s, a) = −(sT Qs + aT Ra) with Q ∈ Rn×n and R ∈ Rm×m be- ing positive semi-definite matrices, Q, R ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The op- timal controller to maximize the sum of future rewards �H t=1 −(sT t Qst+aT t Rat) until the end of horizon H has the form at = −Kst (K ∈ Rm×n) (Bertsekas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The value function is also a quadratic function, V (s) = sT Us+q for some constant q and positive semi-definite matrix U ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In the experiment, the state space is [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Misspecified transition classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Consider a 1D version of LQR with A(x) = (1 + x/10), B(x) = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 − x/10), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6 K 12 9 6 3 0 Return Returns of different policies under true environment Ours MML 1 2 3 4 5 Rank 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 Negative of lower bound (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='00,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='25) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='00) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='25) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='20,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='25) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='20,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='25) Ranking imposed by Eq 6 on policy-model pair (T, ) Model Loss+Distribution Shift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 MBLB MML MOPO D4RL IQM Normalized Score Figure 1: Left: Visualization of true policy value η(T ⋆, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Our algorithm picks the optimal policy, whereas MML picks a suboptimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Middle: Visualization of negative lower bounds lb(T, π) for different policies and models (indexed by the values of (v, u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Right: We show the interquartile mean (IQM) scores of two model-based lower bounds (MML and MBLB) and a recent model-based policy learning algorithm (MOPO) on D4RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Q = 1, R = 1 and noise η ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Our true dy- namics is given by x∗ = 6, and the corresponding optimal policy has K = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Function classes used by Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 1 are finite and computed as follows: (i) the value function class G contains the value functions of 1D LQR with parameters x ∈ {2, 4, 10} and K ∈ {−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='7};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (ii) the transi- tion class T is misspecified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We use the following transition class Tu ∈ T parametrized by u, Tu = �st+1 = A(x∗)st − B(x∗)at, st ∈ [u, u + 1], st+1 = st, otherwise, with u ∈ {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='75, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='25, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='25}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In other words, the capacity of the transition class is limited – each func- tion can only model the true dynamics of a part of the states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (iii) the policy class is given by πv parameterized by v, and πv(s) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1(s − v) + N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='01) with v ∈ {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Intuitively, πv tries to push the state toward s = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Since the state and action spaces are one dimensional, we can compute the density ratio wπ,T efficiently by discretiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The implementation details are deferred to Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We compare our algorithm to minimizing MML loss as described in the OPO algorithm of Voloshin, Jiang, and Yue (2021, Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' MML strictly outperformed VAML (Farahmand, Barreto, and Nikovski 2017) as shown in the experiments of (Voloshin, Jiang, and Yue 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' hence, we only compare to MML in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Figure 1 (Left) shows the return of different poli- cies under the true environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Our method picks the op- timal policy for the true model, whereas MML picks the wrong policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In Figure 1 (Middle), we also visualize dif- ferent terms in the definition of lb(T, π) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note that the model loss for different policy is different (model loss for (v, u) = (0, 0) is significantly larger than (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='25), even if the dynamics are the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' This is because the model loss is evaluated with a different density ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' This highlights the main benefit of our method over the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Since the model class is misspecified, maximizing over the weight function w in the MML loss results in an unrealistically large loss value for some models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' However, if the chosen policy does not visit the part of the state space with a large error, there is no need to incur a high penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' D4RL D4RL (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020) is an offline RL standardized bench- mark designed and commonly used to evaluate the progress of offline RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' This benchmark is standard for evaluating offline policy learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Here, we use a state-of-the-art policy learning algorithm MOPO (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020) to propose a set of policy-transition model tuples – for N policy hyperparameters and K transition models, we can get M × K tuples: {(π1, T1), (π1, T2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=', (πN, TK)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The MOPO algorithm learns an ensemble of transition mod- els and randomly chooses one to sample trajectories during each episode of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Instead, we choose one transition model to generate trajectories for the policy throughout the entire training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In our experiment, we choose M = 1 and K = 5, and train each tuple for 5 random seeds on Hopper and HalfCheetah tasks (see Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We then compute the model-based lower bound for each (πi, Tj), and select the optimal policy that has the highest lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We learn the dynamics using 300k iterations and we train each policy us- ing 100k gradient iterations steps with SAC (Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2018) as the policy gradient algorithm, imitating MOPO (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020) policy gradient update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' MML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Voloshin, Jiang, and Yue (2021) recommended two practical implementations for computing MML lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The implementation parametrizes w(s, a)V (s′) jointly via a new function h(s, a, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We refer readers to Prop 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 from Voloshin, Jiang, and Yue (2021) for a detailed explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We describe how we parametrize this function as follows: Linear: Voloshin, Jiang, and Yue (2021) showed that if T, V, µ are all from the linear function classes, then a model T that minimizes MML loss is both unique and identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' This provides a linear parametrization of h(s, a, s′) = ψ(s, a, s′)T θ, where ψ is a basis function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We choose ψ to be either a squared basis function or a polynomial basis function with degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Kernel: Using a radial basis function (RBF) over S × Dataset Type Env MOPO MML (Squared) MML (Polynomial) MML (RKHS) MBLB (Linear) MBLB (Quad) medium hopper 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='3) 379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 (466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4) 375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6 (459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5) 375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 (459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9) 591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='7 (523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1) 808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 (502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='7) med-expert hopper 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4) 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9 (131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5) 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 (148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0) 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1 (157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9) 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 (134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0) expert hopper 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8 (87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6 (61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 (56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2) 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2 (61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6) 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 (72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5) medium halfcheetah 599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8 (668.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4) 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6 (1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5) 2625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1 (937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2) 3858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2 (1231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1) 3290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 (1753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1) 2484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2 (1526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8) med-expert halfcheetah 486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6 (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1) 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 (137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 (252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5) 343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2 (225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2) 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 (509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5) 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8 (432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0) Table 2: We report the mean and (standard deviation) of selected policy’s simulator environment performance across 5 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' MML and MBLB are used as model-selection procedures where they select the best policy for each seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Our method is choosing the most near-optimal policy across the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' A × S and computing K((s, a, s′), (˜s, ˜a, ˜s′)), Voloshin, Jiang, and Yue (2021) showed that there exists a closed- form solution to compute the maxima of the MML loss (RKHS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Here, there is no need for any gradient update, we only sample s′ from T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' MBLB (Ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For a continuous control task, we compute our model-based lower bound (MBLB) as follows: Compute η(T, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Although it is reasonable to directly use a value function V π T trained during policy learning to compute η(T, π), Paine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2021) points out how this value function often severely over-estimates the ac- tual discounted return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Therefore, we estimate the expected value of policy π using the generalized advantage estima- tor (GAE) (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For a sequence of tran- sitions {st, at, r(st, at), st+1}t∈[0,N], it is defined as: At = �t+N t′=t (γλ)t′−t(r(st′, at′) + γVφ(st′+1) − Vφ(st′)), with λ a fixed hyperparameter and Vφ the value function estimator at the previous optimization iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Then, to estimate the value function, we solve the non-linear regression problem minimizeφ �t+N t′=t (Vφ(st′)− ˆVt′)2 where ˆVt = At+Vφ(st′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We also provide a comparison to using the standard TD-1 Fitted Q Evaluation (FQE) (Le, Voloshin, and Yue 2019) in- stead in Table A1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We find that using GAE provides better policy evaluation estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Behavior density modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We use a state-of-the-art nor- malizing flow probability model to estimate the density of state-action pairs (Papamakarios et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For ρπ T , we sample 10,000 trajectories from T, π, and estimate the cor- responding density;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' for the behavior distribution µ, we use the given dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We empirically decide the number of training epochs that will give the model the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Compute supg∈G |ℓwπ,T (g, T)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We parametrize g either as a linear function of state: g(s) = mT s, or a quadratic func- tion of the state: g(s) = sT Ms + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We use gradient ascent on ℓwπ,T (g, T) to maximize this objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We report the results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' There is gen- eral overlap across seeds for the performance between vari- ous methods, but our approach has the best average perfor- mance or is within the standard deviation of the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We also show that for different choices of how we parameterize the w(s, a)V (s′) distribution (MML) and how we choose the family of g test function (MBLB), we are selecting differ- ent final policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' However, overall, MBLB can pick better- performing final policies with two different parametrizations while MML is choosing lower-performing policies with its three parametrizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We find that our approach of select- ing among the set of policies computed from each of the models used by MOPO consistently outperforms the policy produced by MOPO in the considered tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' To summarize these results, we report the interquartile mean (IQM) scores of each method in Figure 1 (Right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' IQM is an outlier robust metric proposed by Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (2021) to compare deep RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We create the plot by sam- pling with replacement over all runs on all datasets 50000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Though there is significant overlap, our method gen- erally outperforms policies learned from MOPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Conclusion There are many directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The current lb(T, π) implementation with density ratio wπ,T (s, a) is not differentiable: an interesting question is to make this differ- entiable so that we can directly optimize a policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Another interesting question would be to construct estimators for the local misspecification errors ϵρ, ϵµ and ϵV , which could be used to refine the model class to optimize performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' To conclude, this paper studies model-based offline rein- forcement learning with local model misspecification errors, and proves a novel safe policy improvement theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Our theoretical analysis shows the benefit of this tighter analy- sis and approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We illustrate the advantage of our method over prior work in a small linear quadratic example and also demonstrate that it is competitive or has stronger per- formance than recent model-based offline RL methods on policy selection in a set of D4RL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Acknowledgment Research reported in this paper was sponsored in part by NSF grant #2112926, the DEVCOM Army Research Lab- oratory under Cooperative Agreement W911NF-17-2-0196 (ARL IoBT CRA) and a Stanford Hoffman-Yee grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Gov- ernment is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright nota- tion herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' References Agarwal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Schwarzer, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' and Schuurmans, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Gen- DICE: Generalized Offline Estimation of Stationary Values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Missing Proofs High Probability Events In this section, we introduce concentration inequalities and define the high probability events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Define the following quantities L(π, g, T) = E(s,a,s′)∼µ � wπ,T (s, a)(Ex∼T (s,a)[g(x)] − Ex∼T ⋆(s,a)[g(x)]) � , (13) l(π, g, T) = E(s,a,s′)∼D[wπ,T (s, a)(f g T (s, a) − g(s′))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (14) Recall that ι = log(2|G||T ||Π|/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Consider the event E = � |L(π, g, T) − l(π, g, T)| ≤ 2Vmax � ζι n , ∀π ∈ Π, g ∈ G, T ∈ T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (15) In the following, we show that Pr (E) ≥ 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (16) Recall that D = {(si, ai, s′ i)}n i=1 where (si, ai, s′ i) ∼ µ are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' samples from distribution µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For fixed π ∈ Π, g ∈ G, T ∈ T , we have E[ˆl(π, g, T)] = l(π, g, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Meanwhile, note that |wπ,T (s, a)(f g T (s, a) − g(s′))| ≤ ζVmax, (17) E(s,a,s′)∼µ[wπ,T (s, a)2(f g T (s, a) − g(s′))2] (18) ≤ E(s,a,s′)∼ρπ T [wπ,T (s, a)(f g T (s, a) − g(s′))2] ≤ V 2 maxζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (19) By Bernstein inequality, with probability at least 1 − δ/(|G||T ||Π|), |L(π, g, T) − l(π, g, T)| ≤ � 2V 2 maxζ log(2|G||T ||Π|/δ) n + ζVmax 3n log(2|G||T ||Π|/δ) (20) Recall that ι = log(2|G||T ||Π|/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' When n ≥ ζ we have |L(π, g, T) − l(π, g, T)| ≤ 2Vmax � ζι n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (21) Note that when n < ζ, E trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' As a result, applying union bound we prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Proof of Lemma 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' In the following, we consider a fixed policy π and dynamics T ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We use w to denote wπ,T when the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' By basic algebra we get ���E(s,a)∼ρπ T [Gπ T (s, a)] ��� (22) ≤ ����E(s,a)∼ρπ T � I �ρπ T (s, a) ˆµ(s, a) ≤ ζ � Gπ T (s, a) ����� + E(s,a)∼ρπ T � I �ρπ T (s, a) ˆµ(s, a) > ζ � |Gπ T (s, a)| � (23) ≤ ��E(s,a)∼ˆµ[w(s, a)Gπ T (s, a)] �� + VmaxE(s,a)∼ρπ T � I �ρπ T (s, a) ˆµ(s, a) > ζ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (24) Note that E(s,a)∼ˆµ[w(s, a)Gπ T (s, a)] = � s,a ˆµ(s, a)w(s, a)Gπ T (s, a) (25) = � s,a (ˆµ(s, a) − µ(s, a) + µ(s, a))w(s, a)Gπ T (s, a) (26) = � s,a µ(s, a)w(s, a)Gπ T (s, a) + � s,a (ˆµ(s, a) − µ(s, a))w(s, a)Gπ T (s, a) (27) ≤ E(s,a)∼µ[w(s, a)Gπ T (s, a)] + � s,a |ˆµ(s, a) − µ(s, a)|ζVmax (28) ≤ E(s,a)∼µ[w(s, a)Gπ T (s, a)] + ζVmaxTV (ˆµ, µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (29) Continuing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (24) we get ���E(s,a)∼ρπ T [Gπ T (s, a)] ��� (30) ≤ ��E(s,a)∼µ[w(s, a)Gπ T (s, a)] �� + VmaxE(s,a)∼ρπ T � I �ρπ T (s, a) ˆµ(s, a) > ζ �� + ζVmaxTV (ˆµ, µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (31) Consequently, in the following we prove ��E(s,a)∼µ[w(s, a)Gπ T (s, a)] �� ≤ sup g∈G ℓw(g, T) + ϵV (T, π) + 2Vmax � ζι n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let Lw(g, T) = ��E(s,a,s′)∼µ � w(s, a)(Ex∼T (s,a)[g(x)] − Ex∼T ⋆(s,a)[g(x)]) ��� be the population error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Recall that under the high probability event E in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (15), for any g ∈ G and T ∈ T |Lw(g, T) − ℓw(g, T)| ≤ 2Vmax � ζι n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (32) Now by the definition of Gπ T (s, a), for any g ∈ G we have ��E(s,a)∼µ[w(s, a)Gπ T (s, a)] �� (33) = ��E(s,a)∼µ � w(s, a) � Es′∼T (s,a)[V π T ⋆(s′)] − Es′∼T ⋆(s,a)[V π T ⋆(s′)] ���� (34) ≤ ��E(s,a)∼µ � w(s, a) � Es′∼T (s,a)[g(s′)] − Es′∼T ⋆(s,a)[g(s′)] ���� (35) + ��E(s,a)∼µ � w(s, a) � Es′∼T (s,a)[g(s′) − V π T ⋆(s′)] + Es′∼T ⋆(s,a)[g(s′) − V π T ⋆(s′)] ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (36) Define ˆg = argmin g∈G ��E(s,a)∼µ � w(s, a) � Es′∼T (s,a)[g(s′) − V π T ⋆(s′)] + Es′∼T ⋆(s,a)[g(s′) − V π T ⋆(s′)] ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Since g is arbitrarily, continuing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (36) and recalling Definition 2 we get ��E(s,a)∼µ[w(s, a)Gπ T (s, a)] �� (37) ≤ ��E(s,a)∼µ � w(s, a) � Es′∼T (s,a)[ˆg(s′)] − Es′∼T ⋆(s,a)[ˆg(s′)] ���� + ϵV (T, π) (38) ≤ sup g∈G ��E(s,a)∼µ � w(s, a) � Es′∼T (s,a)[g(s′)] − Es′∼T ⋆(s,a)[g(s′)] ���� + ϵV (T, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (39) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (39) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (32) we get, ��E(s,a)∼µ[w(s, a)Gπ T (s, a)] �� ≤ sup g∈G Lw(g, T) + ϵV (T, π) (40) ≤ sup g∈G ℓw(g, T) + ϵV (T, π) + 2Vmax � ζι n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (41) Now plugging in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (31) we get, ���E(s,a)∼ρπ T [Gπ T (s, a)] ��� ≤ sup g∈G ℓw(g, T) + ϵV (T, π) + 2Vmax � ζι n + VmaxE(s,a)∼ρπ T � I �ρπ T (s, a) ˆµ(s, a) > ζ �� + ζVmaxTV (ˆµ, µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Finally, combining with simulation lemma (Lemma 1) we finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Proof of Lemma 5 Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Consider a fixed π ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' When the context is clear, we use ϵρ and ϵµ to denote ϵρ(π) and ϵµ(π) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Consider the dynamics ˆT = argmin T ∈T E(s,a)∼ρπ T ⋆ [TV (T(s, a), T ⋆(s, a))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (42) By the definition of ϵρ we get E(s,a)∼ρπ T ⋆ � TV � ˆT(s, a), T ⋆(s, a) �� ≤ ϵρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Applying Lemma 9 we get ��ρπ ˆT − ρπ T ⋆ �� 1 ≤ ϵρ (1 − γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (43) The rest of the proof is organized in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We bound the three terms in RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (4) respectively as follows η( ˆT, π) ≥ η(T ⋆, π) − Vmax 1 − γ ϵρ, (44) sup g∈G ℓw(g, ˆT) ≤ 2Vmaxϵρ 1 − γ + 2Vmax � ζι n + ζVmaxTV (ˆµ, µ) , (45) E(s,a)∼ρπ ˆ T � I � ρπ ˆT (s, a) ˆµ(s, a) > ζ �� ≤ ϵµ + 3ϵρ (1 − γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (46) Then we combine these inequalities together to prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Step 1: Proving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note that for every T and π, η(T, π) = 1 1−γ ⟨ρπ T , r⟩ where r is the reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Then we have η(T ⋆, π) − η( ˆT, π) = 1 1 − γ � ρπ T ⋆ − ρπ ˆT , r � ≤ 1 1 − γ ��ρπ T ⋆ − ρπ ˆT �� 1 ∥r∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (47) Combining with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (43) we get Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Step 2: Proving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For any fixed function g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let w = wπ, ˆT be a shorthand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Define Lw(g, T) = ��E(s,a,s′)∼µ[w(s, a)(f g T (s, a) − g(s′))] �� to be the population error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Then we have Lw(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' ˆT) = ���E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)∼µ � w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a) � Es′∼ ˆT (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)[g(s′)] − Es′∼T ⋆(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)[g(s′)] ����� ≤ ���E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)∼ˆµ � w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a) � Es′∼ ˆT (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)[g(s′)] − Es′∼T ⋆(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)[g(s′)] ����� + ζVmaxTV (ˆµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' µ) = �����E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)∼ρπ ˆ T � I � ρπ ˆT (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a) ˆµ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a) ≤ ζ � � Es′∼ ˆT (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)[g(s′)] − Es′∼T ⋆(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)[g(s′)] ������� + ζVmaxTV (ˆµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' µ) ≤ VmaxE(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)∼ρπ ˆ T � I � ρπ ˆT (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a) ˆµ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a) ≤ ζ � TV � ˆT(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' T ⋆(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a) �� + ζVmaxTV (ˆµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' µ) ≤ VmaxE(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)∼ρπ T ⋆ � TV � ˆT(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' T ⋆(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a) �� + Vmaxϵρ 1 − γ + ζVmaxTV (ˆµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' µ) (By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (43)) ≤ Vmax � ϵρ + ϵρ 1 − γ � + ζVmaxTV (ˆµ, µ) ≤ 2Vmaxϵρ 1 − γ + ζVmaxTV (ˆµ, µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Under event E we have ℓw(g, ˆT) ≤ Lw(g, ˆT) + 2Vmax � ζι n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (48) Because g is arbitrary, we get Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Step 3: Proving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note that E(s,a)∼ρπ ˆ T � I �ρˆπ T (s, a) ˆµ(s, a) > ζ �� (49) = E(s,a)∼ρπ ˆ T � I � ρπ ˆT (s, a) ρπ T ⋆(s, a) ρπ T ⋆(s, a) ˆµ(s, a) > ζ �� (50) ≤ E(s,a)∼ρπ ˆ T � I � ρπ ˆT (s, a) ρπ T ⋆(s, a) > 2 �� + E(s,a)∼ρπ ˆ T � I �ρπ T ⋆(s, a) ˆµ(s, a) > ζ/2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (51) With the help of Lemma 8, we can upper bound the first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (51) by the total variation between ρπ ˆT and ρπ T ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Combining Lemma 8 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (43) we get E(s,a)∼ρπ ˆ T � I � ρˆπ T (s, a) ρπ T ⋆(s, a) > 2 �� ≤ 2ϵρ 1 − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (52) On the other hand, by combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (43) and the definition of ϵµ we get E(s,a)∼ρπ ˆ T � I �ρπ T ⋆(s, a) ˆµ(s, a) > ζ/2 �� ≤ E(s,a)∼ρπ T ⋆ � I �ρπ T ⋆(s, a) ˆµ(s, a) > ζ/2 �� + ϵρ 1 − γ ≤ ϵµ + ϵρ 1 − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Consequently, we get Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Now we stitch Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (43), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (44) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (45) together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Combining with the definition of lb( ˆT, π) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (4), we have lb( ˆT, π) = η( ˆT, π) − 1 1 − γ � sup g∈G ���ℓwπ,T (g, ˆT) ��� + VmaxE(s,a)∼ρπ T � I � ρπ ˆT (s, a) ˆµ(s, a) > ζ �� + 2ζVmaxTV (ˆµ, µ) � ≥ η(T ⋆, π) − Vmaxϵρ 1 − γ − 2Vmaxϵρ (1 − γ)2 − 2Vmax 1 − γ � ζι n − Vmax 1 − γ � 3ϵρ 1 − γ + ϵµ � − 2ζVmaxTV (ˆµ, µ) 1 − γ ≥ η(T ⋆, π) − 6Vmaxϵρ (1 − γ)2 − Vmaxϵµ 1 − γ − 2Vmax 1 − γ � ζι n − 2ζVmaxTV (ˆµ, µ) 1 − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note that ˆT ∈ T , we have max T ∈T lb(T, π) ≥ lb( ˆT, π), (53) which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Proof of Theorem 4 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let ˆT, ˆπ ← argmaxT ∈T ,π∈Π lb(T, π) be the dynamics and policy that maximizes the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note that ˆπ is the output of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Now under the event E, by Lemma 5, for any policy π we have max T ∈T lb(T, π) ≥ η(T ⋆, π) − 6Vmaxϵρ(π) (1 − γ)2 − Vmaxϵµ(π) 1 − γ − 2Vmax 1 − γ � ζι n − 2ζVmaxTV (ˆµ, µ) 1 − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (54) On the other hand, under the event E, by Lemma 3 we get η(T ⋆, π) ≥ lb( ˆT, ˆπ) − ϵV ( ˆT, ˆπ) 1 − γ − 2Vmax 1 − γ � ζι n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (55) By the optimality of ˆT, ˆπ, we have lb( ˆT, ˆπ) ≥ supT ∈T lb(T, π) for any π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' As a result, combining with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (54) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (55) we get η(T ⋆, ˆπ) ≥ lb( ˆT, ˆπ) − ϵV ( ˆT, ˆπ) 1 − γ − 2Vmax 1 − γ � ζι n (56) ≥ sup π∈Π sup T ∈T lb(T, π) − ϵV ( ˆT, ˆπ) 1 − γ − 2Vmax 1 − γ � ζι n (57) ≥ sup π � η(T ⋆, π) − 6Vmaxϵρ(π) (1 − γ)2 − Vmaxϵµ(π) 1 − γ � − ϵV ( ˆT, ˆπ) 1 − γ − 4Vmax 1 − γ � ζι n − 2ζVmaxTV (ˆµ, µ) 1 − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (58) Proof of Theorem 6 Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note that for any fixed θ ∈ Rd, the transition function for state s1, · · · , sd are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' As a result, Qπ Tθ(si, aj) = Qπ Tθ(si′, aj), ∀i, i′ ∈ [d] for any policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Recall that πθ is the optimal policy of Tθ (with ties breaking uniformly at random).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Therefore, πθ(s0) = 1/A and πθ(si) = πθ(si′), ∀i, i′ ∈ [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' By the definition of the ground-truth dynamics T ⋆ in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (9)-(10), we have Qπθ T ⋆(si, aj) = I [i = j] γ 1−γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Therefore, η(T ⋆, πθ) = γ A d � i=1 Qπθ T ⋆(si, πθ(si)) ≤ γ A max a d � i=1 Qπθ T ⋆(si, a) ≤ γ2 A(1 − γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (59) Since maxπ η(T ⋆, π) = γ2 1−γ , we have max π η(T ⋆, π) − η(T ⋆, πθ) ≥ (A − 1)γ2 A(1 − γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' OPE Error of MML In this section, we show that the off-policy estimation error in Voloshin, Jiang, and Yue (2021) can be large when the dynamics model class is misspecified in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The MML algorithm requires an density ratio class W : S × A → R+ and prove that when wπ,T ∈ W and V π T ⋆ ∈ G, |η(T, π) − η(T ⋆, π)| ≤ γ min T ∈T max w∈W,g∈G |ℓw(g, T)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (60) Unfortunately, this is suboptimal since the error may not converge to zero even given infinite data: Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Consider the set the dynamics class T = {Tθ : θ ∈ Sd−1, θi ≥ 0, ∀i ∈ [d]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let Π = {πx : x ∈ [d]} where πx(si) = ax for 0 ≤ i ≤ d and πx(sg) = πx(sb) = a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let W be the density ratio class induced by π running on {T ⋆} ∪ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Even with G = {V πx T ⋆ : x ∈ [d]} and infinite number of data, we have min T ∈T max w∈W,g∈G |ℓw(g, T)| ≥ γ 8(1 − γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (61) In contrast, the error terms in Theorem 4 converge to 0 when ζ > poly(d, 1/(1 − γ)) and n → ∞ in the same setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Recall that we set the dynamics class T = {Tθ : θ ∈ Sd−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let Π = {πx : x ∈ [d]} where πx(si) = ax for 0 ≤ i ≤ d and πx(sg) = πx(sb) = a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let W be the density ratio induced by π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For any x ∈ [d], we can compute ρπx T ⋆(s0, ai) = (1 − γ)I [i = x] , ρπx T ⋆(si, aj) = γ(1 − γ)I [i = x, j = x] , (62) ρπx T ⋆(sg, aj) = γ2(1 − γ)I [j = 1] , ρπx T ⋆(sb, aj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (63) Let µ be uniform distribution over 3d + d2 state action pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Then we can define W = {wx : x ∈ [d]} where wx(s, a) ≜ 1 1−γ ρπx T ⋆(s,a) µ(s,a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Now for any fixed θ ∈ Sd−1, θ ≥ 0, consider max w∈W,g∈G |ℓw(g, Tθ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (64) Let x = argmini θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We claim that ℓwx(V πx T ⋆ , Tθ) ≥ γ 8(1 − γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Indeed, with infinite data we have ℓwx(V πx T ⋆ , Tθ) = ��E(s,a)∼µ � wx(s, a) � Es′∼T (s,a)[V πx T ⋆ (s′)] − Es′∼T ⋆(s,a)[V πx T ⋆ (s′)] ���� = 1 1 − γ ���E(s,a)∼ρπx T ⋆ �� Es′∼T (s,a)[V πx T ⋆ (s′)] − Es′∼T ⋆(s,a)[V πx T ⋆ (s′)] �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Recall that Tθ = T ⋆ for states s0, sg, sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' As a result, we continue the equation by 1 1 − γ ���E(s,a)∼ρπx T ⋆ �� Es′∼T (s,a)[V πx T ⋆ (s′)] − Es′∼T ⋆(s,a)[V πx T ⋆ (s′)] ����� = γ ��Es′∼T (sx,ax)[V πx T ⋆ (s′)] − Es′∼T ⋆(sx,ax)[V πx T ⋆ (s′)] �� (by the definition of ρ) = γ ���� 1 2(1 + θx)V πx T ⋆ (sg) + 1 2(1 − θx)V πx T ⋆ (sb) − V πx T ⋆ (sg) ���� (by the definition of Tθ) = γ 2 (1 − θx)(V πx T ⋆ (sg) − V πx T ⋆ (sb)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' By basic algebra, V πx T ⋆ (sg) = (1 − γ)−1 and V πx T ⋆ (sb) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' As a result, we get ℓwx(V πx T ⋆ , Tθ) ≥ γ 2(1 − γ)(1 − θx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (65) Recall that x = argmini θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Since θ ∈ Sd−1 and θi ≥ 0, ∀i, we have 1 = �d i=1 θ2 i ≥ dθ2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' As a result, when d > 2 we have θx ≤ 1/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Therefore ℓwx(V πx T ⋆ , Tθ) ≥ γ 2(1 − γ)(1 − θx) ≥ γ 8(1 − γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (66) Helper Lemmas In this section, we present several helper lemmas used in Appendix .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For two distribution p, q over x ∈ X, if we have ∥p − q∥1 ≤ ϵ, then for any ζ > 1, Ex∼p � I �p(x) q(x) > ζ �� ≤ ζ ζ − 1ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Define E(x) = I � p(x) q(x) > ζ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note that under event E(x) we have p(x) > q(x)ζ =⇒ p(x) − q(x) > q(x)(ζ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (67) As a result, ϵ ≥ ∥p − q∥1 ≥ � |p(x) − q(x)|E(x) dx (68) ≥ � (ζ − 1)q(x)E(x) dx = Ex∼q[E(x)](ζ − 1) (69) ≥ (Ex∼p[E(x)] − ϵ)(ζ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (70) By algebraic manipulation we get Ex∼p[E(x)] ≤ ζ ζ−1ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Consider a fixed policy π and two dynamics model T, ¯T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Suppose E(s,a)∼ρπ T � TV � T(s, a), ¯T(s, a) �� ≤ ϵ, we get ��ρπ T − ρπ ¯T �� 1 ≤ 1 1 − γ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (71) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' First of all let G, ¯G be the transition kernel from S × A to S × A induced by T, π and ¯T, π respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Then for any distribution ρ ∈ ∆(S × A) we have ��Gρ − ¯Gρ �� 1 ≤ E(s,a)∼ρ � TV � ¯T(s, a), T(s, a) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (72) Let ρh (or ¯ρh) be the state-action distribution on step h under dynamics T (or ¯T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Then we have ρh − ¯ρh = � Gh − ¯Gh� ρ0 = h−1 � h′=0 ¯Gh−h′−1� G − ¯G � Gh′ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (73) As a result, ∥ρh − ¯ρh∥1 ≤ h−1 � h′=0 ��� ¯Gh−h′−1� G − ¯G � Gh′ρ0 ��� 1 (74) ≤ h−1 � h′=0 ��� � G − ¯G � Gh′ρ0 ��� 1 ≤ h−1 � h′=0 E(s,a)∼ρh′ � TV � ¯T(s, a), T(s, a) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (75) It follows that ��ρπ T − ρπ ¯T �� 1 ≤ (1 − γ) ∞ � h=0 γh ∥ρh − ¯ρh∥1 (76) ≤(1 − γ) ∞ � h=0 γh h−1 � h′=0 E(s,a)∼ρh′ � TV � ¯T(s, a), T(s, a) �� (77) ≤(1 − γ) ∞ � h=0 γh 1 − γ E(s,a)∼ρh � TV � ¯T(s, a), T(s, a) �� (78) = ∞ � h=0 γhE(s,a)∼ρh � TV � ¯T(s, a), T(s, a) �� (79) = 1 1 − γ E(s,a)∼ρπ T � TV � ¯T(s, a), T(s, a) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (80) LQR Experimental Details Data generation The offline dataset is generated by running several πv under the true dynamics with v ∈ {−1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='75, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='25, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='75} and added noise N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5) to the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' As a result, the behavior dataset covers most of the state-action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The dataset contains 2000 trajectories with length 20 from each policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Implementation We compute the density ratio by approximating the behavior distribution µ and the state-action distribution ρπ T respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' By discretizing the state-action space into 10 × 10 bins uniformly, the distribution µ(s, a) is approximated by the frequency of the corresponding bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For ρπ T , we first collect 2000 trajectories of policy π under T and compute the distribution similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Because all the function classes are finite, we enumerate over the function classes to compute lb(T, π) for every pair of dynamics and policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Hyperparameters In the experiments, we use the following hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Cutoff threshold in Line 3 of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 1: ζ = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Random seeds for three runs: 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' State noise: η ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Policy noise: N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Discount factor: γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Mean of initial state: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Noise added to initial state: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Number of trajectories per policy: 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We do not require parameter tuning for optimization procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We tried cutoff threshold with ζ ∈ {10, 20, 50} and number of trajectories in {20, 500, 2000}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Smaller cutoff leads to an over-pessimistic lower bound, and fewer trajectories introduce variance to the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Computing resources These experiments run on a machine with 2 CPUs, 4GB RAM, and Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We don’t require GPU resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We use Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 and numpy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' D4RL Experimental Details Tasks Hopper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The Hopper task is to make a hopper with three joints and four body parts hop forward as fast as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The state space is 11-dimension, the action is a 3-dimensional continuous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' HalfCheetah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The HalfCheetah task is to make a 2D robot with 7 rigid links, including 2 legs and a torso run forward as fast as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The state space is 17-dimension, the action is a 6-dimensional continuous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Model Choice and Hyperparameters For all the dynamics, each model is parametrized as a 4-layer feedforward neural network with 200 hidden units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' For the SAC (Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2018) updates (serving as the policy gradient updates subroutine), the function approximations used for the policy and value function are 2-layer feedforward neural networks with 256 hidden units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' The hyperparameter choices for behavior density modeling are based on the training progress of the normalizing flow model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We pre-select a few (less than 10) combinations of hyperparameters and pick the set that gives us the lowest training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Usually, this is not the best practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' However, the small number of combinations (non-exhaustive search) and small model size reduced our concern for training set overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' MOPO (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 2020): Batch size: 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Rollout horizon: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Lambda: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' MBLB: Random seeds for five runs: 1, 2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Number of trajectories to sample: 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Rollout horizon: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Batch size: 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Cutoff threshold in Line 3 of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 1: ζ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Discount factor γ: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' GAE λ: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' g function latent size: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' MML: Random seeds for five runs: 1, 2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Batch size: 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Basis function class: square, polynomial Ratio-Value function parametrization: linear, reproducing kernel hilbert space (RKHS) For MML, we first need to make a decision on how to parametrize h(s, a, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' If we choose a linear parametrization such as h(s, a, s′) = ψ(s, a, s′)T θ, we need to decide what ψ is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' There are two obvious choices: ψ(x) = [x, x2, 1] (square basis func- tion), or a polynomial basis function with degree 2: given x = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=', xd], ψ(x) = [x2 1, x1x2, x1x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=', x2 2, x2x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=', x2 d], which can be efficiently computed as the upper triangular entries of xxT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' If we choose the ratio-value function parametrization to be RKHS, then we use radial basis function (RBF) as K((s, a, s′), (˜s, ˜a, ˜s′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Computing resources These experiments run on a machine with 4 CPUs, 10GB RAM, and Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We don’t require GPU resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We use Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 and numpy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Algorithms We describe the MML and MBLB algorithms in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Algorithm 2 describes how we compute MBLB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Note that we compute three components of lower bound explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Algorithm 3 describes how we compute MML with linear parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Algorithm 4 describes how we compute MML with RKHS parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Algorithm 2: MBLB: Model-based Lower Bound Input: offline RL data D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' set of dynamics, policy pairs [(π1, T1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=', (πK, TK)], Vmax, γ, ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Output: optimal policy π∗ ˆµ(·, ·) = trainFlow (D) scores = [] for i ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='K do Qπi = trainFQE (Sample (D, Ti, πi), πi) ρTi πi(·, ·) = trainFlow (Sample (D, Ti, πi)) η = E(s,a)∼D[Qπi(s, πi(s))] Initialize (θ) L = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' ∆ = 0 for (s, a, s′) ∈ D do w = max(min( ρ Ti πi(s,a) ˆµ(s,a) , ζ), 0) ℓ = −|w · (Ex∼Ti(s)[gθ(x)] − gθ(s′))| θ = θ + ∇θℓ ∆ = ∆ − Vmax · I � ρ Ti πi(s,a) ˆµ(s,a) > ζ � L = L + ℓ end score = 1 |D|(η + 1 1−γ (∆ + L)) scores ← score end i = argmax(scores) return πi Algorithm 3: MML-Linear: Minimax Model Learning Bound Input: offline RL data D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' set of dynamics, policy pairs [(π1, T1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=', (πK, TK)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Output: optimal policy π∗ scores = [] for i ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='K do Initialize (θ) L = 0 for (s, a, s′) ∈ D do ℓ = −(Ex∼Ti(s)[ψ(s, a, x)T θ] − ψ(s, a, s′)T θ) θ = θ + ∇θℓ L = L + ℓ end score = L |D| scores ← score end i = argmax(scores) return πi Algorithm 4: MML-RKHS: Minimax Model Learning Bound Input: offline RL data D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' set of dynamics, policy pairs [(π1, T1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=', (πK, TK)], kernel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Output: optimal policy π∗ scores = [] for i ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='K do L = 0 for (s, a, s′), (˜s, ˜a, ˜s′) ∈ D do ℓ1 = Ex∼T (s),˜x∼T (˜s)[K((s, a, x), (˜s, ˜a, ˜x))] ℓ2 = −2Ex∼T (s)[K((s, a, x), (˜s, ˜a, ˜s′))] ℓ3 = K((s, a, s′), (˜s, ˜a, ˜s′)) L = L + ℓ1 + ℓ2 + ℓ3 end score = L |D| scores ← score end i = argmax(scores) return πi D4RL Additional Experiments Ablation Study We conduct an ablation study in Table A1 where we evaluate the final performance of the policies selected using either FQE with TD-1 estimation or FQE with GAE estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We observe that using GAE for offline policy selection allows for picking better policies on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' MBLB with RKHS In this section, we derive the closed-form solution to supg∈G ℓw(g, T) when the test function g belongs to a reproducing kernel Hilbert space (RKHS), and empirically evaluate the MBLB method with RKHS parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let K : S ×S → R be a symmetric and positive definite kernel and HK its corresponding RKHS with inner product ⟨·, ·⟩HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Then we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' When G = {g ∈ HK : ⟨g, g⟩HK ≤ 1}, we have sup g∈G ℓw(g, T)2 = Es,a,s′∼D,x∼T (s,a)E˜s,˜a,˜s′∼D,˜x∼T (˜s,˜a) [w(s, a)w(˜s, ˜a)(K(x, ˜x) + K(s′, ˜s′) − K(x, ˜s′) − K(˜x, s′)] (81) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Let Kx ≜ K(x, ·) ∈ HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' By the reproducing property, we have ⟨Kx, Ky⟩HK = K(x, y) and ⟨Kx, g⟩HK = g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' As a Dataset Type Environment FQE (TD-1) FQE (GAE) medium hopper 507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8 (549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6) 533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 (532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6) med-expert hopper 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='3 (146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2) 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1 (157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9) expert hopper 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6) 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='7 (78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='7) medium halfcheetah 1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 (1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9) 2117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 (1215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6) med-expert halfcheetah 302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1 (605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2) 394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9 (632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0) Table A1: We report the mean and (standard deviation) of the selected policy’s environment performance across 3 random seeds using different variants of FQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' sup g∈G ℓw(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' T)2 = sup g:⟨g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='g⟩HK ≤1 Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='s′∼D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='x∼T (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)[w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a)(⟨Kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' g⟩HK − ⟨Ks′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' g⟩HK)]2 (82) = sup g:⟨g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='g⟩HK ≤1 � Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='s′∼D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='x∼T (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)[w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a)(Kx − Ks′)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' g �2 HK (83) = ∥Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='s′∼D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='x∼T (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)[w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a)(Kx − Ks′)]∥2 HK (Cauchy-Schwarz) = � Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='s′∼D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='x∼T (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)[w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a)(Kx − Ks′)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' E˜s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='˜a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='˜s′∼D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='˜x∼T (˜s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='˜a)[w(˜s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' ˜a)(K˜x − K˜s′)] � HK (84) = Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='s′∼D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='x∼T (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)E˜s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='˜a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='˜s′∼D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='˜x∼T (˜s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='˜a)[⟨w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a)(Kx − Ks′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' w(˜s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' ˜a)(K˜x − K˜s′)⟩HK] (85) = Es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='s′∼D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='x∼T (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='a)E˜s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='˜a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='˜s′∼D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='˜x∼T (˜s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='˜a)[w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' a)w(˜s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' ˜a)(K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' ˜x) + K(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' ˜s′) − K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' ˜s′) − K(˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' s′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' (86) Table A2 presents the performance of the MBLB algorithm with RKHS parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' On most of the environments, MBLB-RKHS performs better than/comparable with MML-RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' However, MBLB-Quad consistently outperforms MBLB- RKHS on all the environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' We suspect that MBLB-RKHS could outperform MBLB-Quad with different choices of kernels because the quadratic parameterization can be seen as a special case of RKHS parameterization (with quadratic kernels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Dataset Type Env MOPO MML (Squared) MML (Polynomial) MML (RKHS) MBLB (Linear) MBLB (Quad) MBLB (RKHS) medium hopper 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='3) 379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 (466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4) 375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6 (459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5) 375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 (459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9) 591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='7 (523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1) 808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 (502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='7) 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8 (476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4) med-expert hopper 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8 (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4) 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9 (131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5) 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 (148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0) 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1 (157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9) 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 (134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0) 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1 (144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='3) expert hopper 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8 (87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6 (61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 (56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2) 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2 (61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6) 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 (72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5) 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='9 (61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8) medium halfcheetah 599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8 (668.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4) 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6 (1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5) 2625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1 (937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2) 3858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2 (1231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1) 3290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 (1753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1) 2484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2 (1526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8) 2229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='7 (1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8) med-expert halfcheetah 486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6 (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1) 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5 (137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 (252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5) 343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2 (225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2) 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 (509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='5) 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8 (432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='1 (690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6) Table A2: We report the mean and (standard deviation) of selected policy’s simulator environment performance across 5 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' MML and MBLB are used as model-selection procedures where they select the best policy for each seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' Our method is choosing the most near-optimal policy across the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='0 Normalized Score (τ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} +page_content='00 Fraction of runs with score > τ MBLB MML MOPO Figure A1: Performance profile between three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FJT4oBgHgl3EQfBywq/content/2301.11426v1.pdf'} diff --git a/DNA0T4oBgHgl3EQfAv8K/content/tmp_files/2301.01965v1.pdf.txt b/DNA0T4oBgHgl3EQfAv8K/content/tmp_files/2301.01965v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..be78be9c46677617e80aa3d6c9c8d22120319134 --- /dev/null +++ b/DNA0T4oBgHgl3EQfAv8K/content/tmp_files/2301.01965v1.pdf.txt @@ -0,0 +1,3249 @@ +Inference on the intraday spot volatility from high-frequency order +prices with irregular microstructure noise +Markus Bibinger∗a +aFaculty of Mathematics and Computer Science, Julius-Maximilians-Universität Würzburg, +markus.bibinger@mathematik.uni-wuerzburg.de +Abstract +We consider estimation of the spot volatility in a stochastic boundary model with one-sided mi- +crostructure noise for high-frequency limit order prices. Based on discrete, noisy observations of an +Itô semimartingale with jumps and general stochastic volatility, we present a simple and explicit +estimator using local order statistics. We establish consistency and stable central limit theorems +as asymptotic properties. The asymptotic analysis builds upon an expansion of tail probabilities +for the order statistics based on a generalized arcsine law. In order to use the involved distribution +of local order statistics for a bias correction, an efficient numerical algorithm is developed. We +demonstrate the finite-sample performance of the estimation in a Monte Carlo simulation. +Keywords: +arcsine law, limit order book, market microstructure, nonparametric boundary +model, volatility estimation +MSC Classification: 62M09, 60J65, 60F05 +1. Introduction +Time series of intraday prices are typically described as a discretized path of a continuous-time +stochastic process. To have arbitrage-free markets the log-price process should be a semimartin- +gale. Risk estimation based on high-frequency data at highest available observation frequencies +has to take microstructure frictions into account. Disentangling these market microstructure ef- +fects from dynamics of the long-run price evolution has led to observation models with additive +noise, see, for instance, [9], [2] and [14]. The market microstructure noise, modelling for instance +the oscillation of traded prices between bid and ask order levels in an electronic market, is clas- +sically a centred (white) noise process with expectation equal to zero. These models can explain +many stylized facts of high-frequency data. Having available full limit order books including data +of submissions, cancellations and executions of bid and ask limit orders, however, it is not clear +which time series to consider at all. While challenging the concept of one price process it raises +the question if the information can be exploited more efficiently, in particular to improve risk +quantification. The considered stochastic boundary model for limit order prices of an order book +has been discussed by [5], [15] and Chapter 1.8 of [4]. It preserves the concept of an underlying +efficient, semimartingale log-price which determines long-run price dynamics and an additive, ex- +ogenous noise which models market-specific microstructure frictions. Its key idea is that ask order +prices should (in most cases) lie above the unobservable efficient price and bid prices below the +∗Financial support from the Deutsche Forschungsgemeinschaft (DFG) under grant 403176476 is gratefully ac- +knowledged. +1 +arXiv:2301.01965v1 [math.ST] 5 Jan 2023 + +efficient price. This leads to observation errors which are irregular in the sense of having non-zero +expectation and a distribution with a lower- or upper-bounded support. Considering without loss +of generality a model for (best) ask order prices, we obtain lower-bounded observation errors and +use local minima for the estimation. Modelling (best) bid prices instead would yield a model with +upper-bounded observation errors and local maxima could be used for an analogous estimation. +Both can be combined in practice. Inference on the spot volatility is one of the most important +topics in the financial literature, see, for instance, [17] and the references therein. In this work, +we address spot volatility estimation for the model from [5]. +It is known that the statistical and probabilistic properties of models with irregular noise are +very different than for regular noise and require other methods, see, for instance, [18], [13] and +[19]. Therefore, our estimation methods and asymptotic theory are quite different compared to +the market microstructure literature, while we can still profit from some of the techniques used +there. In [5] an estimator for the quadratic variation of a continuous semimartingale, that is, the +integrated volatility, was proposed with convergence rate n−1/3, based on n discrete observations +with one-sided noise. Optimality of the rate was proved in the standard asymptotic minimax +sense. +A main insight was that this convergence rate is better than the optimal rate, n−1/4, +under regular market microstructure noise. Using local minima over blocks of shrinking lengths +hn ∝ n−2/3 ∝ (nhn)−2, the resulting distribution of local minima is involved and infeasible, such +that in [5] a central limit theorem for the estimator could not be obtained. Our estimator is +related to a localized version of the one from [5], combined with truncation methods to eliminate +jumps of the semimartingale. For the asymptotic theory, however, we follow a different approach +choosing blocks of lengths hn, where hnn2/3 → ∞ slowly. This allows to establish stable central +limit theorems with the best achievable rate, arbitrarily close to n−1/6, in the important special +case of a semimartingale volatility. We exploit this to construct pointwise asymptotic confidence +intervals. +Although the asymptotic theory relies on block lengths that are slightly unbalanced by smooth- +ing out the impact of the noise distribution on the distribution of local minima asymptotically, our +numerical study demonstrates that the confidence intervals work well in realistic scenarios with +block lengths which optimize the estimator’s performance. Robustness to different noise specifi- +cations is an advantage that is naturally implied by our approach. Our estimator is surprisingly +simple, it is a local average of squared differences of block-wise minima times a constant factor +which comes from moments of the half-normal distribution of the minimum of a Brownian motion +over the unit interval. This estimator is consistent. However, the stable central limit theorem at +fast convergence rate requires a subtle bias correction which incorporates a more precise approxi- +mation of the asymptotic distribution of local minima. For that purpose, our analysis is based on +a generalization of the arcsine law which gives the distribution of the proportion of time over some +interval that a Brownian motion is positive. For a numerical computation of the bias-correction +function, we introduce an efficient algorithm. Reducing local minima over many random variables +to iterated minima of two random variables in each step combined with a convolution step, it can +be interpreted as a kind of dynamic programming approach. It turns out to be much more efficient +compared to the natural approximation by a Monte Carlo simulation and is a crucial ingredient +of our numerical application. Our convergence rate is much faster than the optimal rate, n1/8, for +spot volatility estimation under regular noise, see [10]. The main contribution of this work is to +2 + +develop the probabilistic foundation for the asymptotic analysis of the estimator and to establish +the stable central limit theorems, asymptotic confidence and a numerically practicable method. +The methods and proof techniques to deal with jumps are inspired by the truncation methods +pioneered in [16] and summarized in Chapter 13 of [11]. Overall, the strategy and restrictions on +jump processes are to some extent similar, while several details under irregular noise using order +statistics are rather different compared to settings without noise or with regular centred noise as +in [7]. +We introduce and further discuss our model in Section 2. Section 3 presents estimation methods +and Section 4 asymptotic results. The numerical application is considered in Section 5 and a +Monte Carlo simulation study illustrates an appealing finite-sample performance of the method. +All proofs are given in Section 6. +2. Model with lower-bounded, one-sided noise and assumptions +Consider an Itô semimartingale +Xt += +X0 + +� t +0 +as ds + +� t +0 +σs dWs + +� t +0 +� +R +δ(s, z)1{|δ(s,z)|≤1}(µ − ν)(ds, dz) ++ +� t +0 +� +R +δ(s, z)1{|δ(s,z)|>1}µ(ds, dz) , t ≥ 0 , +(1) +with a one-dimensional standard Brownian motion (Wt), defined on some filtered probability +space (ΩX, FX, (FX +t ), PX). For the drift process (at), and the volatility process (σt), we impose +the following quite general assumptions. +Assumption 1. The processes (at)t≥0 and (σt)t≥0 are locally bounded. The volatility process is +strictly positive, inft∈[0,1] σt > 0, PX-almost surely. For all 0 ≤ t + s ≤ 1, t ≥ 0, s ≥ 0, with some +constants Cσ > 0, and α > 0, it holds that +E +� +(σ(t+s) − σt)2� +≤ Cσs2α . +(2) +Condition (2) introduces a regularity parameter α, governing the smoothness of the volatility +process. The parameter α is crucial, since it will naturally influence convergence rates of spot +volatility estimation. Inequality (2) is less restrictive than α-Hölder continuity, since it does not +rule out volatility jumps. This is important as empirical evidence for volatility jumps, in particular +simultaneous price and volatility jumps, has been reported for intraday high-frequency financial +data, see, for instance, [22] and [6]. +The presented theory is moreover for general stochastic +volatilities, allowing as well for rough volatility. +Rough fractional stochastic volatility models +recently became popular and are used, for instance, in the macroscopic model of [8] and [20]. +The jump component of (1) is illustrated as in [11] and related literature, where the predictable +function δ is defined on Ω × R+ × R, and the Poisson random measure µ is compensated by +ν(ds, dz) = λ(dz) ⊗ ds, with a σ-finite measure λ. We impose the following standard condition +with a generalized Blumenthal-Getoor or jump activity index r, 0 ≤ r ≤ 2. +3 + +Assumption 2. Assume that supω,x |δ(t, x)|/γ(x) is locally bounded with a non-negative, deter- +ministic function γ which satisfies +� +R +(γr(x) ∧ 1)λ(dx) < ∞ . +(3) +We use the notation a ∧ b = min(a, b), and a ∨ b = max(a, b), throughout this paper. The +assumption is most restrictive in the case r = 0, when jumps are of finite activity. The larger r, +the more general jump components are allowed. We will develop results under mild restrictions +on r. +The process (Xt), which can be decomposed +Xt = Ct + Jt , +(4) +with the continuous component (Ct), and the càdlàg jump component (Jt), provides a model for +the latent efficient log-price process in continuous time. +High-frequency (best) ask order prices from a limit order book at times tn +i , 0 ≤ i ≤ n, on the +fix time interval [0, 1], cannot be adequately modelled by discrete recordings of (Xt). Instead, we +propose the additive model with lower-bounded, one-sided microstructure noise: +Yi = Xtn +i + ϵi , i = 0, . . . , n, +ϵi +iid +∼ Fη, ϵi ≥ 0 . +(5) +The crucial property of the model is that the support of the noise is lower bounded. It is not that +important, that this boundary is zero, it could be as well a different constant, or even a regularly +varying function over time. We set the bound equal to zero which appears to be the most natural +choice for limit orders. +Assumption 3. The i.i.d. noise (ϵi)0≤i≤n, has a cumulative distribution function (cdf) Fη satis- +fying +Fη(x) = ηx +� +1 + O(1) +� +, as x ↓ 0 . +(6) +This is a nonparametric model in that the extreme value index is −1 for the minimum domain +of attraction close to the boundary. This standard assumption on one-sided noise has been used +by [13] and [19] within different frameworks, as well. We do not require assumptions about the +maximum domain of attraction, moments and the tails of the noise distribution. Parametric exam- +ples which satisfy (6) are, for instance, the uniform distribution on some interval, the exponential +distribution and the standard Pareto distribution with heavy tails. +The i.i.d. assumption on the noise is crucial and generalizations to weakly dependent noise +will require considerable work and new proof concepts. Heterogeneity instead, that is, a time- +dependent noise level η(t), could be included in our asymptotic analysis under mild assumptions. +4 + +3. Construction of spot volatility estimators +We partition the observation interval [0, 1] in h−1 +n +equispaced blocks, h−1 +n +∈ N, and take local +minima on each block. We hence obtain for k = 0, . . . , h−1 +n +− 1, the local, block-wise minima +mk,n = min +i∈In +k +Yi , In +k = {i ∈ {0, . . . , n} : tn +i ∈ (khn, (k + 1)hn)} . +(7) +While h−1 +n +is an integer, nhn is in general not integer-valued. For a simple interpretation, however, +one can think of nhn as an integer-valued sequence which gives the number of noisy observations +per block. A spot volatility estimator could be obtained as a localized version of the estimator from +Eq. (2.9) in [5] for the integrated volatility in the analogous model. The idea is that differences +mk,n − mk−1,n of local minima estimate differences of efficient prices and a sum of squared differ- +ences can be used to estimate the volatility. However, things are not that simple. To determine +the expectation of squared differences of local minima we introduce the function +Ψn(σ2) = +π +2(π − 2)h−1 +n E +�� +min +i∈{0,...,nhn−1} +� +σB i +n + ϵi +� +− +min +i∈{1,...,nhn} +� +σ ˜B i +n + ϵi +��2� +, +(8) +where (Bt) and ( ˜Bt) denote two independent standard Brownian motions. For hnn2/3 → ∞, we +have that +Ψn(σ2) = σ2 + O(1) , +(9) +such that we do not require Ψ−1 +n +for a consistent estimator in this case. Note that we defined Ψn +different compared to [5] with the constant factor π/(π − 2). +When there are no price jumps, a simple consistent estimator for the spot squared volatility +σ2 +τ is given by +ˆσ2 +τ− = +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n +� +mk,n − mk−1,n)2 , +(10) +for suitable sequences hn → 0 and Kn → ∞. Using only observations before time τ, the estimator +is available on-line at time τ ∈ (0, 1] during a trading day. Working with ex-post data over the +whole interval, instead of using only observations before time τ, one may use as well +ˆσ2 +τ+ = +π +2(π − 2)Kn +(⌊h−1 +n τ⌋+Kn)∨(h−1 +n −1) +� +k=⌊h−1 +n τ⌋+1 +h−1 +n +� +mk,n − mk−1,n)2 , +(11) +or an estimator with an average centred around time τ ∈ (0, 1). The difference of the two estimators +(11) and (10) can be used to infer a possible jump in the volatility process at time τ ∈ (0, 1), as +well. +To construct confidence intervals for the spot volatility, it is useful to establish also a spot +5 + +quarticity estimator: +� +σ4τ − = +π +4(3π − 8)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−2 +n +� +mk,n − mk−1,n)4 . +(12) +A spot volatility estimator which is robust with respect to jumps in (Xt) is obtained with +threshold versions of these estimators. We truncate differences of local minima whose absolute +values exceed a threshold un = hκ +n, κ ∈ (0, 1/2), which leads to +ˆσ2,(tr) +τ− += +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n +� +mk,n − mk−1,n)21{|mk,n−mk−1,n|≤un} , +(13) +and analogous versions of the estimators (11) and (12). +4. Asymptotic results +We establish asymptotic results for equidistant observations, tn +i = i/n. We begin with the +asymptotic theory in a setup without jumps in (Xt). +Theorem 1 (Stable central limit theorem for continuous (Xt)). Set hn, such that hnn2/3 → ∞, +and Kn = CKhδ−2α/(1+2α) +n +for arbitrary δ, 0 < δ < 2α/(1 + 2α), and with some constant CK > 0. +If (Xt) is continuous, i.e. Jt = 0 in (4), under Assumptions 1 and 3, the spot volatility estimator +(10) is consistent, ˆσ2 +τ− +P→ σ2 +τ−, and satisfies the stable central limit theorem +K1/2 +n +� +ˆσ2 +τ− − Ψn +� +σ2 +τ− +�� +st +−→ N +� +0, 7π2/4 − 2π/3 − 12 +(π − 2)2 +σ4 +τ− +� +. +(14) +There is only a difference between σ2 +τ and its left limit σ2 +τ− in case of a volatility jump at +time τ. +In particular, the estimator is as well consistent for σ2 +τ, for any fix τ ∈ (0, 1). +The +convergence rate K−1/2 +n +gets arbitrarily close to n−2α/(3+6α), which is optimal in our model. In +the important special case when α = 1/2, for a semimartingale volatility, the rate is arbitrarily +close to n−1/6. This is much faster than the optimal rate of convergence in the model with additive +centred microstructure noise, which is known to be n−1/8, see [10]. The constant in the asymptotic +variance is obtained from several variance and covariance terms including (squared) local minima +and is approximately 2.44. The function Ψn was shown to be monotone and invertible in [5] and +Ψn and its inverse Ψ−1 +n +can be approximated using Monte Carlo simulations, see Section 5.1. The +asymptotic distribution of the estimator does not hinge on the noise level η, different to methods +for centred noise. Hence, we do not require any pre-estimation of noise parameters and the theory +directly extends to a time-varying noise level ηt in (6) under the mild assumption that 0 < ηt < ∞, +for all t. The stable convergence in (14) is stronger than weak convergence and is important, since +the limit distribution is mixed normal depending on the stochastic volatility. We refer to [11], +Section 2.2.1, for an introduction to stable convergence. For a normalized central limit theorem, +we can use the spot quarticity estimator (12). +Proposition 2 (Feasible central limit theorem). Under the conditions of Theorem 1, the spot +quarticity estimator (12) is consistent, such that we get for the spot volatility estimation the +6 + +normalized central limit theorem +K1/2 +n +π − 2 +� +� +σ4τ −(7π2/4 − 2π/3 − 12) +� +ˆσ2 +τ− − Ψn +� +σ2 +τ− +�� +d +−→ N(0, 1) . +(15) +Proposition 2 yields asymptotic confidence intervals for spot volatility estimation. For q ∈ +(0, 1), it holds true that +P +� +σ2 +τ− ∈ +� +Ψ−1 +n +� +ˆσ2 +τ− − +π − 2 +� +� +σ4τ −(7π2/4 − 2π/3 − 12) +K−1/2 +n +Φ−1(1 − q) +� +, +Ψ−1 +n +� +ˆσ2 +τ− + +π − 2 +� +� +σ4τ −(7π2/4 − 2π/3 − 12) +K−1/2 +n +Φ−1(1 − q) +��� +→ 1 − q , +by monotonicity of Ψ−1 +n +with Φ the cdf of the standard normal distribution. Since Ψ−1 +n +is differ- +entiable and the derivative is +� +Ψ−1 +n +�′ = 1 + O(1), the delta method (for stable convergence) yields +as well asymptotic confidence intervals and the central limit theorem +K1/2 +n +� +Ψ−1 +n +� +ˆσ2 +τ− +� +− σ2 +τ− +� +st +−→ N +� +0, 7π2/4 − 2π/3 − 12 +(π − 2)2 +σ4 +τ− +� +. +(16) +We may not simply replace Ψn +� +σ2 +τ− +� +by its first-order approximation σ2 +τ− in (14), since the bias +multiplied with K1/2 +n +does in general not converge to zero. If the condition hnn2/3 → ∞ is violated, +this central limit theorem does not apply. +Theorem 3 (Stable central limit theorem with jumps in (Xt)). Set hn, such that hnn2/3 → ∞, +and Kn = CKhδ−2α/(1+2α) +n +for arbitrary δ, 0 < δ < 2α/(1 + 2α), and with some constant CK > 0. +Under Assumptions 1, 2 and 3, with +r < 2 + 2α +1 + 2α , +(17) +the truncated spot volatility estimator (13) with +κ ∈ +� +1 +2 − r +α +2α + 1, 1 +2 +� +, +(18) +is consistent, ˆσ2,(tr) +τ− +P→ σ2 +τ−, and satisfies the stable central limit theorem +K1/2 +n +� +ˆσ2,(tr) +τ− +− Ψn +� +σ2 +τ− +�� +st +−→ N +� +0, 7π2/4 − 2π/3 − 12 +(π − 2)2 +σ4 +τ− +� +. +(19) +In order to obtain a central limit theorem at (almost) optimal rate, we thus have to impose +mild restrictions on the jump activity. For the standard model with a semimartingale volatility, +i.e. α = 1/2, we need that r < 3/2, and for α = 1 we have the strongest condition that r < 4/3. +These conditions are equivalent to the ones of Theorem 1 in [7], which gives a central limit +theorem for spot volatility estimation under similar assumptions on (Xt), but with slower rate +of convergence for centred microstructure noise. Using a truncated quarticity estimator with the +same thresholding yields again a feasible central limit theorem and asymptotic confidence intervals. +7 + +Remark 1. From a theoretical point of view one might ponder why we do not work out an asymp- +totic theory for hn ∝ n−2/3, when noise and efficient price both influence the asymptotic distribu- +tion of the local minima. However, in this balanced case, the asymptotic distribution is infeasible. +For this reason, [5] could not establish a central limit theorem for their integrated volatility esti- +mator. Moreover, their estimator was only implicitly defined depending on the unknown function +Ψ−1 +n . Even imposing a parametric assumption on the noise as an exponential distribution would +not render a feasible limit theory for hn ∝ n−2/3, see the discussion in [5]. Choosing hn, such +that hnn2/3 → ∞ slowly, yields instead a simple, explicit and consistent estimator and a fea- +sible central limit theorem for spot volatility estimation. In particular, we use Ψn only for the +bias-correction of the simple estimator, while the estimator itself and the (estimated) asymptotic +variance do not hinge on Ψn. Central limit theorems for spot volatility estimators are in general +only available at almost optimal rates, when the variance dominates the squared bias in the mean +squared error, see, for instance, Theorem 13.3.3 and the remarks below in [11]. Therefore, (14) +is the best achievable central limit theorem. Our choice of hn avoids moreover strong assumptions +on the noise that would be inevitable for smaller blocks. Our numerical work will demonstrate that +the presented asymptotic results are useful in practice and can be applied without loosing (much) +efficiency compared to a different selection of blocks. +5. Implementation and simulations +5.1. Monte Carlo approximation of Ψn +Although the function Ψn from (8) tends to the identity asymptotically, it has a crucial role +for a bias correction of our estimator in (14). We can compute the function numerically based +on a Monte Carlo simulation. Hence, we have to compute Ψn(σ2) as a Monte Carlo mean over +many iterations and over a fine grid of values for the squared volatility. +Then, we can also +numerically invert the function and use Ψ−1 +n ( · ). To make this procedure feasible without too +high computational expense we require an algorithm to efficiently sample from the law of the local +minima for some given n and block length hn. +Consider for nhn ∈ N, with Zi +iid +∼ N(0, 1), and the observation errors (ϵk)k≥0, the minimum +M nhn +1 +:= +min +k=1,...,nhn +� σ +√n +k +� +i=1 +Zi + ϵk +� +, +for some fix σ > 0, and for l ∈ {0, . . . , nhn}: +M nhn +l +:= +min +k=l,...,nhn +� σ +√n +k +� +i=0 +Zi + ϵk +� +, +where we set Z0 := 0. Since +Ψn(σ2) = 1 +2 +π +π − 2h−1 +n E +�� +M nhn−1 +0 +− M nhn +1 +�2� +, +with M nhn−1 +0 +generated independently from M nhn +1 +, we want to simulate samples distributed as +M nhn−1 +0 +and M nhn +1 +, respectively. Note that the moments of M nhn−1 +0 +and M nhn +1 +slightly differ +8 + +what can be relevant for moderate values of nhn. As in the simulation of Section 5.2, we im- +plement exponentially distributed observation errors (ϵk), with some given noise level η. In data +applications, we can do the same with an estimated noise level +ˆη = +� 1 +2n +n +� +i=1 +� +Yi − Yi−1 +�2 +�−1/2 += η + OP +� +n−1/2� +. +This estimator works for all noise distributions with finite fourth moments. To simulate the local +minima for given n, hn, η, and squared volatility σ2, in an efficient way we use a specific dynamic +programming principle. Observe that +M nhn +1 += +σ +√nZ1 + min +� +ϵ1, M nhn +2 +� += +σ +√nZ1 + min +� +ϵ1, σ +√nZ2 + min +� +ϵ2, M nhn +3 +�� += +σ +√nZ1 + min +� +. . . min +� +ϵnhn−2, σ +√nZnhn−1 + min +� +ϵnhn−1, σ +√nZnhn + ϵnhn +�� +. . . +� +. +In the baseline noise model, ϵk +iid +∼ Exp(η), the random variable +σ +√nZnhn+ϵnhn has an exponentially +modified Gaussian (EMG) distribution. With any fixed noise distribution, we can easily generate +realizations from this convolution. A pseudo random variable which is distributed as M nhn +1 +is now +generated following the last transformation in the reverse direction. In pseudo code, this reads +1. Generate U_{nh_n}~ EMG(sigma^2/n,eta)~ Exp(eta)+sigma/sqrt(n)*Norm(1) +2. U_{nh_n-1}=min(U_{nh_n},Exp(eta))+sigma/sqrt(n)*Norm(1) +3. iterate until U_1 +where the end point U1 has the target distribution of M nhn +1 +. In each iteration step, we thus take +the minimum of the current state of the process with one independent exponentially distributed +random variable and the convolution with one independent normally distributed random variable. +To sample from the distribution of M nhn−1 +0 +instead, we use the same algorithm and just drop the +convolution with the normal distribution in the last step. +It turns out that this algorithm facilitates a many times faster simulation compared to a +classical simulation starting with a discretized path of (Xt). +Figure 1 plots the result of the Monte Carlo approximation of Ψn(σ2) for n = 23,400 and +n · hn = 15, on a grid of 1500 values of σ2. In this case, hn is quite small, but this configuration +turns out to be useful below in Section 5.2. We know that Ψn(σ2) is monotone, such that the +oscillation of the blue line in Figure 1 is due to the inaccuracy of the Monte Carlo means although +we use N = 100,000 iterations for each grid point. Nevertheless, we can see that the function is +rather close to a linear function with slope 1.046 based on a least squares estimate. The left plot of +Figure 1 draws a comparison to the identity function which is illustrated by the dotted line, while +the plot right-hand side draws a comparison to the linear function with slope 1.046. We see that it +is crucial to correct for the bias in (14) when using such small values of hn. Although the function +Ψn(σ2) is not exactly linear, a simple bias correction dividing estimates by 1.046 is almost as good +as using the more precise numerical inversion based on the Monte Carlo approximation. Since the +Monte Carlo approximations of Ψn(σ2) look close to linear functions in all considered cases, we +9 + +Figure 1: Monte Carlo means to estimate Ψn(σ2) over a fine grid (blue line) for n = 23,400 and n · hn = 15. Left, +the dotted line shows the identity function, right the dotted line is a linear function with slope 1.046. +Table 1: Regression slopes to measure the bias of estimator (10) and deviation Ψn(σ2) − σ2. +n · hn +10 +15 +25 +39 +78 +234 +h−1 +n +2340 +1560 +936 +600 +300 +100 +hn · n2/3 +0.350 +0.524 +0.874 +1.36 +2.73 +8.18 +slope +1.077 +1.046 +1.025 +1.016 +1.008 +1.003 +approx. bias +7.7% +4.6% +2.5% +1.6% +0.8% +0.3% +report the estimated slopes based on least squares and N = 100,000 Monte Carlo iterations for +different values of hn in Table 1 to summarize concisely how far the distance between the function +Ψn(σ2) and the identity is. Simulating all iterations for all grid points with our algorithm takes +only a few hours with a standard computer. +5.2. Simulation study of estimators +We simulate n = 23,400 observations corresponding to one observation per second over a +(NASDAQ) trading day of 6.5 hours. The efficient price process is simulated from the model +dXt = νtσt dWt , +dσ2 +t = 0.0162 · +� +0.8465 − σ2 +t +� +dt + 0.117 · σt dBt , +νt = +� +6 − sin(3πt/4) +� +· 0.002 , t ∈ [0, 1] . +The factor (νt) generates a typical U-shaped intraday volatility pattern. +(Wt, Bt) is a two- +dimensional Brownian motion with leverage d[W, B]t = 0.2 dt. The stochastic volatility component +has several realistic features and the simulated model is in line with recent literature, see [6] and +references therein. Observations with lower-bounded, one-sided microstructure noise are generated +by +Yi = X i +n + ϵi , 0 ≤ i ≤ n , +10 + +0.00012 +80000'0 +f(c3) +0.00004 +00000' +0.00000 +0.00004 +0.00008 +0.000120.00012 +80000'0 +4(c3) +0.00004 +00000' +0.00000 +0.00004 +0.00008 +0.00012 +3Figure 2: True and estimated spot volatility with pointwise confidence sets. +with exponentially distributed noise, ϵi +iid +∼ Exp(η), with η = 10,000. The noise variance is then +rather small, but this is in line with stylized facts of real NASDAQ data as, for instance, those +analysed in [6].1 +The black line in Figure 2 shows a fixed path of the squared volatility. We fix this path for the +following Monte Carlo simulation and generate new observations of (Xt) and (Yi) in each iteration +according to our model. The blue line in Figure 2 gives the estimated volatility by the Monte +Carlo means over N = 50,000 iterations based on n·hn = 15 observations per block using the non- +adjusted estimator (10) with windows which are centred around the block on that we estimate +the spot volatility and with Kn = 180. We plot estimates on each block, where the estimates +close to the boundaries rely on less observations. The red line gives the bias-corrected volatility +estimates using the numerically evaluated function Ψn, based on the algorithm from Section 5.1 +with n · hn = 15 and n = 23,400. We determined the values n · hn = 15 and Kn = 180 as suitable +values to obtain a small mean squared error. In fact, the choice of Kn = 180 is rather large in +favour of a smaller variance what yields a rather smooth estimated spot volatility in Figure 2. +The estimated volatility hence appears smoother compared to the true semimartingale volatility, +but the intraday pattern is well captured by our estimation. We expect that this is typically +an appealing implementation in practice as smaller Kn results in a larger variance. Choosing +Kn = 180 rather large, we have to use quite small block sizes hn, to control the overall bias of +the estimation. Since hn · n2/3 ≈ 0.52 is small, the bias correction becomes crucial here. Still, +1Note that the noise level estimate is analogous to the one used for regular market microstructure noise. Typical +noise levels obtained for trades of e.g. Apple are approx. 15,000 and approx. 4,000 for 3M. For mid quotes or best +ask/bid prices the levels are only slightly larger (variance smaller). +11 + +0.00018 +0.00014 +0.00010 +90000'0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +timeTable 2: Summary statistics of estimation for different values of hn and Kn, MSD = mean standard deviation, +MAB = mean absolute bias, MABC = MAB of bias-corrected estimator. +Kn +120 +180 +240 +nhn +MSD +MAB +MABC +MSD +MAB +MABC +MSD +MAB +MABC +10 +14.6 +7.59 +0.73 +12.0 +7.51 +0.90 +10.5 +7.60 +1.13 +15 +14.4 +4.59 +0.88 +11.8 +4.57 +1.17 +10.3 +4.46 +1.43 +25 +14.3 +2.56 +1.24 +11.8 +2.63 +1.66 +10.3 +2.86 +1.91 +78 +14.7 +2.44 +2.52 +12.3 +3.53 +3.42 +11.0 +4.33 +4.16 +All values multiplied with factor 106. +our asymptotic results work well for this implementation. This can be seen by the comparison of +pointwise empirical 10% and 90% quantiles from the Monte Carlo iterations illustrated by the grey +area and the 10% and 90% quantiles of the limit normal distribution with the asymptotic variance +from (14). The latter are drawn as dotted lines for the blocks with larger distance than Kn/2 +from the boundaries where the variances are of order K−1 +n . Close to the boundaries the empirical +variances increase due to the smaller number of blocks used for the estimates. Moreover, the bias +correction which is almost identical to dividing each estimate by 1.046, correctly scales the simple +estimates which have a significant positive bias for the chosen tuning parameters. Overall, our +asymptotic results provide a good finite-sample fit even though we have hn · n2/3 < 1 here. Note, +however, that σt · η ≈ 100, and our asymptotic expansion requires in fact that hn · n2/3σt · η is +large when taking constants into account. +Table 2 summarizes the performance of the estimation along different choices of nhn and Kn. +We give the following quantities: +1. MSD: the mean standard deviation of N iterations averaged over all grid points; +2. MAB: the mean absolute bias of N iterations averaged over all grid points and for estimator +(10) without any bias correction; +3. MABC: the mean absolute bias of N iterations averaged over all grid points and for estimator +(10) with a simple bias correction dividing estimates by the factors given in Table 1. +All results are based on N = 50,000 Monte Carlo iterations. First of all, the values used for Figure +2 are not unique minimizers of the mean squared error. Several other combinations given in Table +2 render equally well results. Overall, the performance is comparable within a broad range of +block lengths and window sizes. The variances decrease for larger Kn, while the bias increases +with larger Kn for fixed hn. Important for the bias is the total window size, Kn · hn, over that +the volatility is approximated constant for the estimation. The variance only depends on Kn, +changing the block length for fix Kn does not significantly affect the variance. While the MSD is +hence almost constant within the columns of Table 2, the bias after correction, MABC, increases +from top down due to the increasing window size. Without the bias correction two effects interfere +for MAB. Larger blocks reduce the systematic bias due to Ψn(σ2 +t ) − σ2 +t , but the increasing bias +due to the increasing window size prevails for n · hn = 78, and the two larger values of Kn. +12 + +6. Proofs +6.1. Law of the integrated negative part of a Brownian motion +A crucial lemma for our theory is on an upper bound for the cdf of the integrated negative +part of a Brownian motion. We prove a lemma based on a generalization of Lévy’s arc-sine law +by [21]. The result is in line with the conjecture in Eq. (261) of [12] where one finds an expansion +of the density with a precise constant of the leading term. Denote by f+ the positive part and by +f− the negative part of some real-valued function f. +Lemma 4. For a standard Brownian motion (Wt)t≥0, it holds that +P +� � 1 +0 +(Wt)− dt ≤ x +� += O(x1/3), x → 0 . +Proof. Observe the equality in distribution +� 1 +0 (Wt)− dt +d= +� 1 +0 (Wt)+ dt, such that +P +� � 1 +0 +(Wt)− dt ≤ x +� += P +� � 1 +0 +(Wt)+ dt ≤ x +� +, x > 0 . +For any ϵ > 0, the inequality +� 1 +0 +(Wt)+ dt ≥ +� 1 +0 +Wt · 1(Wt > ϵ) dt ≥ ϵ +� 1 +0 +1(Wt > ϵ) dt +leads us to +P +� � 1 +0 +(Wt)+ dt ≤ x +� +≤ P +� +ϵ +� 1 +0 +1(Wt > ϵ) dt ≤ x +� += P +� +1 − +� 1 +0 +1(Wt ≤ ϵ) dt ≤ x/ϵ +� += P +� � 1 +0 +1(Wt ≤ ϵ) dt ≥ 1 − x/ϵ +� +. +Using (15) and (16) from [21], we obtain that +P +� � 1 +0 +1(Wt ≤ ϵ) dt ≥ 1 − x/ϵ +� += 1 +π +� 1 +1−x/ϵ +exp(−ϵ2/(2u)) +� +u(1 − u) +du + 2Φ(ϵ) − 1 , +with Φ the cdf of the standard normal distribution. Thereby, we obtain that +P +� � 1 +0 +(Wt)+ dt ≤ x +� +≤ 1 +π +� 1 +1−x/ϵ +exp(−ϵ2/(2u)) +� +u(1 − u) +du + 2 +� ϵ +0 +exp(−u2/2) +√ +2π +du , +and elementary bounds give the upper bound +P +� � 1 +0 +(Wt)+ dt ≤ x +� +≤ 2 +π +�x +ϵ +1 +� +1 − x/ϵ ++ +2ϵ +√ +2π . +13 + +Choosing ϵ = x1/3, we obtain the upper bound +P +� � 1 +0 +(Wt)+ dt ≤ x +� +≤ 2 +π x1/3 +1 +√ +1 − x2/3 + 2x1/3 +√ +2π . +6.2. Asymptotics of the spot volatility estimation in the continuous case +6.2.1. Proof of Theorem 1 +In the sequel, we write An ≲ Bn for two real sequences, if there exists some n0 ∈ N and a +constant K, such that An ≤ KBn, for all n ≥ n0. +Step 1 +In the first step, we prove the approximation +ˆσ2 +τ− = +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n +� +mk,n − mk−1,n)2 += +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n +� +˜mk,n − ˜m∗ +k−1,n)2 + OP +� +hα∧1/2 +n +� +with +˜mk,n = min +i∈In +k +� +ϵi + σ(k−1)hn(Wtn +i − Wkhn) +� +, and +˜m∗ +k−1,n = min +i∈In +k−1 +� +ϵi − σ(k−1)hn(Wkhn − Wtn +i ) +� +. +We show that for k ∈ {1, . . . , h−1 +n +− 1}, it holds that +mk,n − mk−1,n = ˜mk,n − ˜m∗ +k−1,n + OP +� +h1/2 +n +� +. +(20) +We subtract Xkhn from mk,n and mk−1,n, and use that it holds for all i that +� +Yi − Xkhn +� +− +� +Xtn +i − +� +Xkhn + σ(k−1)hn(Wtn +i − Wkhn) +�� += +� +σ(k−1)hn(Wtn +i − Wkhn) + ϵi +� +. +This implies that +min +i∈In +k +� +Yi−Xkhn +� +−max +i∈In +k +� +Xtn +i − +� +Xkhn+σ(k−1)hn(Wtn +i −Wkhn) +�� +≤ min +i∈In +k +� +σ(k−1)hn(Wtn +i −Wkhn)+ϵi +� +. +Changing the roles of +� +Yi − Xkhn +� +and +� +σ(k−1)hn(Wtn +i − Wkhn) + ϵi +� +, we obtain by the analogous +inequalities and the triangle inequality, with Mt := Xkhn + +� t +khn σ(k−1)hn dWs, that +���mk,n − Xkhn − ˜mk,n +��� ≤ max +i∈In +k +��Xtn +i − Mtn +i +�� ≤ +sup +t∈[khn,(k+1)hn] +��Xt − Mt +�� +≤ +sup +t∈[khn,(k+1)hn] +���Ct − Ckhn − +t +∫ +khn +σ(k−1)hn dWs +��� . +14 + +We write (Ct) for (Xt) to emphasize continuity, see (4). (20) follows from +sup +t∈[khn,(k+1)hn] +���Ct − Ckhn − +t +∫ +khn +σ(k−1)hn dWs +��� = OP(h1/2 +n ) , +(21) +and the analogous estimate for mk−1,n and ˜m∗ +k−1,n. We decompose +sup +t∈[khn,(k+1)hn] +���Ct − Ckhn − +t +∫ +khn +σ(k−1)hn dWs +��� ≤ +sup +t∈[khn,(k+1)hn] +��� +t +∫ +khn +(σs − σ(k−1)hn) dWs +��� ++ +sup +t∈[khn,(k+1)hn] +� t +khn +|as|ds . +Under Assumption 1, we can assume that (σt) and (at) are bounded on [0, 1] by the localization +from Section 4.4.1 in [11]. Using Itô’s isometry and Assumption 1, we obtain that +E +�� � t +khn +(σs − σ(k−1)hn) dWs +�2� += +� t +khn +E +� +(σs − σ(k−1)hn)2� +ds += O +� � t +khn +(s − (k − 1)hn)2α ds +� += O +� +(t − (k − 1)hn)2α+1� +. +By Doob’s martingale maximal inequality and since supt∈[khn,(k+1)hn] +� t +khn |as|ds = OP(hn), it +holds that +sup +t∈[khn,(k+1)hn] +���Ct − Ckhn − +t +∫ +khn +σ(k−1)hn dWs +��� = OP +� +h(1/2+α)∧1 +n +� +. +We conclude that (21) holds, since α > 0. Since +h−1 +n +� +mk,n − mk−1,n +�� +mk,n − ˜mk,n +� += OP +� +hα∧1/2 +n +� +, +and analogously for (mk−1,n − ˜m∗ +k−1,n), we conclude Step 1. +Step 2 +We bound the bias of the spot volatility estimation using Step 1. For ⌊h−1 +n τ⌋ > Kn, we obtain +with the function Ψn from (8) that +E +� +ˆσ2 +τ− − Ψn +� +σ2 +τ− +�� += += +1 +Kn +π +2(π − 2) +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n E +�� +˜mk,n − ˜m∗ +k−1,n)2� +− E +� +Ψn +� +σ2 +τ− +�� ++ O +� +hα∧1/2 +n +� += +1 +Kn +π +2(π − 2) +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +2(π − 2) +π +E +� +Ψn +� +σ2 +(k−1)hn +�� +− E +� +Ψn +� +σ2 +τ− +�� ++ O +� +hα∧1/2 +n +� +≲ +1 +Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +E +� +σ2 +(k−1)hn − σ2 +τ− +� ++ O +� +hα∧1/2 +n +� +≲ +1 +Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +E +� +σ(k−1)hn − στ− +� ++ O +� +hα∧1/2 +n +� +15 + +≲ +1 +Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +� +E +�� +σ(k−1)hn − στ− +�2��1/2 ++ O +� +hα∧1/2 +n +� += O +� +(Kn hn)α� += O +� +hα/(1+2α) +n +� += O +� +K−1/2 +n +� +. +We used that (α ∧ 1/2) > α/(2α + 1) for all α. For the asymptotic upper bounds we used the +binomial formula and Hölder’s inequality to conclude with (2) from Assumption 1. +Step 3 +For (9) and the consistency of ˆσ2 +τ−, we prove that +E +� +ˆσ2 +τ− +� += σ2 +τ− + O(1) . +(22) +Denote by Pσ(k−1)hn the regular conditional probabilities conditioned on σ(k−1)hn, and Eσ(k−1)hn +the expectations with respect to the conditional measures. We obtain by the tower rule that +E +� +h−1 +n +� +˜mk,n − ˜m∗ +k−1,n)2� += E +� +h−1 +n Eσ(k−1)hn +�� +˜mk,n − ˜m∗ +k−1,n)2�� += E +� +Eσ(k−1)hn +�� +h−1/2 +n +˜mk,n)2� ++ Eσ(k−1)hn +�� +h−1/2 +n +˜m∗ +k−1,n)2� +− 2 Eσ(k−1)hn +� +h−1/2 +n +˜mk,n +� +Eσ(k−1)hn +� +h−1/2 +n +˜m∗ +k−1,n +�� +, +by the conditional independence of ˜mk,n and ˜m∗ +k−1,n. +We establish and use an approximation of the tail probabilities of ( ˜mk,n) and ( ˜m∗ +k−1,n), re- +spectively. For x ∈ R, we have that +Pσ(k−1)hn +� +h−1/2 +n +min +i∈In +k +� +ϵi + σ(k−1)hn(Wtn +i − Wkhn) +� +> xσ(k−1)hn +� += Pσ(k−1)hn +� +min +i∈In +k +� +h−1/2 +n +� +Wtn +i − Wkhn +� ++ h−1/2 +n +σ−1 +(k−1)hnϵi +� +> x +� += Eσ(k−1)hn +� ⌊(k+1)nhn⌋ +� +i=⌊knhn⌋+1 +P +� +ϵi > h1/2 +n σ(k−1)hn +� +x − h−1/2 +n +(Wtn +i − Wkhn) +� +|FX�� += Eσ(k−1)hn +� +exp +� ⌊(k+1)nhn⌋ +� +i=⌊knhn⌋+1 +log +� +1 − Fη +� +h1/2 +n σ(k−1)hn +� +x − h−1/2 +n +(Wtn +i − Wkhn) +����� +by the tower rule for conditional expectations, and since ϵi +iid +∼ Fη. It holds that +Wtn +i − Wkhn = +i−⌊knhn⌋ +� +j=1 +˜Uj, ˜Uj +iid +∼ N(0, n−1), j ≥ 2, ˜U1 ∼ N +� +0, tn +⌊knhn⌋+1 − khn +� +, +Uj = h−1/2 +n +˜Uj +iid +∼ N +� +0, (nhn)−1� +, j ≥ 2, U1 ∼ N +� +0, h−1 +n +� +tn +⌊knhn⌋+1 − khn +�� +. +We apply a Riemann sum approximation with a standard Brownian motion (Bt)t≥0. With (6), +and a first-order Taylor expansion of z �→ log(1 − z), we obtain that +Pσ(k−1)hn +� +h−1/2 +n +min +i∈In +k +� +ϵi + σ(k−1)hn(Wtn +i − Wkhn) +� +> xσ(k−1)hn +� += +16 + += Eσ(k−1)hn +� +exp +� +− h1/2 +n σ(k−1)hnη +⌊(k+1)nhn⌋ +� +i=⌊knhn⌋+1 +� +x − +i−⌊knhn⌋ +� +j=1 +Uj +� ++(1 + O(1)) +�� += Eσ(k−1)hn +� +exp +� +− h1/2 +n nhnσ(k−1)hnη +� 1 +0 +(Bt − x)− dt (1 + O(1)) +�� +. +If nh3/2 +n +→ ∞, we deduce that +Pσ(k−1)hn +� +h−1/2 +n +min +i∈In +k +� +ϵi + σ(k−1)hn(Wtn +i − Wkhn) +� +> xσ(k−1)hn +� += P +� +inf +0≤t≤1 Bt ≥ x +� ++ Eσ(k−1)hn +� +1 +� +inf +0≤t≤1 Bt < x +� +exp +� +− h3/2 +n nσ(k−1)hnη +� 1 +0 +(Bt − x)− dt (1 + O(1)) +�� += P +� +inf +0≤t≤1 Bt ≥ x +� ++ P +� +inf +0≤t≤1 Bt < x +� +· O(1) . +(23) +We do not have a lower bound for +� 1 +0 (Bt − x)− dt. However, using that the first entry time Tx of +(Bt) in x, conditional on {inf0≤t≤1 Bt < x}, has a continuous conditional density f(t|Tx < 1), by +Lemma 4 and properties of the Brownian motion we obtain for any δ > 0 that +Eσ(k−1)hn +� +1 +� +inf +0≤t≤1 Bt < x +� +exp +� +− h3/2 +n nσ(k−1)hnη +� 1 +0 +(Bt − x)− dt +�� +≤ exp +� +− +� +h3/2 +n n +�δσ(k−1)hnη +� +P( inf +0≤t≤1 Bt < x) + P +� +inf +0≤t≤1 Bt < x, +� 1 +0 +(Bt − x)− dt ≤ +� +h3/2 +n n +�−1+δ� +≤ +� +exp +� +− +� +h3/2 +n n +�δσ(k−1)hnη +� ++ +� 1 +0 +P +� � 1 +s +(Bt)− dt ≤ +� +h3/2 +n n +�−1+δ� +f(s|Tx < 1) ds +� +P( inf +0≤t≤1 Bt < x) +≤ +� +exp +� +− +� +h3/2 +n n +�δσ(k−1)hnη +� ++ +� 1 +0 +P +� +(1 − s) +� 1 +0 +(Bt)− dt ≤ +� +h3/2 +n n +�−1+δ� +f(s|Tx < 1) ds +� +× P( inf +0≤t≤1 Bt < x) += P( inf +0≤t≤1 Bt < x) · Rn , +with a remainder +Rn = O +�� +h3/2 +n n +�− 1+δ +3 � +. +We applied Lemma 4 in the last step. From the unconditional Lévy distribution of Tx, f(s|Tx < 1) +is explicit, but its precise form does not influence the asymptotic order. Under the condition +nh3/2 +n +→ ∞, the minimum of the Brownian motion over the interval hence dominates the noise +in the distribution of local minima, different than for a choice hn ∝ n−2/3. By the reflection +principle, it holds that +P +� +− inf +0≤t≤1 Bt ≥ x +� += P +� +sup +0≤t≤1 +Bt ≥ x +� += 2P +� +B1 ≥ x +� += P +� +|B1| ≥ x +� +, +(24) +for x ≥ 0. +Using the illustration of moments by integrals over tail probabilities we exploit this, and a +completely analogous estimate for ˜m∗ +k−1,n, to approximate conditional expectations. This yields +17 + +that +Eσ(k−1)hn +� +h−1/2 +n +˜mk,n +� += += +� ∞ +0 +Pσ(k−1)hn +� +h−1/2 +n +˜mk,n > x +� +dx − +� ∞ +0 +Pσ(k−1)hn +� +− h−1/2 +n +˜mk,n > x +� +dx += − +� ∞ +0 +Pσ(k−1)hn +� +σ(k−1)hn sup +0≤t≤1 +Bt > x +� +dx + OP(1) += − +� ∞ +0 +Pσ(k−1)hn +� +σ(k−1)hn|B1| > x +� +dx + OP(1) += −Eσ(k−1)hn +� +σ(k−1)hn|B1| +� ++ OP(1) += − +� +2 +π σ(k−1)hn + OP(1) . +We used (24). An analogous computation yields the same result for ˜m∗ +k−1,n: +Eσ(k−1)hn +� +h−1/2 +n +˜m∗ +k−1,n +� += − +� +2 +π σ(k−1)hn + OP(1) . +For the second conditional moments, we obtain that +Eσ(k−1)hn +� +h−1 +n +� +˜mk,n +�2� += 2 +� ∞ +0 +x Pσ(k−1)hn +� +|h−1/2 +n +˜mk,n| > x +� +dx += 2 +� ∞ +0 +x Pσ(k−1)hn +� +σ(k−1)hn sup +0≤t≤1 +Bt > x +� +dx + OP(1) += 2 +� ∞ +0 +x Pσ(k−1)hn +� +σ(k−1)hn|B1| > x +� +dx + OP(1) += σ2 +(k−1)hn + OP(1) . +The last identity uses the illustration of the second moment of the normal distribution as an +integral over tail probabilities. An analogous computation yields that +Eσ(k−1)hn +� +h−1 +n +� +˜m∗ +k−1,n +�2� += σ2 +(k−1)hn + OP(1) . +This proves (22). +Step 4 +We determine the asymptotic variance of the estimator. Illustrating moments as integrals over +tail probabilities, with the analogous approximation as above, we obtain that +Varσ(k−1)hn +� +˜m2 +k,n +� += Eσ(k−1)hn +� +˜m4 +k,n +� +− +� +Eσ(k−1)hn +� +˜m2 +k,n +��2 += 2σ4 +(k−1)hnh2 +n + OP(h2 +n), +Covσ(k−1)hn +� +˜m2 +k,n, ˜mk,n ˜m∗ +k−1,n +� += Eσ(k−1)hn +� +˜m3 +k,n +� +Eσ(k−1)hn +� +˜m∗ +k−1,n +� +− Eσ(k−1)hn +� +˜m2 +k,n +� +Eσ(k−1)hn +� +˜mk,n +� +Eσ(k−1)hn +� +˜m∗ +k−1,n +� += 2 +π σ4 +(k−1)hnh2 +n + OP(h2 +n), +Varσ(k−1)hn +� +˜mk,n ˜m∗ +k−1,n +� += Eσ(k−1)hn +� +˜m2 +k,n +� +Eσ(k−1)hn +�� +˜m∗ +k−1,n +�2� +18 + +− +� +Eσ(k−1)hn +� +˜mk,n +� +Eσ(k−1)hn +� +˜m∗ +k−1,n +��2 += σ4 +(k−1)hn +� +1 − 4 +π2 +� +h2 +n + OP(h2 +n). +We have used the first four moments of the half-normal distribution and their illustration via +integrals over tail probabilities. The dependence structure between ˜mk,n and ˜m∗ +k,n also affects the +variance of ˆσ2 +τ−. We perform approximation steps for covariances similar as for the moments of +local minima above, using that +h−1 +n Covσ(k−1)hn +� +˜mk,n, ˜m∗ +k,n +� += +� ∞ +−∞ +� ∞ +−∞ +� +Pσ(k−1)hn +� +h−1/2 +n +˜mk,n > x, h−1/2 +n +˜m∗ +k,n > y +� +− Pσ(k−1)hn +� +h−1/2 +n +˜mk,n > x +� +Pσ(k−1)hn +� +h−1/2 +n +˜m∗ +k,n > y +�� +dx dy += +� ∞ +0 +� ∞ +0 +� +Pσ(k−1)hn +� +σ(k−1)hn sup +0≤t≤1 +Bt > x, σ(k−1)hn +� +sup +0≤t≤1 +Bt − B1 +� +> y +� +− Pσ(k−1)hn +� +σ(k−1)hn sup +0≤t≤1 +Bt > x +� +Pσ(k−1)hn +� +σ(k−1)hn +� +sup +0≤t≤1 +Bt − B1 +� +> y +�� +dx dy + OP(1). +This shows that the joint distribution of ( ˜mk,n, ˜m∗ +k,n) relates to the distribution of the minimum +and the difference between minimum and endpoint of Brownian motion over an interval, or equiv- +alently the distribution of the maximum and the difference between maximum and endpoint. The +latter is readily obtained from the joint density of maximum and endpoint which is a well-known +result on stochastic processes. Utilizing this, we obtain that +Covσ(k−1)hn +� +˜mk,n , ˜m∗ +k,n +� += +�1 +2 − 2 +π +� +hn σ2 +(k−1)hn(1 + OP(hα +n)) + OP +� +hn +� +. +The additional remainder of order hα +n in probability is due to the different approximations of (σt) +in ˜mk,n and ˜m∗ +k,n. This implies that +Covσ(k−1)hn +� +˜mk,n ˜m∗ +k−1,n, ˜mk+1,n ˜m∗ +k,n +� += +� +Eσ(k−1)hn +� +˜mk,n ˜m∗ +k,n +� +− Eσ(k−1)hn +� +˜mk,n +� +Eσ(k−1)hn +� +˜m∗ +k,n +�� +Eσ(k−1)hn +� +˜m∗ +k−1,n +� +E +� +˜mk+1,n +� += σ4 +(k−1)hn +� 1 +π − 4 +π2 +� +h2 +n + OP +� +h2 +n +� +. +With analogous steps, we deduce two more covariances which contribute to the asymptotic vari- +ance: +Covσ(k−1)hn +� +˜m2 +k,n, +� +˜m∗ +k,n +�2� += −h2 +n +σ4 +(k−1)hn +2 ++ OP +� +h2 +n +� +, +Covσ(k−1)hn +�� +˜m∗ +k,n +�2, mk ˜m∗ +k−1,n +� += −h2 +n +2 +3π σ4 +(k−1)hn + OP +� +h2 +n +� +. +All covariance terms which enter the asymptotic variance are of one of these forms. +For the +conditional variance given σ2 +τ−, we obtain that +Varσ2 +τ− +� +ˆσ2 +τ− +� += +1 +K2n +π2 +4(π − 2)2 +� +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−2 +n Varσ2 +τ− +� +˜m2 +k,n + ( ˜m∗ +k,n)2 − 2 ˜mk,n ˜m∗ +k−1,n +� +19 + +− +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧2 +4h−2 +n Covσ2 +τ− +� +˜mk,n ˜m∗ +k−1,n , ˜m2 +k−1,n + ( ˜m∗ +k−1,n)2 − 2 ˜mk−1,n ˜m∗ +k−2,n +�� ++ OP +� +K−1 +n +� += +1 +K2n +π2 +4(π − 2)2 +� +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−2 +n +� +2Varσ2 +τ− +� +˜m2 +k,n +� ++ 4Varσ2 +τ− +� +˜mk,n ˜m∗ +k−1,n +� ++ 2 Covσ2 +τ− +� +˜m2 +k,n, ( ˜m∗ +k,n)2� +− 4 Covσ2 +τ− +� +˜m2 +k,n, ˜mk,n ˜m∗ +k−1,n +� +− 4 Covσ2 +τ− +� +( ˜m∗ +k,n)2, ˜mk,n ˜m∗ +k−1,n +�� ++ +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧2 +4h−2 +n +� +2 Covσ2 +τ− +� +˜mk,n ˜m∗ +k−1,n, ˜mk−1,n ˜m∗ +k−2,n +� +− Covσ2 +τ− +� +˜mk,n ˜m∗ +k−1,n, ˜m2 +k−1,n +� +− Covσ2 +τ− +� +˜mk,n ˜m∗ +k−1,n, ( ˜m∗ +k−1,n)2��� ++ OP +� +K−1 +n +� += +1 +Kn +π2 +4(π − 2)2 σ4 +τ− +� +8 − 16 +π2 − 1 − 8 +π + 8 +3π + 2 +� 4 +3π − 16 +π2 +�� ++ OP +� +K−1 +n +� += +1 +Kn +1 +(π − 2)2 +�7π2 +4 +− 2π +3 − 12 +� +σ4 +τ− + OP +� +K−1 +n +� +. +Step 5 +For a central limit theorem, the squared bias needs to be asymptotically negligible compared to +the variance, which is satisfied for Kn = O(h−2α/(1+2α) +n +). By the existence of higher moments of +˜mk,n and ˜m∗ +k−1,n, a Lyapunov-type condition is straightforward, such that asymptotic normality +conditional on σ2 +τ− is implied by a classical central limit theorem for m-dependent triangular +arrays as the one by [3]. A feasible central limit theorem is implied by this conditional asymptotic +normality in combination with FX-stable convergence. +For the stability, we show that αn = +K1/2 +n +� +ˆσ2 +τ− − σ2 +τ− +� +satisfy +E [Zg(αn)] → E [Zg(α)] = E[Z]E [g(α)] , +(25) +for any FX-measurable bounded random variable Z and continuous bounded function g, where +α = σ2 +τ− +1 +(π − 2) +� +7π2 +4 +− 2π +3 − 12 U , +(26) +with U a standard normally distributed random variable which is independent of FX. By the +above approximations it suffices to prove this for the statistics based on ˜mk,n and ˜m∗ +k−1,n from +(20), and Z measurable w.r.t. σ( +� t +0 σs dWs, 0 ≤ t ≤ 1). Set +An = [τ − (Kn + 1)hn, τ] , ˜X(n)t = +� t +0 +1An(s)σ⌊sh−1 +n ⌋hn dWs , ¯X(n)t = Xt − ˜X(n)t . +Denote with Hn the σ-field generated by ¯X(n)t and FX +0 . The sequence +� +Hn +� +n∈N is isotonic with +limit � +n Hn = σ( +� t +0 σs dWs, 0 ≤ t ≤ 1). Since E[Z|Hn] → Z in L1(P) as n → ∞, it is enough to +show that E[Zg(αn)] → E[Z]E[g(α)], for Z being Hn0-measurable for some n0 ∈ N. Observe that +αn includes only increments of local minima based on ˜X(n)t, which are uncorrelated from those +of ¯X(n)t. For all n ≥ n0, we hence obtain that E[Zg(αn)] = E[Z]E[g(αn)] → E[Z]E[g(α)] by a +standard central limit theorem. This shows (25) and completes the proof of (14). +20 + +6.2.2. Proof of Proposition 2 +For the quarticity estimator (12), when ⌊h−1 +n τ⌋ > Kn, we have that +E +�� +σ4τ − − σ4 +τ− +� += +π +4(3π − 8)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−2 +n E +� +˜m4 +k,n + ( ˜m∗ +k−1,n)4 − 4 ˜m3 +k,n ˜m∗ +k−1,n +− 4 ˜mk,n( ˜m∗ +k−1,n)3 + 6 ˜m2 +k,n( ˜m∗ +k−1,n)2� +− E[σ4 +τ−] + O +� +hα∧1/2 +n +� += +� +π +4(3π − 8) +� +6 − 16/π − 16/π + 6 +� +− 1 +� +E[σ4 +τ−] + O(1) += O(1) , +by using the same moments as in the computation of the asymptotic variance. We can bound its +variance by +Var +�� +σ4τ − +� +≤ +π2 +16(3π − 8)2K2n +2Knh−4 +n Var +�� +˜mk,n − ˜m∗ +k−1,n +�4� ++ O +� +K−1 +n +� +≤ +1 +Kn +π2 +8(3π − 8)2 h−4 +n E +�� +˜mk,n − ˜m∗ +k−1,n +�8� ++ O +� +K−1 +n +� +≤ +1 +Kn +π2 +8(3π − 8)2 h−4 +n 256 E +� +˜m8 +k,n +� ++ O +� +K−1 +n +� += O(K−1 +n ) , +what readily implies Proposition 2. +6.3. Asymptotics of the truncated spot volatility estimation with jumps +Denote by +DX +k := mk,n − mk−1,n, k = 1, . . . , h−1 +n +− 1 , +the differences of local minima based on the observations (5), with the general semimartingale (4) +with jumps. Denote by +DC +k := ˜mk,n − ˜m∗ +k−1,n, k = 1, . . . , h−1 +n +− 1 , +the differences of the unobservable local minima considered in Section 6.2. +In particular, the +statistics DC +k are based only on the continuous part (Ct) in (4) such that the jumps are eliminated. +Theorem 3 is implied by Proposition 2, if we can show that +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n +�� +DX +k +�21{|DX +k |≤un} − +� +DC +k +�2� += OP +� +h +α +2α+1 +n +� += OP +� +K−1/2 +n +� +. +We decompose this difference of the truncated estimator, which is based on the available observa- +tions with jumps, and the non-truncated estimator, which uses non-available observations without +21 + +jumps, in the following way: +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n +�� +DX +k +�21{|DX +k |≤un} − +� +DC +k +�2� += +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n +� +1{|DC +k |>cun} +�� +DX +k +�21{|DX +k |≤un} − +� +DC +k +�2� ++ 1{|DC +k |≤cun}1{|DX +k |≤un} +�� +DX +k +�2 − +� +DC +k +�2� +− 1{|DC +k |≤cun}1{|DX +k |>un} +� +DC +k +�2 +� +, +with some arbitrary constant c ∈ (0, 1). We consider the three addends which are different error +terms by +1. large absolute statistics based on the continuous part (Ct); +2. non-truncated statistics which contain (small) jumps; +3. the truncation of also the continuous parts in statistics (DX +k ) which exceed the threshold; +separately. The probability P(|DC +k | > cun) can be bounded using the estimate from (23) and +Gaussian tail bounds. Observe that the remainder in (23) is non-negative. This yields that for +some y > 0, we have that +P +� +h−1/2 +n +�� ˜mk,n +�� > y +� +≤ P +� +sup +0≤t≤1 +Bt > y +� +, +what is intuitive, since the errors (ϵi) are non-negative. We apply the triangular inequality and +then Hölder’s inequality to the expectation of the absolute first error term and obtain for any +p ∈ N that +π +2(π − 2)Kn +E +����� +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n 1{|DC +k |>cun} +�� +DX +k +�21{|DX +k |≤un} − +� +DC +k +�2����� +� +≤ +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n E +� +1{|DC +k |>cun} +��� +� +DX +k +�21{|DX +k |≤un} − +� +DC +k +�2��� +� +≤ +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n +� +P +� +|DC +k | > cun +� +2 +� +u4 +n + E +�� +DC +k +�4���1/2 +≤ +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n +� +P +� +h−1/2 +n +|DC +k | > chκ−1/2 +n +��1/2√ +2 u2 +n +≤ +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n +� +2 P +� +|B1| > c +2hκ−1/2 +n +��1/2√ +2 u2 +n +≤ +√ +2π +(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h2κ−1 +n +exp +� +− c2 +4 h2κ−1 +n +� +22 + += O +� +h(−p+1)(2κ−1) +n +� += O +� +h +α +2α+1 +n +� +. +Since 2κ − 1 < 0 and p arbitrarily large, we conclude that the first error term is asymptotically +negligible. We will use the elementary inequalities +DX +k = min +i∈In +k +� +C i +n + J i +n + ϵi +� +− min +i∈In +k−1 +� +C i +n + J i +n + ϵi +� +≤ min +i∈In +k +� +C i +n + ϵi +� ++ max +i∈In +k +J i +n − min +i∈In +k−1 +� +C i +n + ϵi +� +− min +i∈In +k−1 +J i +n += DC +k + max +i∈In +k +J i +n − min +i∈In +k−1 +J i +n + OP +� +hα∧1/2 +n +� +, +and +DX +k = min +i∈In +k +� +C i +n + J i +n + ϵi +� +− min +i∈In +k−1 +� +C i +n + J i +n + ϵi +� +≥ min +i∈In +k +� +C i +n + ϵi +� ++ min +i∈In +k +J i +n − min +i∈In +k−1 +� +C i +n + ϵi +� +− max +i∈In +k−1 +J i +n += DC +k + min +i∈In +k +J i +n − max +i∈In +k−1 +J i +n + OP +� +hα∧1/2 +n +� +. +Therefore, we can bound |DX +k − DC +k | by +sup +i∈In +k ,j∈In +k−1 +|J i +n − J j +n | ≤ +sup +s∈[khn,(k+1)hn],t∈[(k−1)hn,khn] +|Js − Jt| +≤ +sup +s∈[khn,(k+1)hn] +|Js − Jkhn| + +sup +t∈[(k−1)hn,khn] +|Jkhn − Jt| , +and the remainder term of the approximation for the continuous part which is OP +� +hα∧1/2 +n +� +. Since +the compensated small jumps of a semimartingale admit a martingale structure, Doob’s inequality +for càdlàg L2-martingales can be used to bound these suprema. Based on these preliminaries, we +obtain for the expected absolute value of the second error term that +π +2(π − 2)Kn +E +����� +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n 1{|DC +k |≤cun}1{|DX +k |≤un} +�� +DX +k +�2 − +� +DC +k +�2����� +� +≤ +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n E +� +1{|DC +k |≤cun}1{|DX +k |≤un} +��� +� +DX +k +�2 − +� +DC +k +�2��� +� +≲ +1 +Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n E +� +sup +i∈In +k ,j∈In +k−1 +|J i +n − J j +n |2 ∧ (1 + c)2u2 +n +� +≲ +1 +Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n E +� +sup +t∈[khn,(k+1)hn] +|Jt − Jkhn|2 ∧ u2 +n +� +≲ +1 +Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n E +� +|J(k+1)hn − Jkhn|2 ∧ u2 +n +� += O +� +u2−r +n +� +. +23 + +Applying the elementary inequalities from above, a cross term in the upper bound for +� +DX +k +�2 − +� +DC +k +�2 is of smaller order and directly neglected. It can be handled using the Cauchy-Schwarz +inequality. In the last step, we adopt a bound on the expected absolute thresholded jump incre- +ments from Equation (54) in [1]. For the negligibility of the second error term, we thus get the +condition that +κ(2 − r) ≥ +α +1 + 2α . +(27) +Doob’s inequality yields as well that +P +� +sup +t∈[khn,(k+1)hn] +|Jt − Jkhn| ≥ (1 − c)un +� +≤ E +���J(k+1)hn − Jkhn +��r∧1� +� +(1 − c)un +�r∧1 ++ O(hn) = O +� +hnu−r +n +� +. +For this upper bound, we decomposed the jumps in the sum of large jumps and the martingale of +compensated small jumps to which we apply Doob’s inequality. We derive the following estimate +for the expectation of the third (absolute) error term +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n E +� +1{|DC +k |≤cun}1{|DX +k |>un} +� +DC +k +�2� +≤ +π +2(π − 2)Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n E +� +1{2 sups∈[(k−1)hn,(k+1)hn] |Js−Jkhn|≥(1−c)un} +� +DC +k +�2� +≲ +1 +Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +h−1 +n P +� +sup +t∈[khn,(k+1)hn] +|Jt − Jkhn| ≥ (1 − c)un +� +E +�� +DC +k +�2� +≲ +1 +Kn +⌊h−1 +n τ⌋−1 +� +k=(⌊h−1 +n τ⌋−Kn)∧1 +�E +���J(k+1)hn − Jkhn +��r∧1� +� +(1 − c)un +�r∧1 ++ O(hn) +� += O +� +hnu−r +n +� +. +For the negligibility of the third error term, we thus get the condition that +1 − κr ≥ +α +1 + 2α . +(28) +Since under the conditions of Theorem 3, (27) and (28) are satisfied, the proof is finished by the +negligibility of all addends in the decomposition above. +References +[1] Aït-Sahalia, Y. and Jacod, J. (2010). Is Brownian motion necessary to model high-frequency data? Annals +of Statistics, 38(5), 3093–3128. +[2] Aït-Sahalia, Y. and Jacod, J. (2014). High-frequency financial econometrics. Princeton University Press. +[3] Berk, K. N. (1973). A central limit theorem for m-dependent random variables with unbounded m. Annals +of Probability, 1(2), 352–354. +[4] Bishwal, J.P.N. (2022). Parameter Estimation in Stochastic Volatility Models. Springer, Cham. +24 + +[5] Bibinger, M., Jirak, M. and Reiß, M. (2016). Volatility estimation under one-sided errors with applications +to limit order books. Annals of Applied Probability, 26(5), 2754–2790. +[6] Bibinger, M., Neely, C. and Winkelmann, L. (2019). Estimation of the discontinuous leverage effect: +Evidence from the Nasdaq order book. Journal of Econometrics, 209(2), 158–184. +[7] Bibinger, M. and Winkelmann, L. (2018). Common price and volatility jumps in noisy high-frequency data. +Electronic Journal of Statistics, 12(1), 2018–2073. +[8] El Euch, O., Fukasawa, M. and Rosenbaum, M. (2018). The microstructural foundations of leverage effect +and rough volatility. Finance and Stochastics 22(2), 241–280. +[9] Hansen, P.R. and Lunde, A. (2006). Realized variance and market microstructure noise. Journal of Business +& Economic Statistics, 24(2), 127–161. +[10] Hoffmann, M., Munk A. and Schmidt-Hieber, J. (2012). Adaptive wavelet estimation of the diffusion +coefficient under additive error measurements. Annales de l’IHP Probabilités et statistiques, 48(4), 1186– +1216. +[11] Jacod, J. and Protter, P. (2012). Discretization of processes. Springer. +[12] Svante J. (2007). Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian +areas. Probability Surveys, 4, 80–145. +[13] Jirak M., Meister A. and Reiß, M. (2014). Adaptive function estimation in nonparametric regression with +one-sided errors. Annals of Statistics 42(5), 1970–2002. +[14] Li, Z. M. and Linton, O. (2022). A ReMeDI for microstructure noise, Econometrica, 90(1), 367–389. +[15] Liu Y., Liu Q., Liu Z., and Ding D. (2017). Determining the integrated volatility via limit order books with +multiple records. Quantitative Finance, 17(11), 1697–1714. +[16] Mancini, C. (2009). Non-parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient +and Jumps. Scandinavian Journal of Statistics, 36(2), 270–296. +[17] Mancini, C., Mattiussi, V. and Renò, R. (2015). Spot volatility estimation using delta sequences. Finance +and Stochastics 19(2), 261–293. +[18] Meister, A. and Reiß, M. (2013). Asymptotic equivalence for nonparametric regression with non-regular +errors. Probab. Theory Relat. Fields, 155(1), 201–229. +[19] Reiß, M. and Wahl, M. (2019). Functional estimation and hypothesis testing in nonparametric boundary +models, Bernoulli, 25(4A), 2597–2619. +[20] Rosenbaum, M. and Tomas, M. (2021). From microscopic price dynamics to multidimensional rough volatility +models. Advances in Applied Probability 53(2), 425–462. +[21] Takács, L. (1996). On a generalization of the arc-sine law. Annals of Applied Probability, 6(3), 1035–1040. +[22] Tauchen, G. and Todorov, V. (2011). Volatility jumps. Journal of Business & Economic Statistics, 29(3), +356–371. +25 + diff --git a/DNA0T4oBgHgl3EQfAv8K/content/tmp_files/load_file.txt b/DNA0T4oBgHgl3EQfAv8K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cf0f82e71bb6bbd2443d269b17ad420f3763c1b8 --- /dev/null +++ b/DNA0T4oBgHgl3EQfAv8K/content/tmp_files/load_file.txt @@ -0,0 +1,990 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf,len=989 +page_content='Inference on the intraday spot volatility from high-frequency order prices with irregular microstructure noise Markus Bibinger∗a aFaculty of Mathematics and Computer Science, Julius-Maximilians-Universität Würzburg, markus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='bibinger@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='uni-wuerzburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='de Abstract We consider estimation of the spot volatility in a stochastic boundary model with one-sided mi- crostructure noise for high-frequency limit order prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Based on discrete, noisy observations of an Itô semimartingale with jumps and general stochastic volatility, we present a simple and explicit estimator using local order statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We establish consistency and stable central limit theorems as asymptotic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The asymptotic analysis builds upon an expansion of tail probabilities for the order statistics based on a generalized arcsine law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In order to use the involved distribution of local order statistics for a bias correction, an efficient numerical algorithm is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We demonstrate the finite-sample performance of the estimation in a Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Keywords: arcsine law, limit order book, market microstructure, nonparametric boundary model, volatility estimation MSC Classification: 62M09, 60J65, 60F05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Introduction Time series of intraday prices are typically described as a discretized path of a continuous-time stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' To have arbitrage-free markets the log-price process should be a semimartin- gale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Risk estimation based on high-frequency data at highest available observation frequencies has to take microstructure frictions into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Disentangling these market microstructure ef- fects from dynamics of the long-run price evolution has led to observation models with additive noise, see, for instance, [9], [2] and [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The market microstructure noise, modelling for instance the oscillation of traded prices between bid and ask order levels in an electronic market, is clas- sically a centred (white) noise process with expectation equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' These models can explain many stylized facts of high-frequency data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Having available full limit order books including data of submissions, cancellations and executions of bid and ask limit orders, however, it is not clear which time series to consider at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' While challenging the concept of one price process it raises the question if the information can be exploited more efficiently, in particular to improve risk quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The considered stochastic boundary model for limit order prices of an order book has been discussed by [5], [15] and Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='8 of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' It preserves the concept of an underlying efficient, semimartingale log-price which determines long-run price dynamics and an additive, ex- ogenous noise which models market-specific microstructure frictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Its key idea is that ask order prices should (in most cases) lie above the unobservable efficient price and bid prices below the ∗Financial support from the Deutsche Forschungsgemeinschaft (DFG) under grant 403176476 is gratefully ac- knowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='01965v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='ST] 5 Jan 2023 efficient price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This leads to observation errors which are irregular in the sense of having non-zero expectation and a distribution with a lower- or upper-bounded support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Considering without loss of generality a model for (best) ask order prices, we obtain lower-bounded observation errors and use local minima for the estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Modelling (best) bid prices instead would yield a model with upper-bounded observation errors and local maxima could be used for an analogous estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Both can be combined in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Inference on the spot volatility is one of the most important topics in the financial literature, see, for instance, [17] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In this work, we address spot volatility estimation for the model from [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' It is known that the statistical and probabilistic properties of models with irregular noise are very different than for regular noise and require other methods, see, for instance, [18], [13] and [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Therefore, our estimation methods and asymptotic theory are quite different compared to the market microstructure literature, while we can still profit from some of the techniques used there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In [5] an estimator for the quadratic variation of a continuous semimartingale, that is, the integrated volatility, was proposed with convergence rate n−1/3, based on n discrete observations with one-sided noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Optimality of the rate was proved in the standard asymptotic minimax sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' A main insight was that this convergence rate is better than the optimal rate, n−1/4, under regular market microstructure noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Using local minima over blocks of shrinking lengths hn ∝ n−2/3 ∝ (nhn)−2, the resulting distribution of local minima is involved and infeasible, such that in [5] a central limit theorem for the estimator could not be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Our estimator is related to a localized version of the one from [5], combined with truncation methods to eliminate jumps of the semimartingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For the asymptotic theory, however, we follow a different approach choosing blocks of lengths hn, where hnn2/3 → ∞ slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This allows to establish stable central limit theorems with the best achievable rate, arbitrarily close to n−1/6, in the important special case of a semimartingale volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We exploit this to construct pointwise asymptotic confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Although the asymptotic theory relies on block lengths that are slightly unbalanced by smooth- ing out the impact of the noise distribution on the distribution of local minima asymptotically, our numerical study demonstrates that the confidence intervals work well in realistic scenarios with block lengths which optimize the estimator’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Robustness to different noise specifi- cations is an advantage that is naturally implied by our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Our estimator is surprisingly simple, it is a local average of squared differences of block-wise minima times a constant factor which comes from moments of the half-normal distribution of the minimum of a Brownian motion over the unit interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This estimator is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' However, the stable central limit theorem at fast convergence rate requires a subtle bias correction which incorporates a more precise approxi- mation of the asymptotic distribution of local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For that purpose, our analysis is based on a generalization of the arcsine law which gives the distribution of the proportion of time over some interval that a Brownian motion is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For a numerical computation of the bias-correction function, we introduce an efficient algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Reducing local minima over many random variables to iterated minima of two random variables in each step combined with a convolution step, it can be interpreted as a kind of dynamic programming approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' It turns out to be much more efficient compared to the natural approximation by a Monte Carlo simulation and is a crucial ingredient of our numerical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Our convergence rate is much faster than the optimal rate, n1/8, for spot volatility estimation under regular noise, see [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The main contribution of this work is to 2 develop the probabilistic foundation for the asymptotic analysis of the estimator and to establish the stable central limit theorems, asymptotic confidence and a numerically practicable method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The methods and proof techniques to deal with jumps are inspired by the truncation methods pioneered in [16] and summarized in Chapter 13 of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Overall, the strategy and restrictions on jump processes are to some extent similar, while several details under irregular noise using order statistics are rather different compared to settings without noise or with regular centred noise as in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We introduce and further discuss our model in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Section 3 presents estimation methods and Section 4 asymptotic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The numerical application is considered in Section 5 and a Monte Carlo simulation study illustrates an appealing finite-sample performance of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' All proofs are given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Model with lower-bounded, one-sided noise and assumptions Consider an Itô semimartingale Xt = X0 + � t 0 as ds + � t 0 σs dWs + � t 0 � R δ(s, z)1{|δ(s,z)|≤1}(µ − ν)(ds, dz) + � t 0 � R δ(s, z)1{|δ(s,z)|>1}µ(ds, dz) , t ≥ 0 , (1) with a one-dimensional standard Brownian motion (Wt), defined on some filtered probability space (ΩX, FX, (FX t ), PX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For the drift process (at), and the volatility process (σt), we impose the following quite general assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The processes (at)t≥0 and (σt)t≥0 are locally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The volatility process is strictly positive, inft∈[0,1] σt > 0, PX-almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For all 0 ≤ t + s ≤ 1, t ≥ 0, s ≥ 0, with some constants Cσ > 0, and α > 0, it holds that E � (σ(t+s) − σt)2� ≤ Cσs2α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (2) Condition (2) introduces a regularity parameter α, governing the smoothness of the volatility process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The parameter α is crucial, since it will naturally influence convergence rates of spot volatility estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Inequality (2) is less restrictive than α-Hölder continuity, since it does not rule out volatility jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This is important as empirical evidence for volatility jumps, in particular simultaneous price and volatility jumps, has been reported for intraday high-frequency financial data, see, for instance, [22] and [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The presented theory is moreover for general stochastic volatilities, allowing as well for rough volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Rough fractional stochastic volatility models recently became popular and are used, for instance, in the macroscopic model of [8] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The jump component of (1) is illustrated as in [11] and related literature, where the predictable function δ is defined on Ω × R+ × R, and the Poisson random measure µ is compensated by ν(ds, dz) = λ(dz) ⊗ ds, with a σ-finite measure λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We impose the following standard condition with a generalized Blumenthal-Getoor or jump activity index r, 0 ≤ r ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 3 Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Assume that supω,x |δ(t, x)|/γ(x) is locally bounded with a non-negative, deter- ministic function γ which satisfies � R (γr(x) ∧ 1)λ(dx) < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (3) We use the notation a ∧ b = min(a, b), and a ∨ b = max(a, b), throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The assumption is most restrictive in the case r = 0, when jumps are of finite activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The larger r, the more general jump components are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We will develop results under mild restrictions on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The process (Xt), which can be decomposed Xt = Ct + Jt , (4) with the continuous component (Ct), and the càdlàg jump component (Jt), provides a model for the latent efficient log-price process in continuous time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' High-frequency (best) ask order prices from a limit order book at times tn i , 0 ≤ i ≤ n, on the fix time interval [0, 1], cannot be adequately modelled by discrete recordings of (Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Instead, we propose the additive model with lower-bounded, one-sided microstructure noise: Yi = Xtn i + ϵi , i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' , n, ϵi iid ∼ Fη, ϵi ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (5) The crucial property of the model is that the support of the noise is lower bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' It is not that important, that this boundary is zero, it could be as well a different constant, or even a regularly varying function over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We set the bound equal to zero which appears to be the most natural choice for limit orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' noise (ϵi)0≤i≤n, has a cumulative distribution function (cdf) Fη satis- fying Fη(x) = ηx � 1 + O(1) � , as x ↓ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (6) This is a nonparametric model in that the extreme value index is −1 for the minimum domain of attraction close to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This standard assumption on one-sided noise has been used by [13] and [19] within different frameworks, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We do not require assumptions about the maximum domain of attraction, moments and the tails of the noise distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Parametric exam- ples which satisfy (6) are, for instance, the uniform distribution on some interval, the exponential distribution and the standard Pareto distribution with heavy tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' assumption on the noise is crucial and generalizations to weakly dependent noise will require considerable work and new proof concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Heterogeneity instead, that is, a time- dependent noise level η(t), could be included in our asymptotic analysis under mild assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Construction of spot volatility estimators We partition the observation interval [0, 1] in h−1 n equispaced blocks, h−1 n ∈ N, and take local minima on each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We hence obtain for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' , h−1 n − 1, the local, block-wise minima mk,n = min i∈In k Yi , In k = {i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' , n} : tn i ∈ (khn, (k + 1)hn)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (7) While h−1 n is an integer, nhn is in general not integer-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For a simple interpretation, however, one can think of nhn as an integer-valued sequence which gives the number of noisy observations per block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' A spot volatility estimator could be obtained as a localized version of the estimator from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='9) in [5] for the integrated volatility in the analogous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The idea is that differences mk,n − mk−1,n of local minima estimate differences of efficient prices and a sum of squared differ- ences can be used to estimate the volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' However, things are not that simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' To determine the expectation of squared differences of local minima we introduce the function Ψn(σ2) = π 2(π − 2)h−1 n E �� min i∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=',nhn−1} � σB i n + ϵi � − min i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=',nhn} � σ ˜B i n + ϵi ��2� , (8) where (Bt) and ( ˜Bt) denote two independent standard Brownian motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For hnn2/3 → ∞, we have that Ψn(σ2) = σ2 + O(1) , (9) such that we do not require Ψ−1 n for a consistent estimator in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Note that we defined Ψn different compared to [5] with the constant factor π/(π − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' When there are no price jumps, a simple consistent estimator for the spot squared volatility σ2 τ is given by ˆσ2 τ− = π 2(π − 2)Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n � mk,n − mk−1,n)2 , (10) for suitable sequences hn → 0 and Kn → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Using only observations before time τ, the estimator is available on-line at time τ ∈ (0, 1] during a trading day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Working with ex-post data over the whole interval, instead of using only observations before time τ, one may use as well ˆσ2 τ+ = π 2(π − 2)Kn (⌊h−1 n τ⌋+Kn)∨(h−1 n −1) � k=⌊h−1 n τ⌋+1 h−1 n � mk,n − mk−1,n)2 , (11) or an estimator with an average centred around time τ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The difference of the two estimators (11) and (10) can be used to infer a possible jump in the volatility process at time τ ∈ (0, 1), as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' To construct confidence intervals for the spot volatility, it is useful to establish also a spot 5 quarticity estimator: � σ4τ − = π 4(3π − 8)Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−2 n � mk,n − mk−1,n)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (12) A spot volatility estimator which is robust with respect to jumps in (Xt) is obtained with threshold versions of these estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We truncate differences of local minima whose absolute values exceed a threshold un = hκ n, κ ∈ (0, 1/2), which leads to ˆσ2,(tr) τ− = π 2(π − 2)Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n � mk,n − mk−1,n)21{|mk,n−mk−1,n|≤un} , (13) and analogous versions of the estimators (11) and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Asymptotic results We establish asymptotic results for equidistant observations, tn i = i/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We begin with the asymptotic theory in a setup without jumps in (Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Theorem 1 (Stable central limit theorem for continuous (Xt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Set hn, such that hnn2/3 → ∞, and Kn = CKhδ−2α/(1+2α) n for arbitrary δ, 0 < δ < 2α/(1 + 2α), and with some constant CK > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' If (Xt) is continuous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Jt = 0 in (4), under Assumptions 1 and 3, the spot volatility estimator (10) is consistent, ˆσ2 τ− P→ σ2 τ−, and satisfies the stable central limit theorem K1/2 n � ˆσ2 τ− − Ψn � σ2 τ− �� st −→ N � 0, 7π2/4 − 2π/3 − 12 (π − 2)2 σ4 τ− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (14) There is only a difference between σ2 τ and its left limit σ2 τ− in case of a volatility jump at time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In particular, the estimator is as well consistent for σ2 τ, for any fix τ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The convergence rate K−1/2 n gets arbitrarily close to n−2α/(3+6α), which is optimal in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In the important special case when α = 1/2, for a semimartingale volatility, the rate is arbitrarily close to n−1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This is much faster than the optimal rate of convergence in the model with additive centred microstructure noise, which is known to be n−1/8, see [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The constant in the asymptotic variance is obtained from several variance and covariance terms including (squared) local minima and is approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The function Ψn was shown to be monotone and invertible in [5] and Ψn and its inverse Ψ−1 n can be approximated using Monte Carlo simulations, see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The asymptotic distribution of the estimator does not hinge on the noise level η, different to methods for centred noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Hence, we do not require any pre-estimation of noise parameters and the theory directly extends to a time-varying noise level ηt in (6) under the mild assumption that 0 < ηt < ∞, for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The stable convergence in (14) is stronger than weak convergence and is important, since the limit distribution is mixed normal depending on the stochastic volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We refer to [11], Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='1, for an introduction to stable convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For a normalized central limit theorem, we can use the spot quarticity estimator (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Proposition 2 (Feasible central limit theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Under the conditions of Theorem 1, the spot quarticity estimator (12) is consistent, such that we get for the spot volatility estimation the 6 normalized central limit theorem K1/2 n π − 2 � � σ4τ −(7π2/4 − 2π/3 − 12) � ˆσ2 τ− − Ψn � σ2 τ− �� d −→ N(0, 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (15) Proposition 2 yields asymptotic confidence intervals for spot volatility estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For q ∈ (0, 1), it holds true that P � σ2 τ− ∈ � Ψ−1 n � ˆσ2 τ− − π − 2 � � σ4τ −(7π2/4 − 2π/3 − 12) K−1/2 n Φ−1(1 − q) � , Ψ−1 n � ˆσ2 τ− + π − 2 � � σ4τ −(7π2/4 − 2π/3 − 12) K−1/2 n Φ−1(1 − q) ��� → 1 − q , by monotonicity of Ψ−1 n with Φ the cdf of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Since Ψ−1 n is differ- entiable and the derivative is � Ψ−1 n �′ = 1 + O(1), the delta method (for stable convergence) yields as well asymptotic confidence intervals and the central limit theorem K1/2 n � Ψ−1 n � ˆσ2 τ− � − σ2 τ− � st −→ N � 0, 7π2/4 − 2π/3 − 12 (π − 2)2 σ4 τ− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (16) We may not simply replace Ψn � σ2 τ− � by its first-order approximation σ2 τ− in (14), since the bias multiplied with K1/2 n does in general not converge to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' If the condition hnn2/3 → ∞ is violated, this central limit theorem does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Theorem 3 (Stable central limit theorem with jumps in (Xt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Set hn, such that hnn2/3 → ∞, and Kn = CKhδ−2α/(1+2α) n for arbitrary δ, 0 < δ < 2α/(1 + 2α), and with some constant CK > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Under Assumptions 1, 2 and 3, with r < 2 + 2α 1 + 2α , (17) the truncated spot volatility estimator (13) with κ ∈ � 1 2 − r α 2α + 1, 1 2 � , (18) is consistent, ˆσ2,(tr) τ− P→ σ2 τ−, and satisfies the stable central limit theorem K1/2 n � ˆσ2,(tr) τ− − Ψn � σ2 τ− �� st −→ N � 0, 7π2/4 − 2π/3 − 12 (π − 2)2 σ4 τ− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (19) In order to obtain a central limit theorem at (almost) optimal rate, we thus have to impose mild restrictions on the jump activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For the standard model with a semimartingale volatility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' α = 1/2, we need that r < 3/2, and for α = 1 we have the strongest condition that r < 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' These conditions are equivalent to the ones of Theorem 1 in [7], which gives a central limit theorem for spot volatility estimation under similar assumptions on (Xt), but with slower rate of convergence for centred microstructure noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Using a truncated quarticity estimator with the same thresholding yields again a feasible central limit theorem and asymptotic confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 7 Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' From a theoretical point of view one might ponder why we do not work out an asymp- totic theory for hn ∝ n−2/3, when noise and efficient price both influence the asymptotic distribu- tion of the local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' However, in this balanced case, the asymptotic distribution is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For this reason, [5] could not establish a central limit theorem for their integrated volatility esti- mator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Moreover, their estimator was only implicitly defined depending on the unknown function Ψ−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Even imposing a parametric assumption on the noise as an exponential distribution would not render a feasible limit theory for hn ∝ n−2/3, see the discussion in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Choosing hn, such that hnn2/3 → ∞ slowly, yields instead a simple, explicit and consistent estimator and a fea- sible central limit theorem for spot volatility estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In particular, we use Ψn only for the bias-correction of the simple estimator, while the estimator itself and the (estimated) asymptotic variance do not hinge on Ψn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Central limit theorems for spot volatility estimators are in general only available at almost optimal rates, when the variance dominates the squared bias in the mean squared error, see, for instance, Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='3 and the remarks below in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Therefore, (14) is the best achievable central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Our choice of hn avoids moreover strong assumptions on the noise that would be inevitable for smaller blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Our numerical work will demonstrate that the presented asymptotic results are useful in practice and can be applied without loosing (much) efficiency compared to a different selection of blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Implementation and simulations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Monte Carlo approximation of Ψn Although the function Ψn from (8) tends to the identity asymptotically, it has a crucial role for a bias correction of our estimator in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We can compute the function numerically based on a Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Hence, we have to compute Ψn(σ2) as a Monte Carlo mean over many iterations and over a fine grid of values for the squared volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Then, we can also numerically invert the function and use Ψ−1 n ( · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' To make this procedure feasible without too high computational expense we require an algorithm to efficiently sample from the law of the local minima for some given n and block length hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Consider for nhn ∈ N, with Zi iid ∼ N(0, 1), and the observation errors (ϵk)k≥0, the minimum M nhn 1 := min k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=',nhn � σ √n k � i=1 Zi + ϵk � , for some fix σ > 0, and for l ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' , nhn}: M nhn l := min k=l,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=',nhn � σ √n k � i=0 Zi + ϵk � , where we set Z0 := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Since Ψn(σ2) = 1 2 π π − 2h−1 n E �� M nhn−1 0 − M nhn 1 �2� , with M nhn−1 0 generated independently from M nhn 1 , we want to simulate samples distributed as M nhn−1 0 and M nhn 1 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Note that the moments of M nhn−1 0 and M nhn 1 slightly differ 8 what can be relevant for moderate values of nhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' As in the simulation of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2, we im- plement exponentially distributed observation errors (ϵk), with some given noise level η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In data applications, we can do the same with an estimated noise level ˆη = � 1 2n n � i=1 � Yi − Yi−1 �2 �−1/2 = η + OP � n−1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This estimator works for all noise distributions with finite fourth moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' To simulate the local minima for given n, hn, η, and squared volatility σ2, in an efficient way we use a specific dynamic programming principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Observe that M nhn 1 = σ √nZ1 + min � ϵ1, M nhn 2 � = σ √nZ1 + min � ϵ1, σ √nZ2 + min � ϵ2, M nhn 3 �� = σ √nZ1 + min � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' min � ϵnhn−2, σ √nZnhn−1 + min � ϵnhn−1, σ √nZnhn + ϵnhn �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In the baseline noise model, ϵk iid ∼ Exp(η), the random variable σ √nZnhn+ϵnhn has an exponentially modified Gaussian (EMG) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' With any fixed noise distribution, we can easily generate realizations from this convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' A pseudo random variable which is distributed as M nhn 1 is now generated following the last transformation in the reverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In pseudo code, this reads 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Generate U_{nh_n}~ EMG(sigma^2/n,eta)~ Exp(eta)+sigma/sqrt(n)*Norm(1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' U_{nh_n-1}=min(U_{nh_n},Exp(eta))+sigma/sqrt(n)*Norm(1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' iterate until U_1 where the end point U1 has the target distribution of M nhn 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In each iteration step, we thus take the minimum of the current state of the process with one independent exponentially distributed random variable and the convolution with one independent normally distributed random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' To sample from the distribution of M nhn−1 0 instead, we use the same algorithm and just drop the convolution with the normal distribution in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' It turns out that this algorithm facilitates a many times faster simulation compared to a classical simulation starting with a discretized path of (Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Figure 1 plots the result of the Monte Carlo approximation of Ψn(σ2) for n = 23,400 and n · hn = 15, on a grid of 1500 values of σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In this case, hn is quite small, but this configuration turns out to be useful below in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We know that Ψn(σ2) is monotone, such that the oscillation of the blue line in Figure 1 is due to the inaccuracy of the Monte Carlo means although we use N = 100,000 iterations for each grid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Nevertheless, we can see that the function is rather close to a linear function with slope 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='046 based on a least squares estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The left plot of Figure 1 draws a comparison to the identity function which is illustrated by the dotted line, while the plot right-hand side draws a comparison to the linear function with slope 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We see that it is crucial to correct for the bias in (14) when using such small values of hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Although the function Ψn(σ2) is not exactly linear, a simple bias correction dividing estimates by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='046 is almost as good as using the more precise numerical inversion based on the Monte Carlo approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Since the Monte Carlo approximations of Ψn(σ2) look close to linear functions in all considered cases, we 9 Figure 1: Monte Carlo means to estimate Ψn(σ2) over a fine grid (blue line) for n = 23,400 and n · hn = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Left, the dotted line shows the identity function, right the dotted line is a linear function with slope 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Table 1: Regression slopes to measure the bias of estimator (10) and deviation Ψn(σ2) − σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' n · hn 10 15 25 39 78 234 h−1 n 2340 1560 936 600 300 100 hn · n2/3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='524 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='874 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='73 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='18 slope 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='077 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='046 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='008 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='003 approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' bias 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='7% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='6% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='5% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='6% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='3% report the estimated slopes based on least squares and N = 100,000 Monte Carlo iterations for different values of hn in Table 1 to summarize concisely how far the distance between the function Ψn(σ2) and the identity is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Simulating all iterations for all grid points with our algorithm takes only a few hours with a standard computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Simulation study of estimators We simulate n = 23,400 observations corresponding to one observation per second over a (NASDAQ) trading day of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='5 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The efficient price process is simulated from the model dXt = νtσt dWt , dσ2 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='0162 · � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='8465 − σ2 t � dt + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='117 · σt dBt , νt = � 6 − sin(3πt/4) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='002 , t ∈ [0, 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The factor (νt) generates a typical U-shaped intraday volatility pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (Wt, Bt) is a two- dimensional Brownian motion with leverage d[W, B]t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The stochastic volatility component has several realistic features and the simulated model is in line with recent literature, see [6] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Observations with lower-bounded, one-sided microstructure noise are generated by Yi = X i n + ϵi , 0 ≤ i ≤ n , 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content="00012 80000'0 f(c3) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content="00004 00000' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='00004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='00008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='000120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content="00012 80000'0 4(c3) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content="00004 00000' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='00004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='00008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='00012 3Figure 2: True and estimated spot volatility with pointwise confidence sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' with exponentially distributed noise, ϵi iid ∼ Exp(η), with η = 10,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The noise variance is then rather small, but this is in line with stylized facts of real NASDAQ data as, for instance, those analysed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='1 The black line in Figure 2 shows a fixed path of the squared volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We fix this path for the following Monte Carlo simulation and generate new observations of (Xt) and (Yi) in each iteration according to our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The blue line in Figure 2 gives the estimated volatility by the Monte Carlo means over N = 50,000 iterations based on n·hn = 15 observations per block using the non- adjusted estimator (10) with windows which are centred around the block on that we estimate the spot volatility and with Kn = 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We plot estimates on each block, where the estimates close to the boundaries rely on less observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The red line gives the bias-corrected volatility estimates using the numerically evaluated function Ψn, based on the algorithm from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='1 with n · hn = 15 and n = 23,400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We determined the values n · hn = 15 and Kn = 180 as suitable values to obtain a small mean squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In fact, the choice of Kn = 180 is rather large in favour of a smaller variance what yields a rather smooth estimated spot volatility in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The estimated volatility hence appears smoother compared to the true semimartingale volatility, but the intraday pattern is well captured by our estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We expect that this is typically an appealing implementation in practice as smaller Kn results in a larger variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Choosing Kn = 180 rather large, we have to use quite small block sizes hn, to control the overall bias of the estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Since hn · n2/3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='52 is small, the bias correction becomes crucial here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Still, 1Note that the noise level estimate is analogous to the one used for regular market microstructure noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Typical noise levels obtained for trades of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Apple are approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 15,000 and approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 4,000 for 3M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For mid quotes or best ask/bid prices the levels are only slightly larger (variance smaller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='00018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='00014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content="00010 90000'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='0 timeTable 2: Summary statistics of estimation for different values of hn and Kn, MSD = mean standard deviation, MAB = mean absolute bias, MABC = MAB of bias-corrected estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Kn 120 180 240 nhn MSD MAB MABC MSD MAB MABC MSD MAB MABC 10 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='73 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='90 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='13 15 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='88 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='57 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='17 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='43 25 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='24 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='66 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='91 78 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='52 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='42 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='16 All values multiplied with factor 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' our asymptotic results work well for this implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This can be seen by the comparison of pointwise empirical 10% and 90% quantiles from the Monte Carlo iterations illustrated by the grey area and the 10% and 90% quantiles of the limit normal distribution with the asymptotic variance from (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The latter are drawn as dotted lines for the blocks with larger distance than Kn/2 from the boundaries where the variances are of order K−1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Close to the boundaries the empirical variances increase due to the smaller number of blocks used for the estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Moreover, the bias correction which is almost identical to dividing each estimate by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='046, correctly scales the simple estimates which have a significant positive bias for the chosen tuning parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Overall, our asymptotic results provide a good finite-sample fit even though we have hn · n2/3 < 1 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Note, however, that σt · η ≈ 100, and our asymptotic expansion requires in fact that hn · n2/3σt · η is large when taking constants into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Table 2 summarizes the performance of the estimation along different choices of nhn and Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We give the following quantities: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' MSD: the mean standard deviation of N iterations averaged over all grid points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' MAB: the mean absolute bias of N iterations averaged over all grid points and for estimator (10) without any bias correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' MABC: the mean absolute bias of N iterations averaged over all grid points and for estimator (10) with a simple bias correction dividing estimates by the factors given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' All results are based on N = 50,000 Monte Carlo iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' First of all, the values used for Figure 2 are not unique minimizers of the mean squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Several other combinations given in Table 2 render equally well results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Overall, the performance is comparable within a broad range of block lengths and window sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The variances decrease for larger Kn, while the bias increases with larger Kn for fixed hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Important for the bias is the total window size, Kn · hn, over that the volatility is approximated constant for the estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The variance only depends on Kn, changing the block length for fix Kn does not significantly affect the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' While the MSD is hence almost constant within the columns of Table 2, the bias after correction, MABC, increases from top down due to the increasing window size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Without the bias correction two effects interfere for MAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Larger blocks reduce the systematic bias due to Ψn(σ2 t ) − σ2 t , but the increasing bias due to the increasing window size prevails for n · hn = 78, and the two larger values of Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Proofs 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Law of the integrated negative part of a Brownian motion A crucial lemma for our theory is on an upper bound for the cdf of the integrated negative part of a Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We prove a lemma based on a generalization of Lévy’s arc-sine law by [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The result is in line with the conjecture in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (261) of [12] where one finds an expansion of the density with a precise constant of the leading term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Denote by f+ the positive part and by f− the negative part of some real-valued function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For a standard Brownian motion (Wt)t≥0, it holds that P � � 1 0 (Wt)− dt ≤ x � = O(x1/3), x → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Observe the equality in distribution � 1 0 (Wt)− dt d= � 1 0 (Wt)+ dt, such that P � � 1 0 (Wt)− dt ≤ x � = P � � 1 0 (Wt)+ dt ≤ x � , x > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For any ϵ > 0, the inequality � 1 0 (Wt)+ dt ≥ � 1 0 Wt · 1(Wt > ϵ) dt ≥ ϵ � 1 0 1(Wt > ϵ) dt leads us to P � � 1 0 (Wt)+ dt ≤ x � ≤ P � ϵ � 1 0 1(Wt > ϵ) dt ≤ x � = P � 1 − � 1 0 1(Wt ≤ ϵ) dt ≤ x/ϵ � = P � � 1 0 1(Wt ≤ ϵ) dt ≥ 1 − x/ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Using (15) and (16) from [21], we obtain that P � � 1 0 1(Wt ≤ ϵ) dt ≥ 1 − x/ϵ � = 1 π � 1 1−x/ϵ exp(−ϵ2/(2u)) � u(1 − u) du + 2Φ(ϵ) − 1 , with Φ the cdf of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Thereby, we obtain that P � � 1 0 (Wt)+ dt ≤ x � ≤ 1 π � 1 1−x/ϵ exp(−ϵ2/(2u)) � u(1 − u) du + 2 � ϵ 0 exp(−u2/2) √ 2π du , and elementary bounds give the upper bound P � � 1 0 (Wt)+ dt ≤ x � ≤ 2 π �x ϵ 1 � 1 − x/ϵ + 2ϵ √ 2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 13 Choosing ϵ = x1/3, we obtain the upper bound P � � 1 0 (Wt)+ dt ≤ x � ≤ 2 π x1/3 1 √ 1 − x2/3 + 2x1/3 √ 2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Asymptotics of the spot volatility estimation in the continuous case 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Proof of Theorem 1 In the sequel, we write An ≲ Bn for two real sequences, if there exists some n0 ∈ N and a constant K, such that An ≤ KBn, for all n ≥ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Step 1 In the first step, we prove the approximation ˆσ2 τ− = π 2(π − 2)Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n � mk,n − mk−1,n)2 = π 2(π − 2)Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n � ˜mk,n − ˜m∗ k−1,n)2 + OP � hα∧1/2 n � with ˜mk,n = min i∈In k � ϵi + σ(k−1)hn(Wtn i − Wkhn) � , and ˜m∗ k−1,n = min i∈In k−1 � ϵi − σ(k−1)hn(Wkhn − Wtn i ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We show that for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' , h−1 n − 1}, it holds that mk,n − mk−1,n = ˜mk,n − ˜m∗ k−1,n + OP � h1/2 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (20) We subtract Xkhn from mk,n and mk−1,n, and use that it holds for all i that � Yi − Xkhn � − � Xtn i − � Xkhn + σ(k−1)hn(Wtn i − Wkhn) �� = � σ(k−1)hn(Wtn i − Wkhn) + ϵi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This implies that min i∈In k � Yi−Xkhn � −max i∈In k � Xtn i − � Xkhn+σ(k−1)hn(Wtn i −Wkhn) �� ≤ min i∈In k � σ(k−1)hn(Wtn i −Wkhn)+ϵi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Changing the roles of � Yi − Xkhn � and � σ(k−1)hn(Wtn i − Wkhn) + ϵi � , we obtain by the analogous inequalities and the triangle inequality, with Mt := Xkhn + � t khn σ(k−1)hn dWs, that ���mk,n − Xkhn − ˜mk,n ��� ≤ max i∈In k ��Xtn i − Mtn i �� ≤ sup t∈[khn,(k+1)hn] ��Xt − Mt �� ≤ sup t∈[khn,(k+1)hn] ���Ct − Ckhn − t ∫ khn σ(k−1)hn dWs ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 14 We write (Ct) for (Xt) to emphasize continuity, see (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (20) follows from sup t∈[khn,(k+1)hn] ���Ct − Ckhn − t ∫ khn σ(k−1)hn dWs ��� = OP(h1/2 n ) , (21) and the analogous estimate for mk−1,n and ˜m∗ k−1,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We decompose sup t∈[khn,(k+1)hn] ���Ct − Ckhn − t ∫ khn σ(k−1)hn dWs ��� ≤ sup t∈[khn,(k+1)hn] ��� t ∫ khn (σs − σ(k−1)hn) dWs ��� + sup t∈[khn,(k+1)hn] � t khn |as|ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Under Assumption 1, we can assume that (σt) and (at) are bounded on [0, 1] by the localization from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='1 in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Using Itô’s isometry and Assumption 1, we obtain that E �� � t khn (σs − σ(k−1)hn) dWs �2� = � t khn E � (σs − σ(k−1)hn)2� ds = O � � t khn (s − (k − 1)hn)2α ds � = O � (t − (k − 1)hn)2α+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' By Doob’s martingale maximal inequality and since supt∈[khn,(k+1)hn] � t khn |as|ds = OP(hn), it holds that sup t∈[khn,(k+1)hn] ���Ct − Ckhn − t ∫ khn σ(k−1)hn dWs ��� = OP � h(1/2+α)∧1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We conclude that (21) holds, since α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Since h−1 n � mk,n − mk−1,n �� mk,n − ˜mk,n � = OP � hα∧1/2 n � , and analogously for (mk−1,n − ˜m∗ k−1,n), we conclude Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Step 2 We bound the bias of the spot volatility estimation using Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For ⌊h−1 n τ⌋ > Kn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' we obtain with the function Ψn from (8) that E � ˆσ2 τ− − Ψn � σ2 τ− �� = = 1 Kn π 2(π − 2) ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n E �� ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n − ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='− E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='Ψn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='τ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='+ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='hα∧1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2(π − 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k=(⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−Kn)∧1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2(π − 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='Ψn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='(k−1)hn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='− E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='Ψn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='τ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='+ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='hα∧1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='≲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k=(⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−Kn)∧1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='(k−1)hn − σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='τ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='+ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='hα∧1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='≲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k=(⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−Kn)∧1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='σ(k−1)hn − στ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='+ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='hα∧1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='≲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k=(⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−Kn)∧1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='σ(k−1)hn − στ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='�2��1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='+ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='hα∧1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='= O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='(Kn hn)α� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='= O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='hα/(1+2α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='= O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='K−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We used that (α ∧ 1/2) > α/(2α + 1) for all α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For the asymptotic upper bounds we used the binomial formula and Hölder’s inequality to conclude with (2) from Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Step 3 For (9) and the consistency of ˆσ2 τ−, we prove that E � ˆσ2 τ− � = σ2 τ− + O(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (22) Denote by Pσ(k−1)hn the regular conditional probabilities conditioned on σ(k−1)hn, and Eσ(k−1)hn the expectations with respect to the conditional measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We obtain by the tower rule that E � h−1 n � ˜mk,n − ˜m∗ k−1,n)2� = E � h−1 n Eσ(k−1)hn �� ˜mk,n − ˜m∗ k−1,n)2�� = E � Eσ(k−1)hn �� h−1/2 n ˜mk,n)2� + Eσ(k−1)hn �� h−1/2 n ˜m∗ k−1,n)2� − 2 Eσ(k−1)hn � h−1/2 n ˜mk,n � Eσ(k−1)hn � h−1/2 n ˜m∗ k−1,n �� , by the conditional independence of ˜mk,n and ˜m∗ k−1,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We establish and use an approximation of the tail probabilities of ( ˜mk,n) and ( ˜m∗ k−1,n), re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For x ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' we have that Pσ(k−1)hn � h−1/2 n min i∈In k � ϵi + σ(k−1)hn(Wtn i − Wkhn) � > xσ(k−1)hn � = Pσ(k−1)hn � min i∈In k � h−1/2 n � Wtn i − Wkhn � + h−1/2 n σ−1 (k−1)hnϵi � > x � = Eσ(k−1)hn � ⌊(k+1)nhn⌋ � i=⌊knhn⌋+1 P � ϵi > h1/2 n σ(k−1)hn � x − h−1/2 n (Wtn i − Wkhn) � |FX�� = Eσ(k−1)hn � exp � ⌊(k+1)nhn⌋ � i=⌊knhn⌋+1 log � 1 − Fη � h1/2 n σ(k−1)hn � x − h−1/2 n (Wtn i − Wkhn) ����� by the tower rule for conditional expectations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' and since ϵi iid ∼ Fη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' It holds that Wtn i − Wkhn = i−⌊knhn⌋ � j=1 ˜Uj, ˜Uj iid ∼ N(0, n−1), j ≥ 2, ˜U1 ∼ N � 0, tn ⌊knhn⌋+1 − khn � , Uj = h−1/2 n ˜Uj iid ∼ N � 0, (nhn)−1� , j ≥ 2, U1 ∼ N � 0, h−1 n � tn ⌊knhn⌋+1 − khn �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We apply a Riemann sum approximation with a standard Brownian motion (Bt)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' With (6), and a first-order Taylor expansion of z �→ log(1 − z), we obtain that Pσ(k−1)hn � h−1/2 n min i∈In k � ϵi + σ(k−1)hn(Wtn i − Wkhn) � > xσ(k−1)hn � = 16 = Eσ(k−1)hn � exp � − h1/2 n σ(k−1)hnη ⌊(k+1)nhn⌋ � i=⌊knhn⌋+1 � x − i−⌊knhn⌋ � j=1 Uj � +(1 + O(1)) �� = Eσ(k−1)hn � exp � − h1/2 n nhnσ(k−1)hnη � 1 0 (Bt − x)− dt (1 + O(1)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' If nh3/2 n → ∞, we deduce that Pσ(k−1)hn � h−1/2 n min i∈In k � ϵi + σ(k−1)hn(Wtn i − Wkhn) � > xσ(k−1)hn � = P � inf 0≤t≤1 Bt ≥ x � + Eσ(k−1)hn � 1 � inf 0≤t≤1 Bt < x � exp � − h3/2 n nσ(k−1)hnη � 1 0 (Bt − x)− dt (1 + O(1)) �� = P � inf 0≤t≤1 Bt ≥ x � + P � inf 0≤t≤1 Bt < x � O(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (23) We do not have a lower bound for � 1 0 (Bt − x)− dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' using that the first entry time Tx of (Bt) in x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' conditional on {inf0≤t≤1 Bt < x},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' has a continuous conditional density f(t|Tx < 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' by Lemma 4 and properties of the Brownian motion we obtain for any δ > 0 that Eσ(k−1)hn � 1 � inf 0≤t≤1 Bt < x � exp � − h3/2 n nσ(k−1)hnη � 1 0 (Bt − x)− dt �� ≤ exp � − � h3/2 n n �δσ(k−1)hnη � P( inf 0≤t≤1 Bt < x) + P � inf 0≤t≤1 Bt < x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' � 1 0 (Bt − x)− dt ≤ � h3/2 n n �−1+δ� ≤ � exp � − � h3/2 n n �δσ(k−1)hnη � + � 1 0 P � � 1 s (Bt)− dt ≤ � h3/2 n n �−1+δ� f(s|Tx < 1) ds � P( inf 0≤t≤1 Bt < x) ≤ � exp � − � h3/2 n n �δσ(k−1)hnη � + � 1 0 P � (1 − s) � 1 0 (Bt)− dt ≤ � h3/2 n n �−1+δ� f(s|Tx < 1) ds � × P( inf 0≤t≤1 Bt < x) = P( inf 0≤t≤1 Bt < x) · Rn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' with a remainder Rn = O �� h3/2 n n �− 1+δ 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We applied Lemma 4 in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' From the unconditional Lévy distribution of Tx, f(s|Tx < 1) is explicit, but its precise form does not influence the asymptotic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Under the condition nh3/2 n → ∞, the minimum of the Brownian motion over the interval hence dominates the noise in the distribution of local minima, different than for a choice hn ∝ n−2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' By the reflection principle, it holds that P � − inf 0≤t≤1 Bt ≥ x � = P � sup 0≤t≤1 Bt ≥ x � = 2P � B1 ≥ x � = P � |B1| ≥ x � , (24) for x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Using the illustration of moments by integrals over tail probabilities we exploit this, and a completely analogous estimate for ˜m∗ k−1,n, to approximate conditional expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This yields 17 that Eσ(k−1)hn � h−1/2 n ˜mk,n � = = � ∞ 0 Pσ(k−1)hn � h−1/2 n ˜mk,n > x � dx − � ∞ 0 Pσ(k−1)hn � − h−1/2 n ˜mk,n > x � dx = − � ∞ 0 Pσ(k−1)hn � σ(k−1)hn sup 0≤t≤1 Bt > x � dx + OP(1) = − � ∞ 0 Pσ(k−1)hn � σ(k−1)hn|B1| > x � dx + OP(1) = −Eσ(k−1)hn � σ(k−1)hn|B1| � + OP(1) = − � 2 π σ(k−1)hn + OP(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We used (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' An analogous computation yields the same result for ˜m∗ k−1,n: Eσ(k−1)hn � h−1/2 n ˜m∗ k−1,n � = − � 2 π σ(k−1)hn + OP(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For the second conditional moments, we obtain that Eσ(k−1)hn � h−1 n � ˜mk,n �2� = 2 � ∞ 0 x Pσ(k−1)hn � |h−1/2 n ˜mk,n| > x � dx = 2 � ∞ 0 x Pσ(k−1)hn � σ(k−1)hn sup 0≤t≤1 Bt > x � dx + OP(1) = 2 � ∞ 0 x Pσ(k−1)hn � σ(k−1)hn|B1| > x � dx + OP(1) = σ2 (k−1)hn + OP(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The last identity uses the illustration of the second moment of the normal distribution as an integral over tail probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' An analogous computation yields that Eσ(k−1)hn � h−1 n � ˜m∗ k−1,n �2� = σ2 (k−1)hn + OP(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This proves (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Step 4 We determine the asymptotic variance of the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Illustrating moments as integrals over tail probabilities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' with the analogous approximation as above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' we obtain that Varσ(k−1)hn � ˜m2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � = Eσ(k−1)hn � ˜m4 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � − � Eσ(k−1)hn � ˜m2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ��2 = 2σ4 (k−1)hnh2 n + OP(h2 n),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Covσ(k−1)hn � ˜m2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � = Eσ(k−1)hn � ˜m3 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � Eσ(k−1)hn � ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � − Eσ(k−1)hn � ˜m2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � Eσ(k−1)hn � ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � Eσ(k−1)hn � ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � = 2 π σ4 (k−1)hnh2 n + OP(h2 n),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Varσ(k−1)hn � ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � = Eσ(k−1)hn � ˜m2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � Eσ(k−1)hn �� ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n �2� 18 − � Eσ(k−1)hn � ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � Eσ(k−1)hn � ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ��2 = σ4 (k−1)hn � 1 − 4 π2 � h2 n + OP(h2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We have used the first four moments of the half-normal distribution and their illustration via integrals over tail probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The dependence structure between ˜mk,n and ˜m∗ k,n also affects the variance of ˆσ2 τ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We perform approximation steps for covariances similar as for the moments of local minima above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' using that h−1 n Covσ(k−1)hn � ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' ˜m∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � = � ∞ −∞ � ∞ −∞ � Pσ(k−1)hn � h−1/2 n ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n > x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' h−1/2 n ˜m∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n > y � − Pσ(k−1)hn � h−1/2 n ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n > x � Pσ(k−1)hn � h−1/2 n ˜m∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n > y �� dx dy = � ∞ 0 � ∞ 0 � Pσ(k−1)hn � σ(k−1)hn sup 0≤t≤1 Bt > x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' σ(k−1)hn � sup 0≤t≤1 Bt − B1 � > y � − Pσ(k−1)hn � σ(k−1)hn sup 0≤t≤1 Bt > x � Pσ(k−1)hn � σ(k−1)hn � sup 0≤t≤1 Bt − B1 � > y �� dx dy + OP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This shows that the joint distribution of ( ˜mk,n, ˜m∗ k,n) relates to the distribution of the minimum and the difference between minimum and endpoint of Brownian motion over an interval, or equiv- alently the distribution of the maximum and the difference between maximum and endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The latter is readily obtained from the joint density of maximum and endpoint which is a well-known result on stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Utilizing this, we obtain that Covσ(k−1)hn � ˜mk,n , ˜m∗ k,n � = �1 2 − 2 π � hn σ2 (k−1)hn(1 + OP(hα n)) + OP � hn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The additional remainder of order hα n in probability is due to the different approximations of (σt) in ˜mk,n and ˜m∗ k,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This implies that Covσ(k−1)hn � ˜mk,n ˜m∗ k−1,n, ˜mk+1,n ˜m∗ k,n � = � Eσ(k−1)hn � ˜mk,n ˜m∗ k,n � − Eσ(k−1)hn � ˜mk,n � Eσ(k−1)hn � ˜m∗ k,n �� Eσ(k−1)hn � ˜m∗ k−1,n � E � ˜mk+1,n � = σ4 (k−1)hn � 1 π − 4 π2 � h2 n + OP � h2 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' With analogous steps, we deduce two more covariances which contribute to the asymptotic vari- ance: Covσ(k−1)hn � ˜m2 k,n, � ˜m∗ k,n �2� = −h2 n σ4 (k−1)hn 2 + OP � h2 n � , Covσ(k−1)hn �� ˜m∗ k,n �2, mk ˜m∗ k−1,n � = −h2 n 2 3π σ4 (k−1)hn + OP � h2 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' All covariance terms which enter the asymptotic variance are of one of these forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For the conditional variance given σ2 τ−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' we obtain that Varσ2 τ− � ˆσ2 τ− � = 1 K2n π2 4(π − 2)2 � ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−2 n Varσ2 τ− � ˜m2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n + ( ˜m∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n)2 − 2 ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � 19 − ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧2 4h−2 n Covσ2 τ− � ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' ˜m2 k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n + ( ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n)2 − 2 ˜mk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ˜m∗ k−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n �� + OP � K−1 n � = 1 K2n π2 4(π − 2)2 � ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−2 n � 2Varσ2 τ− � ˜m2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � + 4Varσ2 τ− � ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � + 2 Covσ2 τ− � ˜m2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' ( ˜m∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n)2� − 4 Covσ2 τ− � ˜m2 k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � − 4 Covσ2 τ− � ( ˜m∗ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n �� + ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧2 4h−2 n � 2 Covσ2 τ− � ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' ˜mk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ˜m∗ k−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � − Covσ2 τ− � ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' ˜m2 k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n � − Covσ2 τ− � ˜mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' ( ˜m∗ k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n)2��� + OP � K−1 n � = 1 Kn π2 4(π − 2)2 σ4 τ− � 8 − 16 π2 − 1 − 8 π + 8 3π + 2 � 4 3π − 16 π2 �� + OP � K−1 n � = 1 Kn 1 (π − 2)2 �7π2 4 − 2π 3 − 12 � σ4 τ− + OP � K−1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Step 5 For a central limit theorem, the squared bias needs to be asymptotically negligible compared to the variance, which is satisfied for Kn = O(h−2α/(1+2α) n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' By the existence of higher moments of ˜mk,n and ˜m∗ k−1,n, a Lyapunov-type condition is straightforward, such that asymptotic normality conditional on σ2 τ− is implied by a classical central limit theorem for m-dependent triangular arrays as the one by [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' A feasible central limit theorem is implied by this conditional asymptotic normality in combination with FX-stable convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For the stability, we show that αn = K1/2 n � ˆσ2 τ− − σ2 τ− � satisfy E [Zg(αn)] → E [Zg(α)] = E[Z]E [g(α)] , (25) for any FX-measurable bounded random variable Z and continuous bounded function g, where α = σ2 τ− 1 (π − 2) � 7π2 4 − 2π 3 − 12 U , (26) with U a standard normally distributed random variable which is independent of FX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' By the above approximations it suffices to prove this for the statistics based on ˜mk,n and ˜m∗ k−1,n from (20), and Z measurable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' σ( � t 0 σs dWs, 0 ≤ t ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Set An = [τ − (Kn + 1)hn, τ] , ˜X(n)t = � t 0 1An(s)σ⌊sh−1 n ⌋hn dWs , ¯X(n)t = Xt − ˜X(n)t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Denote with Hn the σ-field generated by ¯X(n)t and FX 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The sequence � Hn � n∈N is isotonic with limit � n Hn = σ( � t 0 σs dWs, 0 ≤ t ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Since E[Z|Hn] → Z in L1(P) as n → ∞, it is enough to show that E[Zg(αn)] → E[Z]E[g(α)], for Z being Hn0-measurable for some n0 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Observe that αn includes only increments of local minima based on ˜X(n)t, which are uncorrelated from those of ¯X(n)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For all n ≥ n0, we hence obtain that E[Zg(αn)] = E[Z]E[g(αn)] → E[Z]E[g(α)] by a standard central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This shows (25) and completes the proof of (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Proof of Proposition 2 For the quarticity estimator (12), when ⌊h−1 n τ⌋ > Kn, we have that E �� σ4τ − − σ4 τ− � = π 4(3π − 8)Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−2 n E � ˜m4 k,n + ( ˜m∗ k−1,n)4 − 4 ˜m3 k,n ˜m∗ k−1,n − 4 ˜mk,n( ˜m∗ k−1,n)3 + 6 ˜m2 k,n( ˜m∗ k−1,n)2� − E[σ4 τ−] + O � hα∧1/2 n � = � π 4(3π − 8) � 6 − 16/π − 16/π + 6 � − 1 � E[σ4 τ−] + O(1) = O(1) , by using the same moments as in the computation of the asymptotic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We can bound its variance by Var �� σ4τ − � ≤ π2 16(3π − 8)2K2n 2Knh−4 n Var �� ˜mk,n − ˜m∗ k−1,n �4� + O � K−1 n � ≤ 1 Kn π2 8(3π − 8)2 h−4 n E �� ˜mk,n − ˜m∗ k−1,n �8� + O � K−1 n � ≤ 1 Kn π2 8(3π − 8)2 h−4 n 256 E � ˜m8 k,n � + O � K−1 n � = O(K−1 n ) , what readily implies Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Asymptotics of the truncated spot volatility estimation with jumps Denote by DX k := mk,n − mk−1,n, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' , h−1 n − 1 , the differences of local minima based on the observations (5), with the general semimartingale (4) with jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Denote by DC k := ˜mk,n − ˜m∗ k−1,n, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' , h−1 n − 1 , the differences of the unobservable local minima considered in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In particular, the statistics DC k are based only on the continuous part (Ct) in (4) such that the jumps are eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Theorem 3 is implied by Proposition 2, if we can show that π 2(π − 2)Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n �� DX k �21{|DX k |≤un} − � DC k �2� = OP � h α 2α+1 n � = OP � K−1/2 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We decompose this difference of the truncated estimator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' which is based on the available observa- tions with jumps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' and the non-truncated estimator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' which uses non-available observations without 21 jumps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' in the following way: π 2(π − 2)Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n �� DX k �21{|DX k |≤un} − � DC k �2� = π 2(π − 2)Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n � 1{|DC k |>cun} �� DX k �21{|DX k |≤un} − � DC k �2� + 1{|DC k |≤cun}1{|DX k |≤un} �� DX k �2 − � DC k �2� − 1{|DC k |≤cun}1{|DX k |>un} � DC k �2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' with some arbitrary constant c ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We consider the three addends which are different error terms by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' large absolute statistics based on the continuous part (Ct);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' non-truncated statistics which contain (small) jumps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' the truncation of also the continuous parts in statistics (DX k ) which exceed the threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' The probability P(|DC k | > cun) can be bounded using the estimate from (23) and Gaussian tail bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Observe that the remainder in (23) is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' This yields that for some y > 0, we have that P � h−1/2 n �� ˜mk,n �� > y � ≤ P � sup 0≤t≤1 Bt > y � , what is intuitive, since the errors (ϵi) are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We apply the triangular inequality and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='then Hölder’s inequality to the expectation of the absolute first error term and obtain for any ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='p ∈ N that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2(π − 2)Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k=(⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−Kn)∧1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n 1{|DC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k |>cun} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='DX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='�21{|DX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k |≤un} − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='DC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='�2����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2(π − 2)Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k=(⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−Kn)∧1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='1{|DC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k |>cun} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='DX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='�21{|DX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k |≤un} − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='DC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='�2��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2(π − 2)Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k=(⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−Kn)∧1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k=(⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−Kn)∧1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='h−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='|DC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k | > chκ−1/2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2 P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='|B1| > c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2hκ−1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='��1/2√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2 u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='(π − 2)Kn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='k=(⌊h−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n τ⌋−Kn)∧1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='h2κ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='− c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='4 h2κ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='= O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='h(−p+1)(2κ−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='= O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='2α+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Since 2κ − 1 < 0 and p arbitrarily large, we conclude that the first error term is asymptotically negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We will use the elementary inequalities DX k = min i∈In k � C i n + J i n + ϵi � − min i∈In k−1 � C i n + J i n + ϵi � ≤ min i∈In k � C i n + ϵi � + max i∈In k J i n − min i∈In k−1 � C i n + ϵi � − min i∈In k−1 J i n = DC k + max i∈In k J i n − min i∈In k−1 J i n + OP � hα∧1/2 n � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' and DX k = min i∈In k � C i n + J i n + ϵi � − min i∈In k−1 � C i n + J i n + ϵi � ≥ min i∈In k � C i n + ϵi � + min i∈In k J i n − min i∈In k−1 � C i n + ϵi � − max i∈In k−1 J i n = DC k + min i∈In k J i n − max i∈In k−1 J i n + OP � hα∧1/2 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Therefore, we can bound |DX k − DC k | by sup i∈In k ,j∈In k−1 |J i n − J j n | ≤ sup s∈[khn,(k+1)hn],t∈[(k−1)hn,khn] |Js − Jt| ≤ sup s∈[khn,(k+1)hn] |Js − Jkhn| + sup t∈[(k−1)hn,khn] |Jkhn − Jt| , and the remainder term of the approximation for the continuous part which is OP � hα∧1/2 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Since the compensated small jumps of a semimartingale admit a martingale structure, Doob’s inequality for càdlàg L2-martingales can be used to bound these suprema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' Based on these preliminaries,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' we obtain for the expected absolute value of the second error term that π 2(π − 2)Kn E ����� ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n 1{|DC k |≤cun}1{|DX k |≤un} �� DX k �2 − � DC k �2����� � ≤ π 2(π − 2)Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n E � 1{|DC k |≤cun}1{|DX k |≤un} ��� � DX k �2 − � DC k �2��� � ≲ 1 Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n E � sup i∈In k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='j∈In k−1 |J i n − J j n |2 ∧ (1 + c)2u2 n � ≲ 1 Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n E � sup t∈[khn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='(k+1)hn] |Jt − Jkhn|2 ∧ u2 n � ≲ 1 Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n E � |J(k+1)hn − Jkhn|2 ∧ u2 n � = O � u2−r n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' 23 Applying the elementary inequalities from above, a cross term in the upper bound for � DX k �2 − � DC k �2 is of smaller order and directly neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' It can be handled using the Cauchy-Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' In the last step, we adopt a bound on the expected absolute thresholded jump incre- ments from Equation (54) in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For the negligibility of the second error term, we thus get the condition that κ(2 − r) ≥ α 1 + 2α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (27) Doob’s inequality yields as well that P � sup t∈[khn,(k+1)hn] |Jt − Jkhn| ≥ (1 − c)un � ≤ E ���J(k+1)hn − Jkhn ��r∧1� � (1 − c)un �r∧1 + O(hn) = O � hnu−r n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For this upper bound, we decomposed the jumps in the sum of large jumps and the martingale of compensated small jumps to which we apply Doob’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' We derive the following estimate for the expectation of the third (absolute) error term π 2(π − 2)Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n E � 1{|DC k |≤cun}1{|DX k |>un} � DC k �2� ≤ π 2(π − 2)Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n E � 1{2 sups∈[(k−1)hn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='(k+1)hn] |Js−Jkhn|≥(1−c)un} � DC k �2� ≲ 1 Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 h−1 n P � sup t∈[khn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content='(k+1)hn] |Jt − Jkhn| ≥ (1 − c)un � E �� DC k �2� ≲ 1 Kn ⌊h−1 n τ⌋−1 � k=(⌊h−1 n τ⌋−Kn)∧1 �E ���J(k+1)hn − Jkhn ��r∧1� � (1 − c)un �r∧1 + O(hn) � = O � hnu−r n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' For the negligibility of the third error term, we thus get the condition that 1 − κr ≥ α 1 + 2α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' (28) Since under the conditions of Theorem 3, (27) and (28) are satisfied, the proof is finished by the negligibility of all addends in the decomposition above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNA0T4oBgHgl3EQfAv8K/content/2301.01965v1.pdf'} +page_content=' References [1] Aït-Sahalia, Y.' metadata={'source': 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Electrical-Thermal Co-Simulation +Method with LTS for Multiscale Structures +Kai Zhu, Graduate Student Member, IEEE, and Shunchuan Yang, Senior Member, IEEE +Abstract—In this article, an efficient transient electrical- +thermal co-simulation method based on the finite element method +(FEM) and the discontinuous Galerkin time-domain (DGTD) +method is developed for electrical-thermal coupling analysis of +multiscale structures. Two Independent meshes are adopted by +the steady electrical analysis and the transient thermal simulation +to avoid redundant overhead. In order to enhance the feasibility +and efficiency of solving multiscale and sophisticated structures, a +local time stepping (LTS) technique coupled with an interpolation +method is incorporated into the co-simulation method. Several +numerical examples from simple structures to complex multi- +scale PDN structures are carried out to demonstrate the accuracy +and efficiency of the proposed method by comparing with +the COMSOL. Finally, two practical numerical examples are +considered to confirm the performance of the proposed method +for complex and multiscale structures. +Index Terms—Discontinuous Galerkin time-domain (DGTD) +method, electrical-thermal co-simulation, finite element method +(FEM), local time stepping (LTS), PDN. +I. INTRODUCTION +W +ITH the development of semiconductor technique and +packaging technology over the past few decades, the +typical size of components in integrated circuits (ICs) has +kept shrinking while the integration density sees an upward +trend. It is well known that the Joule heating effect in +ICs is a challenging issue which has attracted substantial +attention from researchers. Since the electrical malfunction +is frequently related to temperature increment and improper +power distribution, a reasonable design both in circuit structure +and thermal sinks becomes crucially important. Taking the +through-silicon via (TSV) for instance, it plays a key role in +2.5D/3D ICs design [1], [2], for enabling the high-speed signal +processing by smoothing paths for continuing the interconnect +scaling. However, currents flowing through TSVs leads to local +temperature rise, then potentially influences the transistors +switching states and induces circuit failures or performance +deterioration [3], [4]. Generally, electromigration, estimated +by the Black’s equation, works as the primary cause of circuit +failure [5], [6]. From them we can obtain that the mean time of +This work was supported in part by the National Natural Science Foundation +of China under Grant 62141405, 62101020, 62071125, in part by Defense In- +dustrial Technology Development Program under Grant JCKY2019601C005, +in part by Pre-Research Project under Grant J2019-VIII-0009-0170 and +Fundamental Research Funds for the Central Universities. (Corresponding +author: Shunchuan Yang.) +K. Zhu is with the School of Electronic and Information Engineering, +Beihang University, Beijing, 100083, China (e-mail: zhukai7@buaa.edu.cn). +S. Yang is with the Research Institute for Frontier Science and the School of +Electronic and Information Engineering, Beihang University, Beijing, 100083, +China (e-mail: scyang@buaa.edu.cn). +Manuscript received xxx; revised xxx. +failure abides by a negative exponential multiplier relationship +[7]. Therefore, an efficient and accurate electrical-thermal co- +simulation algorithm can be indispensable for ICs design. +There have been a great deal of numerical algorithms +developed for solving thermal or electrical problems with +respective advantages and deficiencies either in accuracy or +efficiency. The analytical method, such as the equivalent circuit +model, can be adopted in some circumstances and efficiency +improvements can be obtained. However, it suffers from the +lack of generality [8]. The finite volume method (FVM) can +be adopted to analyze the heat transfer problems [9], which +introduces the numerical flux to represent the information +exchange between adjacent subdivision elements [10]. The +widely used finite difference method (FDM) benefits from +simplicity and efficiency [11], [12]. However, it is subject +to staircase errors caused by structured meshes [13]. The +finite method time domain (FETD) is practically restrained for +solving a large matrix equation at each time step, which can +be computationally intensive [14], [15] and an ill-conditioned +matrix may be obtained. The domain decomposition method +(DDM) coupled by the finite element tearing and intercon- +necting (FETI) can contribute to alleviating the computation +burden [16], [17], which have been applied for coping with +large scale problems [18]. +The discontinuous galerkin time-domain (DGTD) method +[19], [20] has attracted much attention and developed rapidly +for inheriting the advantages of the FVTD method and the +FETD method. It can be regarded as an element-level DDM +[21], which implies an innate parallel characteristic and pro- +vides the possibility of utilizing adaptive orders or types of +basis functions in different elements [22]. +In this article, we introduce an electrical-thermal co- +simulation scheme, which integrates the electrical analysis and +thermal simulation through an iteration procedure. The electri- +cal analysis is based on the finite element method (FEM) for its +capacity of modeling sophisticated structures [14]. The thermal +simulation can be divided into two phases and is based on the +DGTD method. It is noteworthy that the thermal conduction +equation is a parabolic partial differential equation, which is +difficult to be solved directly by the traditional DGTD method +[20]. In order to address this issue, an auxiliary equation need +to be introduced to degrade the parabolic partial differential +equation to a hyperbolic partial differential equation, which +can be solved directly by the DGTD method [23]. +The DGTD method generally leads to a series of compact +linear systems, the dimension of which is equal to the degrees +of freedom within the corresponding element, and those matrix +equations are required to be solved at each time step. The merit +arXiv:2301.00088v1 [math.NA] 31 Dec 2022 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +2 +of this method over the FETD method lies in that the matrices +dimension is quite small, which indicates the inverses of ele- +mental matrices can be calculated readily and stored before the +time iteration begins. The computational complexity mainly +depends upon the number of elements, the order of chosen +local basis functions, and the simulation time steps. If explicit +time discretization scheme is adopted, the time step is limited +rigorously by the minimal size of the discretized elements +according to the Courant–Friedrichs–Lewy (CFL) stability +condition [24]. For multiscale structures, the disparities of +the mesh size in different regions can be large. If the global +time step (GTS) scheme is adopted, the time step is restricted +to the global smallest mesh size to guarantee the stability, +thereby leading to substantial computational overhead. Some +implicit-explicit methods have been developed to alleviate +this issue, where an implicit time-marching scheme is used +in regions with fine mesh, while the explicit time-marching +scheme is adopted in coarse mesh regions [25], [26]. However, +these methods dramatically increase the computational time +and memory consumption for large scale problems and the +highly disparate mesh element sizes may cause ill-conditioned +problem when the implicit time-marching scheme is applied. +Therefore, this method cannot cope with the problems effi- +ciently encountered in analyzing multiscale structures [27]. +To tackle the aforementioned challenge, a local time step- +ping (LTS) scheme is integrated into the electrical-thermal co- +simulation method, with which the computational efficiency +can be significantly improved and the capability of simulating +multiscale and locally refined structures can be facilitated [28]. +The structure can be divided into several groups according +to different mesh sizes, then elements in each group advance +in time with local time steps [29], [30], which can reduce +simulation time while maintaining accurate solutions. It is +worth noting that the time step in each group is required to +be selected carefully in order to guarantee the global stability +and accuracy requirements. Then, a rigorous analysis of the +stability of the LTS scheme is introduced based on the Von- +Neuman method [31], [32]. +The article is organized as follows. In section II, the detailed +formulation for electrical and thermal problem is presented. +Then, the co-simulation procedure as well as the concept and +implementation of the LTS technique is developed. In section +III, several simple examples are presented to demonstrate +the accuracy and efficiency of the proposed scheme, as well +as the efficiency enhanced by the LTS technique compared +with the GTS scheme. In section IV, the proposed scheme is +applied to some practical structures to verify the capability +of co-simulation for some practical structures. Finally, some +conclusions are drawn in Section V. +II. THEORIES AND FORMULATIONS +In this section, the detailed formulations for current conti- +nuity equation and heat conduction equation are introduced. +Then the coupling simulation flow algorithm is elaborated, +including the application of independent meshes for electrical +and thermal simulations and the LTS technique developed for +coping with multiscale structures. +A. Formulations of Electrical and Thermal Analysis +The current continuity equation is considered in the elec- +trostatic analysis, which can be written as +∇ · (σ∇φ + ε∇∂φ +∂t ) = 0, +(1) +where σ and ε are the electrical conductivity and the per- +mittivity of the medium, respectively. The Dirichlet boundary +condition subjected to the governing equation can be expressed +as +φ = φ0, +(2) +The impedance boundary condition adopted for modeling +lossy conductors can be imposed on the surface of the con- +ductor with the form +ˆn · σ∇φ = +φ +RS . +(3) +where ˆn is the unit normal vector pointing outward from +the boundary of the computational domain, R denotes the +surface impedance of a conductor, S represents the area of +the boundary surface. In our implementation, the potential +distribution is considered constant during an interval, and the +steady-state solution is analyzed through the FEM method. +As for the thermal simulation, the temperature evolution of +a spatial point is governed by the transient heat conduction +equation, which can be written as +ρc∂T +∂t = ∇ · (k∇T) + Q, +(4) +where ρ represents the density of the medium, c denotes +the heat capacity, k is the thermal conductivity, Q represents +the heat source, respectively. To solve (4), the corresponding +boundary conditions include the Dirichlet boundary condition +T = T0, +(5) +and the convective boundary condition +ˆn · (k∇T) = −h (T − Ta) , +(6) +where h is the convective heat transfer coefficient, Ta denotes +the ambient temperature. Since the traditional DGTD method +is incapable to solve the parabolic differential equation di- +rectly, an auxiliary vector variable q is introduced to transform +(4) to a hyperbolic differential equation [20], which can be +rewritten as +q = −k∇T, +(7) +ρc∂T +∂t = −∇ · q + Q. +(8) +By implementing the Galerkin’s spatial testing procedure +to (7) and (8) in the ith subdomain, the formulation can be +obtained as +� +Vi +Nk +� +qx + k ∂T +∂x +� +dV = 0, +(9) +� +Vi +Nk +� +ρc∂T +∂t +∇ · q − Q +� +dV = 0, +(10) + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +3 +where Nk denotes the kth test basis function, qx represents +the component of q in the x-direction, and the components in +other two directions can be obtained in a similar manner. With +the application of spatial integration and Gauss’s theorem, (9) +and (10) can be rewritten as +� +Vi +NkqxdV = k +� +Vi +T ∂Nk +∂x dV +− k +4 +� +f=1 +� +∂Vi +nxT ∗NkdS, +(11) +� +Vi +Nkρc∂T +∂t dV = +� +Vi +(q · ∇Nk + NkQ) dV +− +4 +� +f=1 +� +∂Vi +Nk ˆn · q∗dS, +(12) +where T ∗ and q∗ are the numerical fluxes which represent the +information exchange between adjacent elements, which can +be expressed as the linear combination of variables in adjacent +elements like (13). +ˆn · q∗ = D0 +�ˆn · qi + ˆn · qj� ++ D1 +�ˆn · qi − ˆn · qj� ++ D2 +� +T i − T j� +, +T ∗ = D3 +� +T i + T j� ++ D4 +� +T i − T j� ++ D5 +�ˆn · qi − ˆn · qj� +. +(13) +where T i and T j denote the temperature in self element and +external neighboring element, respectively. The same notations +are adopted for q, and Di (i = 0, . . . 5) are constants, which +depend on the chosen numerical flux form. In our imple- +mentation, the upwind flux is adopted for better convergence +properties [33]. Therefore, the coefficients can be defined as +D0 = D3 = 0.5, D2 = −4 and D1 = D4 = D5 = 0. In addi- +tion, for the numerical flux on convective boundary surfaces, +the coefficients are revised as D3 = D4 = 0.5, D2 = −h and +D0 = D1 = D5 = 0 correspondingly. +To establish the semi-discrete matrix system, (13) is applied +to (11) and (12), with T and q approximated by nodal basis +functions, which leads to +Miqi +x = kSi +xTi − +4 +� +f=1 +� +k(D3 + D4)Gi +xTi ++k(D3 − D4)Gj +xTj� +, +(14) +ρcMi ∂Ti +∂t = Si +xqi +x + Si +yqi +y + Si +zqi +z + Qi +− +4� +f=1 +� +D0Gi +xqi +x + D0Gi +yqi +y + D0Gi +zqi +z + D0Gj +xqj +x ++D0Gj +yqj +y + D0Gj +zqj +z + D2CiTi − D2CjTj� +. +(15) +where Mi denotes the local mass matrix and Si +x, Si +y and Si +z are +the local stiffness matrices, the detailed expression of matrices +and vectors in (14) and (15) can be written as +� +Mi� +kl = +� +Vi +N i +kN i +l dV, +(16) +� +Si +x +� +kl = +� +Vi +∂N i +k +∂x N i +l dV, +(17) +� +Qi� +k = +� +Vi +N i +kQdV, +(18) +� +Gi +x +� +kl = +� +∂Vi +nxN i +kN i +l dS, +(19) +� +Gj +x +� +kl = +� +∂Vi +nxN i +kN j +l dS, +(20) +� +Ci� +kl = +� +∂Vi +N i +kN i +l dS, +(21) +� +Cj� +kl = +� +∂Vi +N i +kN j +l dS. +(22) +where nx denotes the component in the x-direction of the +outward normal vector. The detailed forms of other matrices +including Si +y, Si +z, Gi +y, Gi +z, Gj +y and Gj +z can be obtained simi- +larly. +Since the obtained matrix equation is still semi-discrete, the +derivative in the temporal dimension is required to be dis- +cretized. The backward difference is unconditionally stable but +yields a global matrix operation thereby losing the advantage +of the DGTD method. Therefore, the forward difference is +adopted in our implementation, where the time derivative term +can be approximated by +∂T +∂t = T (t + ∆t) − T (t) +∆t ++ O (∆t) . +(23) +to obtain the finial matrix equation in the thermal simulation. +The forward difference is conditionally stable, with the +convergence dependent on selected time step and related +system coefficients. In order to guarantee the stability, the +Courant–Friedrichs–Lewy (CFL) condition is required to be +satisfied, hence finding a valid approach for estimating the +time step bound is of vital importance. In this implementation, +the Von-Neuman stability analysis is introduced to validate the +stability of the chosen time step. Firstly, a column vector is +constructed to include all the unknowns of the system equation +at a specific time. For instance, unknowns related to heat flux +can be written as Uq = +�� +q1 +x, q1 +y, q1 +z +� +, . . . , +� +qN +x , qN +y , qN +z +��T , +where qi +x, qi +y and qi +z denote the components of the heat flux of +the ith element in different directions, respectively. N denotes +the number of split elements. If the number of basis func- +tions is M, qi +x can be represented as +� +qi,1 +x , qi,2 +x , . . . , qi,M +x +� +. +Similarly, the column vector for T can be written as UT = +� +T1, . . . , TN�T . According to (14), Uq at tn can be obtained +by +Uq (tn) = AqUT (tn) , +(24) +By substituting (24) into (15), the time-marching relation- +ship between U (tn+1) and U (tn) can be rewritten in a +compact matrix form +UT (tn+1) = AT UT (tn) . +(25) +where the dimension of Aq and AT are 3MN × 3MN and +MN × MN, respectively, which assemble the information of +all element matrices and corresponding numerical flux. The +concrete form of A can vary with the discretizing parameters + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +4 +adopted in the DGTD method, including the chosen time step +(∆t), the order and form of the basis function (M), the form +of numerical flux (q∗, T ∗), and the properties of material and +meshes. +The stability of the system can be analyzed by computing +the MN eigenvalues of AT (λi, i = 1, . . . , MN). If all +the eigenvalues are located inside the unit circle, it can be +concluded that it is stable. +B. The Electrical-Thermal Co-Simulation Procedure +The flowchart of the procedure of the electrical-thermal co- +simulation is represented in Algorithm 1. The initialization +includes inputting the material parameters required in the +electrical and thermal simulation, such as the electrical con- +ductivity for electrical analysis, material density, heat capacity +and thermal conductivity for thermal simulation. In addition, +the time step, the simulation duration, and the relationship +coefficients between material parameters and temperature need +to be taken into consideration. Over each iteration, the elec- +trical problem is solved through the FEM solver with the +voltage and current density distribution obtained. Then, the +dissipated power calculated in each element is considered as +the source for subsequent thermal simulation. The dissipated +power during a time step period can be written as +Q = σ| ⃗E|2 = σ|∇φ|2. +(26) +In this article, the influence of temperature on electrical +conductivity is considered, and its value can be calculated by +the fitted interpolation function, the fourth-order form can be +written as +σ(T) = +4 +� +n=0 +AnT n +T0 ≤ T ≤ T1. +(27) +where An (n = 0, . . . 4) are the fitting coefficients of mate- +rial properties, [T0, T1] denotes the interpolation interval, the +related coefficients of the materials used are shown in Table +I [34], [35]. It is worth noting that the co-simulation method +can also be used when other parameters vary with temperature +without sacrificing its generality. +TABLE I +TEMPERATURE INTERPOLATION COEFFICIENTS OF TWO MATERIALS +Cu +Poly-Si +A0 +2.91 × 108 +7.45 × 104 +A1 +−1.56 × 106 +−1.08 × 102 +A2 +3.70 × 103 +1.01 × 10−1 +A3 +−3.93 +−5.17 × 10−5 +A4 +−1.56 × 103 +1.07 × 10−7 +After the temperature distribution at a time step is obtained, +the material parameters are updated within each split element +based on this distribution and the interpolation function con- +necting material properties and temperature if the simulation +is not finished. Otherwise, the time iteration phase completes. +Although some structures are electrically insulated and can +be ignored in electrical analysis, they are still required to be +considered for the heat transfer effect to ensure that the tem- +perature distribution analysis in the overall region is accurate. +Moreover, fine meshes for the electrical analysis may be used +to accurately model complex structures, while relatively coarse +meshes can be adopted in the thermal analysis for the slow +pace of change. There can also be large gaps of mesh densities +in different parts of the structure. If the structure is meshed in a +unified density for electrical analysis and thermal simulation, +the time step can be restricted to extremely small values to +guaranteed stability, thereby leading to a great amount of +computational overhead, even the time consumption can be +unacceptable. To address this issue, two independent meshes +for the electrical and thermal simulation are used to improve +the flexibility of the algorithm and to avoid the redundant +degrees of freedom (DoFs). A mapping relationship between +meshes is built and a interpolation method is applied for +imposing heat source and updating the material coefficients +in the time-marching process. +With the growing of unknowns, the construction of mapping +relationship bridging the discontinuities between electrical and +thermal meshes can be increasingly time-consuming. In order +to alleviate this problem, two tree structures can be constructed +in advance and split elements in electrical and thermal analysis +are stored in leaf nodes to accelerate the traversal process. In +this implement, two full octrees are adopted for simplicity, +where within each level every node has 8 children, and the +computational domain is divided into 512 blocks if the depth +is set to 4, as illustrated in Fig. 1. The jth node of the +ith level is denoted as Ci,j, where a node element includes +the location information (tetrahedron indexes, space boundary) +and pointers to each of its 8 children. +... +... +Child +Child +Child +0,0 +C +1,0 +C +1,4 +C +1,7 +C +... +... +2,0 +C +3,0 +C +3,4 +C +... +3,7 +C +... +…... +Key +Key +Key +... +(a) +1,0 +C +1,1 +C +1,2 +C +1,3 +C +1,4 +C +1,5 +C +2,0 +C +2,1 +C +2,2 +C +2,3 +C +2,4 +C +2,5 +C +3,0 +C +Level 1 +Level 2 +Level 3 +(b) +Fig. 1. +Illustration of the construction of octree (a) Topology of the tree, (b) +Geometric space representation for different levels. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +5 +Algorithm 1 Transient electrical-thermal co-simulation +Input: Parameters of material properties +Control factors of time stepping +Output: Distribution of temperature and electric potential +1: Initiate t = 0 and σ = σ0 +2: repeat +3: +Analyze current density through electrical analysis; +4: +Calculate produced Joule heat; +5: +Simulate temperature through thermal analysis; +6: +Update σ, set t = t + ∆t; +7: until t >= tmax +8: Output T and φ. +C. The Explicit Local Time Stepping +For a multiscale structure, elements can be partitioned into +groups and different time steps are allowed in separate groups +based on the explicit local time stepping (LTS) scheme, with +time step in each group constrained by the local mesh size. +Therefore, the total number of equation calculation needed +for advancing the numerical solution from tn to tn + ∆t +will decrease in comparison with using the global smallest +time step. For simplicity, assuming the split elements are +divided into four groups, the average sizes in each group +are h, h/2, h/4 and h/8, respectively. Since the time step +in each group is required to satisfy the stability condition, +which can be chosen as ∆t1 = 2∆t2 = 4∆t3 = 8∆t4, with +∆ti (i = 1, . . . , 4) denote the time step for the ith group. The +implementation of time stepping is shown schematically in +Fig. 2. The recursive procedure can be given as +1) Elements in each group advance one step with local time +steps ∆ti (i = 1, . . . , 4), the sequence of the advance- +ment is Group #1 (averagely coarsest mesh), followed +by Group #2, Group #3, and finally Group #4, as shown +in Fig. 2(a). +2) Elements in Group#4 (averagely finest mesh) advance one +step with local time step ∆t4, then elements in Group#4 +and Group#3 are at the same time level, as shown in Fig. +2(b). +3) Elements in Group#3 and Group#4 advance one step in +sequence with local time steps ∆t3 and ∆t4, as illustrated +in Fig. 2(c). +4) Elements in Group#4 advance one step with local time +step ∆t4, then the elements in Group#2, Group#3, and +Group#4 are at the same time level, as outlined in Fig. +2(d). +5) Elements in Group#2, Group#3, Group#4 advance one +steps with local time steps ∆ti (i = 2, 3, 4), as presented +in Fig. 2(e). +6) The processes in 2–4 repeat until all elements reach the +final interval. +While analyzing an element in Group#i with a neighboring +element located in Group#j (j < i), the temperature of the +latter may be unknown at the current time step, which leads +to a difficulty in constructing the numerical flux. Taking Fig. +2(g) for instance, if the element in Group#4 has an adjacent +element located in group 1-3, the temperature of the adjacent +Group 1 +t +size +Group 2 +t +size +Group 3 +Group 4 +t +size +t +size +△t1 +(a) +(b) +(c) +(d) +t +size +t +size +t +size +t +size +(e) +(f) +(g) +(h) +△t1/2 +△t1/4 +△t1/8 +△t1/2 +3△t1/8 +△t1 +△t1/2 +△t1 +3△t1/4 +△t1 +△t1/2 +△t1/4 +△t1 +5△t1/8 +3△t1/4 +△t1 +7△t1/8 +△t1 +△t1 +Fig. 2. +Example of the LTS stepping process for four groups with time steps +∆t1 = 2∆t2 = 4∆t3 = 8∆t4. +element is unknown at 7∆t1/8. However, after the process (1) +presented in Fig. 2(a) finished, the temperature of elements +in Group#1 at ∆t1 has been obtained, the similar situation +is also for elements in Group#2 and Group#3. Therefore, a +linear interpolation strategy can be adopted to estimate the +temperature of adjacent elements in different groups. While +considering the element in Group#i at n∆ti, the temperature +of the adjacent element in Group#j at this time can be +interpolated by +T j +app = +� +T j (nratio) , +j ≥ i +(1 − C)T j (nlow) + CT j (nlow + 1) , +j < i +(28) +where the parameters are given by +nratio = n∆ti +∆tj +, +(29) +nlow = +�n∆ti +∆tj +� +, +(30) +C = [n% (∆tj/∆ti)] / (∆tj/∆ti) . +(31) +and ⌊a⌋ denotes the maximum integer less than a. In the +thermal analysis, the interpolation strategy for adjacent ele- +ments in different groups is also needed to handle the auxiliary +variable q. There are two types of interpolation methods for +q. The first type is similar to that for T in (28), and nearly +no extra storage is required except for some assigned for +symbol marks. The other type consists of two substeps, firstly +obtaining the interpolation temperature for adjacent elements +by (28), then (14) is solved to get the approximated qx, and the + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +6 +same process for qy and qz. The second interpolation strategy +is applied in our implementation, with a higher accuracy +obtained, notwithstanding extra computational resources at an +acceptable level are required. +For the stability analysis of the LTS technique, the similar +Von-Neuman method is also introduced. To better fulfill the +matrix-filling process, the column vector including the un- +knowns are divided according to their groups owning different +time steps. Taking four groups in Fig. 2 for example, interme- +diate unknowns of elements in different group at a time are +stored in four arrays noted as Ui +q and Ui +T (i = 1, . . . , 4), then +the time march from tn to tn + ∆ti in the ith group can be +written in a compact form as +Uq (tn) = AqiUT (tn) , +(32) +UT (tn + ∆ti) = AT iUT (t0) + UT (tn) , +(33) +where t0 denotes the initial simulation time. If there are Ni +elements in Group#i, the dimensions of Aqi and AT i are +3MNi × 3MN and MNi × MN, respectively. After the +substeps in each group according to recursive order shown +in Fig. 2 are finished, the time-marching from tn to tn + ∆t1 +in each group can be filled into a compact form +UT (tn + ∆t1) = AiUT (t0) , i = 1, . . . , 4. +(34) +Based on the mapping relationship of the local node indexes to +the global node indexes, Ai can be sorted into the final matrix +A. Given that all the eigenvalues of A are located inside the +unit circle, the proposed LTS method can be regarded as stable. +III. VERIFICATION AND DEMONSTRATION OF THE +STABILITY, ACCURACY AND EFFICIENCY +In this section, several basic numerical examples are pro- +vided to verify the efficiency and accuracy of the proposed +electrical-thermal co-simulation method by comparing with +the COMSOL software. The accuracy and stability of the LTS +method are also verified, then the speedup performance is +investigated for different time step ratios and mesh densities. +All computations in this section are performed on the computer +with Intel i9-10900 2.8 GHz CPU and 32 GB memory. +A. Accuracy and Stability Verification +A copper block with the dimension of 1.2×6.6×0.6 mm3 is +tested to demonstrate the proposed co-simulation scheme, and +the thermal property of the copper is shown in Table II. The +convection boundary condition is applied on all six surfaces +of the structure with h = 1000 W/(m2K), and the ambient +temperature is set as Ta = 300 K. Two Gaussian pulses are +imposed on one face of the structure sequentially, with the +form of +VGauss(t) = V0e−(t−t0)2/τ 2 +(35) +where V0 = 0.02 V, τ 2 = 0.1, t0 = 0.6 and 3.6, respectively. It +is noteworthy that V0 represents the peak voltage of the pulse, +t0 is the time when the pulse reaches its peak, τ controls the +width of the pulse. +The temporary temperature at P1 (0.6, 3.3, 0.3) (mm) is +illustrated in Fig. 3(a), it can be found that results obtained +from the proposed method show excellent agreement with +those from the COMSOL. To have a better clarification, the +relative error of the observation point at each time point is +outlined in Fig. 3(b), which is defined as +RE = (Tp − Tc) /Tc +(36) +where Tp and Tc denote the results obtained from the proposed +method and the COMSOL, respectively. +TABLE II +THERMAL PROPERTIES OF DIFFERENT MATERIALS +Thermal +conductivity +(W · m−1 · K−1) +Heat +capacity +(J · kg−1 · K−1) +Density +(kg · m−3) +Copper +400 +385 +8.7 × 103 +Nickel +91 +440 +8.9 × 103 +Al2O3 +10 +750 +3.9 × 103 +Silicon +130 +700 +2.3 × 103 +0 +2 +4 +6 +Time (s) +300 +320 +340 +360 +380 +400 +Temperature (K) +0.01 +0.02 +0.03 +0.04 +0.05 +Voltage (V) +COMSOL +Proposed +V +(a) +0 +2 +4 +6 +Time (s) +-6 +-3 +0 +3 +6 +8 +Relative Error +10-6 +Error +(b) +Fig. 3. +Simulation of the temperature with voltage pulses imposed. (a) +Temperature at P1 (0.6, 3.3, 0.3) (mm) obtained from the proposed scheme +and the COMSOL, (b) The relative error. +-1 +-0.5 +0 +0.5 +1 +Real( +i) +-1 +-0.5 +0 +0.5 +1 +Imag( +i) +0.9 1 +-0.2 +0 +0.2 +Fig. 4. +The distribution of the eigenvalues of AT with ∆t = 2 × 10−5 s. +To test the stability, eigenvalues of AT in this context are +analyzed and presented in Fig. 4. Since all the eigenvalues are +located inside the unit circle, the stability in this circumstance +can be concluded. +In Fig. 5, the temperature distribution of the plane z = 0.3 +(mm) of the structure at t = 4 (s) obtained from these two + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +7 +methods are presented. Then, the electric potential distribution +of the same plane at t = 4 (s) is shown in Fig. 6. For a +fair comparison, the structure is split into a similar number +of tetrahedrons, and the same time step is adopted for the +proposed scheme and the COMSOL. The CPU time and +memory consumption are only about 265 s and 15 MB for +the proposed scheme. By comparison, the figures are 2014 s +and 3.1 GB for the COMSOL, which implies an efficiency +improvement of the proposed scheme. +Temperature (K) +389.5 +389.0 +(a) +(b) +389.1 +389.2 +389.3 +389.4 +y +(mm) +x (mm) +0 +1 +2 +3 +4 +5 +6 +0.5 +1 +y +(mm) +x (mm) +0 +1 +2 +3 +4 +5 +6 +0.5 +1 +Fig. 5. +Temperature distribution of the plane z = 0.03 mm at 4 s obtained +from (a) the COMSOL, (b) the proposed scheme. + (mV) +10 +0 +(a) +(b) +2 +4 +6 +8 +y +(mm) +x (mm) +0 +1 +2 +3 +4 +5 +6 +0.5 +1 +y +(mm) +x (mm) +0 +1 +2 +3 +4 +5 +6 +0.5 +1 +12 +14 + +Fig. 6. +Electric potential distribution of the plane z = 0.03 mm at 4 s +obtained from (a) the COMSOL, (b) The proposed scheme. +0 +2 +4 +6 +Time (s) +300 +320 +340 +360 +380 +Temperature (K) +0 +0.01 +0.02 +0.03 +Voltage (V) +COMSOL +Proposed +V +(a) +0 +2 +4 +6 +Time (s) +-8 +-6 +-4 +-2 +0 +2 +4 +Relative Error +10-6 +Error +(b) +Fig. 7. +Simulation of temperature with a step voltage imposed. (a) +Temperature at P1 (0.6, 3.3, 0.3) (mm) obtained from the proposed scheme +and the COMSOL, (b) The relative error. +In addition, a ladder signal is imposed for testing, the results +and relative errors are shown in Fig. 7. Similar to the previous +situation, results obtained from the proposed scheme agree +well with the COMSOL. +B. Efficiency Improvement by the LTS Method +To further demonstrate the stability and efficiency improve- +ment of the LTS technique, the structure of two conductors +composed of copper and nickel covered by a silicon box is +considered, as illustrated in Fig. 8. The thermal properties of +the media are listed in Table II. In this implementation, a volt- +age pulse varying over time is imposed on one face of the cop- +per, which can be written as V1 = 0.04 + VGauss sin (300πt) +(V), where VGauss is described in (35) with V0 = 0.03, +τ = 0.01, and t0 = 0.001. For the electrical analysis, only +the copper is considered and discretized. For the thermal +analysis, the whole structure is discretized into tetrahedrons +and then divided into three groups according to material. The +convection boundary condition is applied on six surfaces of +the silicon block to represent the thermal transfer between the +object and the environment, with h = 1000 W/(m2 · K) and +the ambient temperature Ta = 300 K. +Group II +0.86 +0.58 +Group III +0.18 +0.06 +0.66 +0.12 +Nickle +Copper +Silicon +Unit: mm +0.34 +Electrical +mesh +Heat +mesh +Mapping +1t + +2t + +3t + +Group I +Port +Sink +Fig. 8. +Geometry of the conductors and silicon box, and illustration of the +independent meshes adopted in electrical and thermal analysis in different +groups. +0 +0.005 +0.01 +0.015 +0.02 +Time (s) +300 +320 +340 +360 +380 +400 +Temperature (K) +2 +4 +6 +8 +10 +Voltage (mV) +COMSOL-p1 +COMSOL-p2 +COMSOL-p3 +Proposed-p1 +Proposed-p2 +Proposed-p3 +V +(a) +0 +0.005 +0.01 +0.015 +0.02 +Time (s) +-10 +-8 +-6 +-4 +-2 +0 +2 +Relative Error +10-4 +Error-p1 +Error-p2 +Error-p3 +(b) +0 +0.005 +0.01 +0.015 +0.02 +Time (s) +-5 +0 +5 +10 +Relative Error +10-4 +Error-p1 +Error-p2 +Error-p3 +(c) +0 +1 +2 +3 +4 +5 +Number of tetrahedron +105 +0 +1 +2 +3 +4 +Simulation Time (s) +104 +5 +5.5 +6 +Ratio +Proposed-GTS +Proposed-LTS +Ratio +(d) +Fig. 9. +Simulation of the temperature with the voltage pulses imposed. +(a) Temperature at P1 (-0.06, 0.33, 0.03) (mm), P2 (-0.52, 0.33, 0.03) (mm) +and P3 (-0.29, 0.33, 0.03) (mm) obtained from the proposed-LTS scheme +and COMSOL, (b) The relative error between the proposed-GTS scheme and +the COMSOL, (c) The relative error between the proposed-LTS scheme and +the COMSOL, (d) The comparison of the CPU time for the proposed-GTS +scheme and the proposed-LTS scheme with the structures discretized into +different number of tetrahedrons. + +0:06+00 +2.0 +00+90.5 +lewbetaineS.0 00+90.0 +0'4 +0.0 +8.0 +00+98.10'06+00 +2.0 +. +00+00.50'06+000:5 +06+000.Q88 +380°2 +LGbGLLG6 (K)LGGLLG +0.8 +380' +380'5 +380'3 +380°4 +(K) +38020.0 +o's +04 +oe +8.0 +b!0.Q88 +380°2 +LGbGLLG6 (K)JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +8 +Since the temperature changes more dramatically over time +for the Joule heating caused by conducting current in the +copper, it is discretized into finer elements and recognized +as Group#3. The larger volume of the outer layer composed +of silicon is desired to be discretized into coarser elements to +save computational cost. Three observing probes are placed +at P1 (-0.06, 0.33, 0.03) (mm), P2 (-0.52, 0.33, 0.03) (mm), +and P3 (-0.29, 0.33, 0.03) (mm) to record the temperature +variation. In Fig. 9(a), the temperature of probes obtained from +the proposed-LTS scheme and the COMSOL are presented. An +excellent agreement can be observed as expected. +The CPU time and memory consumption of the COMSOL, +and the proposed scheme with global time stepping, as well as +the proposed scheme with local time stepping are compared +in Table III, which indicates the efficiency improvement of +the proposed scheme. For the proposed-LTS scheme, time +steps adopted in different groups have the relationship ∆t3 = +2∆t2 = 6∆t1, with the minimum time step ∆t1 = 5 × 10−8 +s. For the COMSOL and the proposed-GTS scheme, the time +step is selected as ∆t = 5 × 10−8 s. The relative error of +the proposed-GTS scheme and the proposed-LTS scheme in +comparison with the COMSOL are shown in Fig. 9(b) and +Fig. 9(c), respectively. It can be observed that about four times +speed up is achieved with a reasonable accuracy loss. +In this occasion, the eigenvalues of AT are illustrated in +Fig. 11(a), since all the eigenvalues are located inside the unit +circle, the stability can be guaranteed theoretically. When time +steps are chosen as ∆t3 = 5∆t2 = 100∆t1 with ∆t1 = +2.5 × 10−8 s, the eigenvalues distribution is shown in Fig. +11(b), and the stability can still be verified. +Temperature (K) +370 +372 +374 +376 +378 +380 +382 +(a) +y +(mm) +x (mm) +0 +0.2 +0.1 +0.4 +0.6 +0.8 +0.3 +0.2 +0.4 +0.5 +(b) +y +(mm) +x (mm) +0 +0.2 +0.1 +0.4 +0.6 +0.8 +0.3 +0.2 +0.4 +0.5 +Fig. 10. +Temperature distribution of the plane z = 0.03 mm at 0.01 s +obtained from (a) the COMSOL, (b) the proposed-LTS scheme with ∆t3 = +2∆t2 = 6∆t1. +TABLE III +COMPARISON OF THE COMPUTATIONAL COST BETWEEN THE +PROPOSED-GTS SCHEME, THE PROPOSED-LTS SCHEME AND THE +COMSOL +Tetrahedrons +Memory +CPU Time (s) +COMSOL +1016 +3.7 GB +9239 +Proposed-GTS +1357 +25 MB +698 +Proposed-LTS +1357 +27 MB +172 +When the structure is discretized into finer meshes, the sim- +ulation time is compared in Fig. 9(d), it can be obtained that +the speedup ratio, which is defined by the ratio of the runtime +of the two schemes, is relatively stable with varied number of +tetrahedrons if the time step relationship is constant. However, +the saved time has seen an upward trend with the number +of tetrahedrons growing. As for the memory consumption, +when 418,546 tetrahedrons are generated in this context, the +memory consumption is about 3 GB. Therefore, the capability +of the proposed-LTS scheme applied for simulating multiscale +problem can be demonstrated. +-1 +-0.5 +0 +0.5 +1 +Real( +i) +-1 +-0.5 +0 +0.5 +1 +Imag( +i) +0.9 1 +-0.2 +0 +0.2 +(a) +-1 +-0.5 +0 +0.5 +1 +Real( +i) +-1 +-0.5 +0 +0.5 +1 +Imag( +i) +0.9 1 +-0.2 +0 +0.2 +(b) +Fig. 11. +The eigenvalues distribution of AT for the proposed-LTS scheme +with different time step relationship. (a) ∆t3 = 2∆t2 = 6∆t1, (b) ∆t3 = +5∆t2 = 100∆t1. +IV. CO-SIMULATION EXAMPLES +In this section, two representative PDN structures are pre- +sented to demonstrate the capability of the proposed method. +By comparing the results of electrical–thermal coupling sim- +ulation with the results of electrical simulation, the effect of +temperature variation on potential distribution is analyzed. +A. PDN structure with power planes +A simplified PDN structure is considered in this example +[36], as shown in Fig. 12, which can be recognized as a +combination of units. The overhead and cross-sectional view +of the grid unit are presented in Fig. 13 (a) and Fig. 13 (b), +respectively. The dimension of the cross section of conductor +grid is 10×1 mm2, the radius and height of micro bumps are +10 and 20 mm, respectively, with the figures for connecting +conductors between different grid layers are 7 and 30 mm. The +dimension of vias connecting the neighboring two conductor +layers is 5×5×0.6 mm3. For simplicity, the conductor layers, +and the vias, as well as the micro bumps are all made of copper +and the structure is covered by a silicon rectangular with the +dimension of 920 × 1040 × 75.4 mm3. +Similar to previous example, a periodic voltage pulse is +imposed on the upper surfaces of micro bumps, as shown in +Fig. 14 (a), while the connecting conductors at the bottom +layer are recognized as the ground. The related parameters of +the voltage pulses are shown in Table IV. For the electrical +analysis, only the units through which current flows are +considered, including micro bumps, power planes and con- +necting conductors. For the thermal analysis, all the structures +are discretized into tetrahedrons and then divided into three +regions, including power region, ground region and silicon +region. The time steps adopted in different regions have the + +X +30:0 +3150 +340 +31e:0 +318°0 +380°0 +385°0 +LGLG3100 +315'0 +3140 +318'0 +380'0 +385'0JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +9 +relationship ∆t3 = 2∆t2 = 4∆t1, with the minimum time +step ∆t1 = 2 × 10−11 s. The convection boundary condition +is applied on the six surfaces of silicon block to represent the +thermal transfer between the structure and the environment, +with h = 20 W/(m2 · K) and the ambient temperature +Ta = 300 K. +Unit I +Fig. 12. +Illustration of the PDN with power planes. +10 +10 +8 + + +(a) +10 +10 +8 +Unit: mm +Ground +pad +Power +pad +(b) +1 +0.6 +5 +20 +20 +14 +30 +Unit: mm +bump +Via +p3 +p6 +p7 +p4 +p5 +p1 +p2 +(c) +Fig. 13. +The overhead and side view of the PDN structure. (a) one type of +unit cell, (b) another type of unit cell for the conductor grid, (c) the size of +the structure and the location of seven observing probes. +There are 25,1889 and 545,915 tetrahedrons generated in +electrical and thermal simulation, respectively, which result in +2,282,092 unknowns during an interval. In order to compare +the temperature rise effect in different sections, seven observe +probes are placed to record the temperature variation, with +the coordinates of these probes listed in Table VI, which are +also marked in Fig. 13(c): (1) P1, P3 and P5 are on the vias +connecting conductor layers; (2) P2 and P4 are on the power +grids; (3) P6 and P7 are on the connecting conductors between +different grid layers. +The temporary temperature at probes on three different units +obtained from the proposed-LTS scheme is shown in Fig. +14 (b)–(d). Then, the voltage difference distribution between +considering thermal effect and without heat impact is con- +sidered in Fig. 15. Due to the simulation time constraints, the +temperature rises are not conspicuous, but it can be anticipated +to keep rising at subsequent time. The total number of time +steps for the finest group is 16,000, and the simulation costs +41,960 s in total and 4.1 GB memory, with 1,365 s spent on +pre-processing and 40,595 s on time stepping, respectively. +TABLE IV +THE SPATIAL COORDINATES OF SEVEN PROBES (UNITS: mm) +P1 +P2 +P3 +P4 +P5 +P6 +P7 +x +-9 +-9 +-9 +-9 +-9 +-9 +-9 +y +79.5 +77 +77 +72 +79.5 +79.5 +77 +z +-0.3 +0.5 +1.3 +2.1 +2.9 +-1.2 +-1.2 +0 +50 +100 +150 +200 +Time (ns) +10 +20 +30 +40 +50 +Voltage (V) +(a) +(b) +(c) +(d) +Fig. 14. +The imposed voltage pulse and transient temperature at probes on +different units. (a) the imposed pulse, (b) temperature on the first unit, (c) the +second unit, (d) the third unit. + +T at Ps +T +T at P6 +T at P- +302 +300 +0 +100 +200 +Time (ns)1 +300306 +T at Pi +T at P2 +T at P3 +304 +T at n306 +P4 +() +T at Ps +304 +_T at P7 +302 +300 +0 +100 +200 +Time (ns)300310 +T at Pi +T at P2 +308 +T at P3 +TT at Ps +302 +_T at P6 +T +T at P- +301 +300 +0 +100 +200 +Time (ns)300304 +4rTatPi +T at P2 +TatP3 +303 +T at nJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +10 + (mV) +2.9 +2.0 +1.0 +0.0 +-1.0 +-2.1 +Fig. 15. +Electric potential difference between considering and ignoring +thermal effects at t = 300 ns. +B. PDN structure with chips +To further demonstrate the capability of the proposed +scheme, a PDN structure with 4 simplified chips is tested +in this section, as shown in Fig. 16 (a). The PDN structure +consists of three conductor layers, with the size of each layer +identical to the former example. The size information of the +chips are shown in Fig. 16 (b). In view of the scale difference +of the structure, it can be divided vertically into three regions +including chip region, connecting conductor region and PDN +region. The structure is made of copper for simplicity. +Similar to the former example, for the electrical analysis, a +periodic voltage signal is imposed on the upper surface of the +chips, which is represented in Fig. 17 (a), while the bottom +of the conductor layer is recognized as the ground. For the +thermal analysis, time steps adopted in different groups have +the relationship ∆t3 = 2∆t2 = 6∆t1, with the minimum +time step ∆t1 = 8 × 10−11 s. The convection boundary +condition is applied on the outer surfaces of the structure +to represent the thermal transfer with the environment, with +h = 20 W/(m2 · K) and the ambient temperature Ta = 300 +K. +(a) +Unit: mm +170 +150 +30 +1 +3 +Region I +Region II +Region III +(b) +Fig. 16. +Illustration of the PDN structure with chips. (a) 3-D diagram, (b) +overhead overview and size information. +0 +2000 +4000 +6000 +Time (ns) +4 +6 +8 +10 +12 +Voltage (V) +(a) +0 +2000 +4000 +6000 +Time (ns) +300 +302 +304 +306 +308 +T (K) +T at p1 +T at p2 +T at p3 +T at p4 +T at p5 +T at p6 +T at p7 +(b) +0 +2000 +4000 +6000 +Time (ns) +300 +301 +302 +303 +304 +305 +T (K) +T at p1 +T at p2 +T at p3 +T at p4 +T at p5 +T at p6 +T at p7 +(c) +0 +2000 +4000 +6000 +Time (ns) +300 +300.2 +300.4 +300.6 +300.8 +301 +T (K) +T at p1 +T at p2 +T at p3 +T at p4 +T at p5 +T at p6 +T at p7 +(d) +Fig. 17. +The imposed voltage pulse and transient temperature at probes on +different layers. (a) the imposed pulse, (b) temperature on the third layer, (c) +the second layer, (d) the first layer. +There are 232,172 and 110,682 tetrahedrons generated in the +electrical and thermal simulation, respectively, which results +in 530,146 unknowns during an interval. The total number of +time steps for the finest group is 80,000, which costs 59,597 s +in total and 1.1 GB memory, with 72 s spent on pre-processing +and 59,525 s on time stepping. In order to compare the +temperature rises in different layers, seven observing probes +are placed on each layer to record the temperature variation, +with the coordinates of probes on the top layer identical to the +former example, as listed in Table VI. +(a) +(b) +(c) +(d) +Fig. 18. +Temperature profiles of plane y = 85 mm of the PDN structure at +four instances. (a) 2560 ns, (b) 5120 ns, (c) 6400 ns, (d) colormap. +(a) +(b) +Fig. 19. +Current density amplitude of plane y = 85 mm of the PDN structure +at 6400 ns. +The temporary temperature at the probes on different layers +obtained from the proposed-LTS scheme is shown in Fig. 17 +(b)–(d). Fig. 18 shows the temperature profiles at the plane + +303 +(D) +3:02 +301 +3001.8e+09 +1.5e+9 +Amplitude (A/m) +1.0e+9 +5.0e+8 +8.4e-04I.S +0.1- +0.0 +J'O +5'O +e.sJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 +11 +(a) +(b) +Fig. 20. +Temperature profile of the exterior surface of the PDN structure at +t = 6400 ns. +y = 85 mm at four instances, 2560, 5120, and 6400 ns. The +current density amplitude distribution of this plane at 6400 ns +is also presented in Fig. 19. It can be obtained that temperature +rise concentrates in small areas for greater current density and +spread away from those locations. +Then, the voltage difference distribution between consid- +ering thermal effect and without heat impact is considered +in Fig. 20. Despite the maximum number is quite small for +the limited time in this simulation, it is foreseeable that the +influence will keep accumulating over time, which indicates +the indispensability of taking thermal effect into consideration. +V. CONCLUSION +In this article, a transient electrical–thermal co-simulation +scheme has been developed based on the FEM and the DGTD +method. In the thermal simulation, an auxiliary variable is +introduced to degrade the parabolic equation to a hyperbolic +one, which can be solved by DGTD method directly. By +adopting different discretized volumes and independent grids +for the electrical solver and the thermal solver, redundant +computational overhead can be avoided. On the premise of +guaranteeing the stability, a flexible explicit LTS technique +based on interpolation method is incorporated into the solver +to improve the capability of solving multi-scale problem. With +the LTS technique, the sophisticated structure can be divided +into groups and different time steps are allowed in separate +groups. Two numerical examples are provided to demonstrate +the validity, flexibility, as well as the efficiency improvement +by the LTS technique in comparison with COMSOL. Further- +more, the electrical–thermal behavior of two multiscale PDN +systems is analyzed. Oriented to the increasingly miniaturized +and multiscale electronic devices, the proposed co-simulation +algorithm provides an accurate and effective alternative to +analyze their potential distribution and thermal effects in real +time. +REFERENCES +[1] Luke Hu, Chun-Hung Chen, and Steven Hsu, “Optimization and Char- +acterization of the Metal Cap Layout above Through-Silicon Via to +Improve Copper Dishing and Protrusion Effect for the Application of +3-D Integrated Circuits,” IEEE Trans. Compon. Packag. Manuf. Technol., +vol. 8, no. 12, pp. 2222-2226, Dec. 2018. +[2] J. U. Knickerbocker et al., “3-D silicon integration and silicon packaging +technology using silicon through-vias,” IEEE J. Solid-State Circuits, vol. +41, no. 8, pp. 1718–1725, Aug. 2006. +[3] M. Pedram and S. Nazarian, “Thermal modeling, analysis, and manage- +ment in VLSI circuits: Principles and methods,” Proc. IEEE, vol. 94, no. +8, pp. 1487–1501, Aug. 2006. +[4] H. Oh, G. S. May, and M. S. 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Technol., vol. 4, no. 2, pp. 323-331, Feb. 2014. + diff --git a/E9AyT4oBgHgl3EQfSfeg/content/tmp_files/load_file.txt b/E9AyT4oBgHgl3EQfSfeg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc641d980cf351a3945026ef1736704264692dc0 --- /dev/null +++ b/E9AyT4oBgHgl3EQfSfeg/content/tmp_files/load_file.txt @@ -0,0 +1,1004 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf,len=1003 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 8, AUGUST 2021 1 A Transient Electrical-Thermal Co-Simulation Method with LTS for Multiscale Structures Kai Zhu, Graduate Student Member, IEEE, and Shunchuan Yang, Senior Member, IEEE Abstract—In this article, an efficient transient electrical- thermal co-simulation method based on the finite element method (FEM) and the discontinuous Galerkin time-domain (DGTD) method is developed for electrical-thermal coupling analysis of multiscale structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Two Independent meshes are adopted by the steady electrical analysis and the transient thermal simulation to avoid redundant overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In order to enhance the feasibility and efficiency of solving multiscale and sophisticated structures, a local time stepping (LTS) technique coupled with an interpolation method is incorporated into the co-simulation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Several numerical examples from simple structures to complex multi- scale PDN structures are carried out to demonstrate the accuracy and efficiency of the proposed method by comparing with the COMSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Finally, two practical numerical examples are considered to confirm the performance of the proposed method for complex and multiscale structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Index Terms—Discontinuous Galerkin time-domain (DGTD) method, electrical-thermal co-simulation, finite element method (FEM), local time stepping (LTS), PDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' INTRODUCTION W ITH the development of semiconductor technique and packaging technology over the past few decades, the typical size of components in integrated circuits (ICs) has kept shrinking while the integration density sees an upward trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' It is well known that the Joule heating effect in ICs is a challenging issue which has attracted substantial attention from researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Since the electrical malfunction is frequently related to temperature increment and improper power distribution, a reasonable design both in circuit structure and thermal sinks becomes crucially important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Taking the through-silicon via (TSV) for instance, it plays a key role in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5D/3D ICs design [1], [2], for enabling the high-speed signal processing by smoothing paths for continuing the interconnect scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' However, currents flowing through TSVs leads to local temperature rise, then potentially influences the transistors switching states and induces circuit failures or performance deterioration [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Generally, electromigration, estimated by the Black’s equation, works as the primary cause of circuit failure [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' From them we can obtain that the mean time of This work was supported in part by the National Natural Science Foundation of China under Grant 62141405, 62101020, 62071125, in part by Defense In- dustrial Technology Development Program under Grant JCKY2019601C005, in part by Pre-Research Project under Grant J2019-VIII-0009-0170 and Fundamental Research Funds for the Central Universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (Corresponding author: Shunchuan Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=') K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Zhu is with the School of Electronic and Information Engineering, Beihang University, Beijing, 100083, China (e-mail: zhukai7@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Yang is with the Research Institute for Frontier Science and the School of Electronic and Information Engineering, Beihang University, Beijing, 100083, China (e-mail: scyang@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Manuscript received xxx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' revised xxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' failure abides by a negative exponential multiplier relationship [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Therefore, an efficient and accurate electrical-thermal co- simulation algorithm can be indispensable for ICs design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' There have been a great deal of numerical algorithms developed for solving thermal or electrical problems with respective advantages and deficiencies either in accuracy or efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The analytical method, such as the equivalent circuit model, can be adopted in some circumstances and efficiency improvements can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' However, it suffers from the lack of generality [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The finite volume method (FVM) can be adopted to analyze the heat transfer problems [9], which introduces the numerical flux to represent the information exchange between adjacent subdivision elements [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The widely used finite difference method (FDM) benefits from simplicity and efficiency [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' However, it is subject to staircase errors caused by structured meshes [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The finite method time domain (FETD) is practically restrained for solving a large matrix equation at each time step, which can be computationally intensive [14], [15] and an ill-conditioned matrix may be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The domain decomposition method (DDM) coupled by the finite element tearing and intercon- necting (FETI) can contribute to alleviating the computation burden [16], [17], which have been applied for coping with large scale problems [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The discontinuous galerkin time-domain (DGTD) method [19], [20] has attracted much attention and developed rapidly for inheriting the advantages of the FVTD method and the FETD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' It can be regarded as an element-level DDM [21], which implies an innate parallel characteristic and pro- vides the possibility of utilizing adaptive orders or types of basis functions in different elements [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In this article, we introduce an electrical-thermal co- simulation scheme, which integrates the electrical analysis and thermal simulation through an iteration procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The electri- cal analysis is based on the finite element method (FEM) for its capacity of modeling sophisticated structures [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The thermal simulation can be divided into two phases and is based on the DGTD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' It is noteworthy that the thermal conduction equation is a parabolic partial differential equation, which is difficult to be solved directly by the traditional DGTD method [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In order to address this issue, an auxiliary equation need to be introduced to degrade the parabolic partial differential equation to a hyperbolic partial differential equation, which can be solved directly by the DGTD method [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The DGTD method generally leads to a series of compact linear systems, the dimension of which is equal to the degrees of freedom within the corresponding element, and those matrix equations are required to be solved at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The merit arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='00088v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='NA] 31 Dec 2022 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 8, AUGUST 2021 2 of this method over the FETD method lies in that the matrices dimension is quite small, which indicates the inverses of ele- mental matrices can be calculated readily and stored before the time iteration begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The computational complexity mainly depends upon the number of elements, the order of chosen local basis functions, and the simulation time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' If explicit time discretization scheme is adopted, the time step is limited rigorously by the minimal size of the discretized elements according to the Courant–Friedrichs–Lewy (CFL) stability condition [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' For multiscale structures, the disparities of the mesh size in different regions can be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' If the global time step (GTS) scheme is adopted, the time step is restricted to the global smallest mesh size to guarantee the stability, thereby leading to substantial computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Some implicit-explicit methods have been developed to alleviate this issue, where an implicit time-marching scheme is used in regions with fine mesh, while the explicit time-marching scheme is adopted in coarse mesh regions [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' However, these methods dramatically increase the computational time and memory consumption for large scale problems and the highly disparate mesh element sizes may cause ill-conditioned problem when the implicit time-marching scheme is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Therefore, this method cannot cope with the problems effi- ciently encountered in analyzing multiscale structures [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' To tackle the aforementioned challenge, a local time step- ping (LTS) scheme is integrated into the electrical-thermal co- simulation method, with which the computational efficiency can be significantly improved and the capability of simulating multiscale and locally refined structures can be facilitated [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The structure can be divided into several groups according to different mesh sizes, then elements in each group advance in time with local time steps [29], [30], which can reduce simulation time while maintaining accurate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' It is worth noting that the time step in each group is required to be selected carefully in order to guarantee the global stability and accuracy requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Then, a rigorous analysis of the stability of the LTS scheme is introduced based on the Von- Neuman method [31], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In section II, the detailed formulation for electrical and thermal problem is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Then, the co-simulation procedure as well as the concept and implementation of the LTS technique is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In section III, several simple examples are presented to demonstrate the accuracy and efficiency of the proposed scheme, as well as the efficiency enhanced by the LTS technique compared with the GTS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In section IV, the proposed scheme is applied to some practical structures to verify the capability of co-simulation for some practical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Finally, some conclusions are drawn in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' THEORIES AND FORMULATIONS In this section, the detailed formulations for current conti- nuity equation and heat conduction equation are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Then the coupling simulation flow algorithm is elaborated, including the application of independent meshes for electrical and thermal simulations and the LTS technique developed for coping with multiscale structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Formulations of Electrical and Thermal Analysis The current continuity equation is considered in the elec- trostatic analysis, which can be written as ∇ · (σ∇φ + ε∇∂φ ∂t ) = 0, (1) where σ and ε are the electrical conductivity and the per- mittivity of the medium, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The Dirichlet boundary condition subjected to the governing equation can be expressed as φ = φ0, (2) The impedance boundary condition adopted for modeling lossy conductors can be imposed on the surface of the con- ductor with the form ˆn · σ∇φ = φ RS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (3) where ˆn is the unit normal vector pointing outward from the boundary of the computational domain, R denotes the surface impedance of a conductor, S represents the area of the boundary surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In our implementation, the potential distribution is considered constant during an interval, and the steady-state solution is analyzed through the FEM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' As for the thermal simulation, the temperature evolution of a spatial point is governed by the transient heat conduction equation, which can be written as ρc∂T ∂t = ∇ · (k∇T) + Q, (4) where ρ represents the density of the medium, c denotes the heat capacity, k is the thermal conductivity, Q represents the heat source, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' To solve (4), the corresponding boundary conditions include the Dirichlet boundary condition T = T0, (5) and the convective boundary condition ˆn · (k∇T) = −h (T − Ta) , (6) where h is the convective heat transfer coefficient, Ta denotes the ambient temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Since the traditional DGTD method is incapable to solve the parabolic differential equation di- rectly, an auxiliary vector variable q is introduced to transform (4) to a hyperbolic differential equation [20], which can be rewritten as q = −k∇T, (7) ρc∂T ∂t = −∇ · q + Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (8) By implementing the Galerkin’s spatial testing procedure to (7) and (8) in the ith subdomain, the formulation can be obtained as � Vi Nk � qx + k ∂T ∂x � dV = 0, (9) � Vi Nk � ρc∂T ∂t +∇ · q − Q � dV = 0, (10) JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 8, AUGUST 2021 3 where Nk denotes the kth test basis function, qx represents the component of q in the x-direction, and the components in other two directions can be obtained in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' With the application of spatial integration and Gauss’s theorem, (9) and (10) can be rewritten as � Vi NkqxdV = k � Vi T ∂Nk ∂x dV − k 4 � f=1 � ∂Vi nxT ∗NkdS, (11) � Vi Nkρc∂T ∂t dV = � Vi (q · ∇Nk + NkQ) dV − 4 � f=1 � ∂Vi Nk ˆn · q∗dS, (12) where T ∗ and q∗ are the numerical fluxes which represent the information exchange between adjacent elements, which can be expressed as the linear combination of variables in adjacent elements like (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' ˆn · q∗ = D0 �ˆn · qi + ˆn · qj� + D1 �ˆn · qi − ˆn · qj� + D2 � T i − T j� , T ∗ = D3 � T i + T j� + D4 � T i − T j� + D5 �ˆn · qi − ˆn · qj� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (13) where T i and T j denote the temperature in self element and external neighboring element, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The same notations are adopted for q, and Di (i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 5) are constants, which depend on the chosen numerical flux form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In our imple- mentation, the upwind flux is adopted for better convergence properties [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Therefore, the coefficients can be defined as D0 = D3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5, D2 = −4 and D1 = D4 = D5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In addi- tion, for the numerical flux on convective boundary surfaces, the coefficients are revised as D3 = D4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5, D2 = −h and D0 = D1 = D5 = 0 correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' To establish the semi-discrete matrix system, (13) is applied to (11) and (12), with T and q approximated by nodal basis functions, which leads to Miqi x = kSi xTi − 4 � f=1 � k(D3 + D4)Gi xTi +k(D3 − D4)Gj xTj� , (14) ρcMi ∂Ti ∂t = Si xqi x + Si yqi y + Si zqi z + Qi − 4� f=1 � D0Gi xqi x + D0Gi yqi y + D0Gi zqi z + D0Gj xqj x +D0Gj yqj y + D0Gj zqj z + D2CiTi − D2CjTj� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (15) where Mi denotes the local mass matrix and Si x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Si y and Si z are the local stiffness matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' the detailed expression of matrices and vectors in (14) and (15) can be written as � Mi� kl = � Vi N i kN i l dV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (16) � Si x � kl = � Vi ∂N i k ∂x N i l dV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (17) � Qi� k = � Vi N i kQdV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (18) � Gi x � kl = � ∂Vi nxN i kN i l dS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (19) � Gj x � kl = � ∂Vi nxN i kN j l dS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (20) � Ci� kl = � ∂Vi N i kN i l dS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (21) � Cj� kl = � ∂Vi N i kN j l dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (22) where nx denotes the component in the x-direction of the outward normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The detailed forms of other matrices including Si y, Si z, Gi y, Gi z, Gj y and Gj z can be obtained simi- larly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Since the obtained matrix equation is still semi-discrete, the derivative in the temporal dimension is required to be dis- cretized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The backward difference is unconditionally stable but yields a global matrix operation thereby losing the advantage of the DGTD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Therefore, the forward difference is adopted in our implementation, where the time derivative term can be approximated by ∂T ∂t = T (t + ∆t) − T (t) ∆t + O (∆t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (23) to obtain the finial matrix equation in the thermal simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The forward difference is conditionally stable, with the convergence dependent on selected time step and related system coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In order to guarantee the stability, the Courant–Friedrichs–Lewy (CFL) condition is required to be satisfied, hence finding a valid approach for estimating the time step bound is of vital importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In this implementation, the Von-Neuman stability analysis is introduced to validate the stability of the chosen time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Firstly, a column vector is constructed to include all the unknowns of the system equation at a specific time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' For instance, unknowns related to heat flux can be written as Uq = �� q1 x, q1 y, q1 z � , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' , � qN x , qN y , qN z ��T , where qi x, qi y and qi z denote the components of the heat flux of the ith element in different directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' N denotes the number of split elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' If the number of basis func- tions is M, qi x can be represented as � qi,1 x , qi,2 x , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' , qi,M x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Similarly, the column vector for T can be written as UT = � T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' , TN�T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' According to (14), Uq at tn can be obtained by Uq (tn) = AqUT (tn) , (24) By substituting (24) into (15), the time-marching relation- ship between U (tn+1) and U (tn) can be rewritten in a compact matrix form UT (tn+1) = AT UT (tn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (25) where the dimension of Aq and AT are 3MN × 3MN and MN × MN, respectively, which assemble the information of all element matrices and corresponding numerical flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The concrete form of A can vary with the discretizing parameters JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 8, AUGUST 2021 4 adopted in the DGTD method, including the chosen time step (∆t), the order and form of the basis function (M), the form of numerical flux (q∗, T ∗), and the properties of material and meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The stability of the system can be analyzed by computing the MN eigenvalues of AT (λi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' , MN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' If all the eigenvalues are located inside the unit circle, it can be concluded that it is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The Electrical-Thermal Co-Simulation Procedure The flowchart of the procedure of the electrical-thermal co- simulation is represented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The initialization includes inputting the material parameters required in the electrical and thermal simulation, such as the electrical con- ductivity for electrical analysis, material density, heat capacity and thermal conductivity for thermal simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In addition, the time step, the simulation duration, and the relationship coefficients between material parameters and temperature need to be taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Over each iteration, the elec- trical problem is solved through the FEM solver with the voltage and current density distribution obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Then, the dissipated power calculated in each element is considered as the source for subsequent thermal simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The dissipated power during a time step period can be written as Q = σ| ⃗E|2 = σ|∇φ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (26) In this article, the influence of temperature on electrical conductivity is considered, and its value can be calculated by the fitted interpolation function, the fourth-order form can be written as σ(T) = 4 � n=0 AnT n T0 ≤ T ≤ T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (27) where An (n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 4) are the fitting coefficients of mate- rial properties, [T0, T1] denotes the interpolation interval, the related coefficients of the materials used are shown in Table I [34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' It is worth noting that the co-simulation method can also be used when other parameters vary with temperature without sacrificing its generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' TABLE I TEMPERATURE INTERPOLATION COEFFICIENTS OF TWO MATERIALS Cu Poly-Si A0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='91 × 108 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='45 × 104 A1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='56 × 106 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='08 × 102 A2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='70 × 103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='01 × 10−1 A3 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='93 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='17 × 10−5 A4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='56 × 103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='07 × 10−7 After the temperature distribution at a time step is obtained, the material parameters are updated within each split element based on this distribution and the interpolation function con- necting material properties and temperature if the simulation is not finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Otherwise, the time iteration phase completes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Although some structures are electrically insulated and can be ignored in electrical analysis, they are still required to be considered for the heat transfer effect to ensure that the tem- perature distribution analysis in the overall region is accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Moreover, fine meshes for the electrical analysis may be used to accurately model complex structures, while relatively coarse meshes can be adopted in the thermal analysis for the slow pace of change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' There can also be large gaps of mesh densities in different parts of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' If the structure is meshed in a unified density for electrical analysis and thermal simulation, the time step can be restricted to extremely small values to guaranteed stability, thereby leading to a great amount of computational overhead, even the time consumption can be unacceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' To address this issue, two independent meshes for the electrical and thermal simulation are used to improve the flexibility of the algorithm and to avoid the redundant degrees of freedom (DoFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' A mapping relationship between meshes is built and a interpolation method is applied for imposing heat source and updating the material coefficients in the time-marching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' With the growing of unknowns, the construction of mapping relationship bridging the discontinuities between electrical and thermal meshes can be increasingly time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In order to alleviate this problem, two tree structures can be constructed in advance and split elements in electrical and thermal analysis are stored in leaf nodes to accelerate the traversal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In this implement, two full octrees are adopted for simplicity, where within each level every node has 8 children, and the computational domain is divided into 512 blocks if the depth is set to 4, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The jth node of the ith level is denoted as Ci,j, where a node element includes the location information (tetrahedron indexes, space boundary) and pointers to each of its 8 children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Child Child Child 0,0 C 1,0 C 1,4 C 1,7 C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 2,0 C 3,0 C 3,4 C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 3,7 C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Key Key Key .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (a) 1,0 C 1,1 C 1,2 C 1,3 C 1,4 C 1,5 C 2,0 C 2,1 C 2,2 C 2,3 C 2,4 C 2,5 C 3,0 C Level 1 Level 2 Level 3 (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Illustration of the construction of octree (a) Topology of the tree, (b) Geometric space representation for different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 8, AUGUST 2021 5 Algorithm 1 Transient electrical-thermal co-simulation Input: Parameters of material properties Control factors of time stepping Output: Distribution of temperature and electric potential 1: Initiate t = 0 and σ = σ0 2: repeat 3: Analyze current density through electrical analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 4: Calculate produced Joule heat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 5: Simulate temperature through thermal analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 6: Update σ, set t = t + ∆t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 7: until t >= tmax 8: Output T and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The Explicit Local Time Stepping For a multiscale structure, elements can be partitioned into groups and different time steps are allowed in separate groups based on the explicit local time stepping (LTS) scheme, with time step in each group constrained by the local mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Therefore, the total number of equation calculation needed for advancing the numerical solution from tn to tn + ∆t will decrease in comparison with using the global smallest time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' For simplicity, assuming the split elements are divided into four groups, the average sizes in each group are h, h/2, h/4 and h/8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Since the time step in each group is required to satisfy the stability condition, which can be chosen as ∆t1 = 2∆t2 = 4∆t3 = 8∆t4, with ∆ti (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' , 4) denote the time step for the ith group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The implementation of time stepping is shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The recursive procedure can be given as 1) Elements in each group advance one step with local time steps ∆ti (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' , 4), the sequence of the advance- ment is Group #1 (averagely coarsest mesh), followed by Group #2, Group #3, and finally Group #4, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 2) Elements in Group#4 (averagely finest mesh) advance one step with local time step ∆t4, then elements in Group#4 and Group#3 are at the same time level, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 3) Elements in Group#3 and Group#4 advance one step in sequence with local time steps ∆t3 and ∆t4, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 4) Elements in Group#4 advance one step with local time step ∆t4, then the elements in Group#2, Group#3, and Group#4 are at the same time level, as outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 5) Elements in Group#2, Group#3, Group#4 advance one steps with local time steps ∆ti (i = 2, 3, 4), as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 2(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 6) The processes in 2–4 repeat until all elements reach the final interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' While analyzing an element in Group#i with a neighboring element located in Group#j (j < i), the temperature of the latter may be unknown at the current time step, which leads to a difficulty in constructing the numerical flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Taking Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 2(g) for instance, if the element in Group#4 has an adjacent element located in group 1-3, the temperature of the adjacent Group 1 t size Group 2 t size Group 3 Group 4 t size t size △t1 (a) (b) (c) (d) t size t size t size t size (e) (f) (g) (h) △t1/2 △t1/4 △t1/8 △t1/2 3△t1/8 △t1 △t1/2 △t1 3△t1/4 △t1 △t1/2 △t1/4 △t1 5△t1/8 3△t1/4 △t1 7△t1/8 △t1 △t1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Example of the LTS stepping process for four groups with time steps ∆t1 = 2∆t2 = 4∆t3 = 8∆t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' element is unknown at 7∆t1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' However, after the process (1) presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 2(a) finished, the temperature of elements in Group#1 at ∆t1 has been obtained, the similar situation is also for elements in Group#2 and Group#3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Therefore, a linear interpolation strategy can be adopted to estimate the temperature of adjacent elements in different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' While considering the element in Group#i at n∆ti, the temperature of the adjacent element in Group#j at this time can be interpolated by T j app = � T j (nratio) , j ≥ i (1 − C)T j (nlow) + CT j (nlow + 1) , j < i (28) where the parameters are given by nratio = n∆ti ∆tj , (29) nlow = �n∆ti ∆tj � , (30) C = [n% (∆tj/∆ti)] / (∆tj/∆ti) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (31) and ⌊a⌋ denotes the maximum integer less than a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In the thermal analysis, the interpolation strategy for adjacent ele- ments in different groups is also needed to handle the auxiliary variable q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' There are two types of interpolation methods for q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The first type is similar to that for T in (28), and nearly no extra storage is required except for some assigned for symbol marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The other type consists of two substeps, firstly obtaining the interpolation temperature for adjacent elements by (28), then (14) is solved to get the approximated qx, and the JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 8, AUGUST 2021 6 same process for qy and qz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The second interpolation strategy is applied in our implementation, with a higher accuracy obtained, notwithstanding extra computational resources at an acceptable level are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' For the stability analysis of the LTS technique, the similar Von-Neuman method is also introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' To better fulfill the matrix-filling process, the column vector including the un- knowns are divided according to their groups owning different time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Taking four groups in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 2 for example, interme- diate unknowns of elements in different group at a time are stored in four arrays noted as Ui q and Ui T (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' , 4), then the time march from tn to tn + ∆ti in the ith group can be written in a compact form as Uq (tn) = AqiUT (tn) , (32) UT (tn + ∆ti) = AT iUT (t0) + UT (tn) , (33) where t0 denotes the initial simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' If there are Ni elements in Group#i, the dimensions of Aqi and AT i are 3MNi × 3MN and MNi × MN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' After the substeps in each group according to recursive order shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 2 are finished, the time-marching from tn to tn + ∆t1 in each group can be filled into a compact form UT (tn + ∆t1) = AiUT (t0) , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (34) Based on the mapping relationship of the local node indexes to the global node indexes, Ai can be sorted into the final matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Given that all the eigenvalues of A are located inside the unit circle, the proposed LTS method can be regarded as stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' VERIFICATION AND DEMONSTRATION OF THE STABILITY, ACCURACY AND EFFICIENCY In this section, several basic numerical examples are pro- vided to verify the efficiency and accuracy of the proposed electrical-thermal co-simulation method by comparing with the COMSOL software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The accuracy and stability of the LTS method are also verified, then the speedup performance is investigated for different time step ratios and mesh densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' All computations in this section are performed on the computer with Intel i9-10900 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='8 GHz CPU and 32 GB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Accuracy and Stability Verification A copper block with the dimension of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='6×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='6 mm3 is tested to demonstrate the proposed co-simulation scheme, and the thermal property of the copper is shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The convection boundary condition is applied on all six surfaces of the structure with h = 1000 W/(m2K), and the ambient temperature is set as Ta = 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Two Gaussian pulses are imposed on one face of the structure sequentially, with the form of VGauss(t) = V0e−(t−t0)2/τ 2 (35) where V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='02 V, τ 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='1, t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' It is noteworthy that V0 represents the peak voltage of the pulse, t0 is the time when the pulse reaches its peak, τ controls the width of the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The temporary temperature at P1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='3) (mm) is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 3(a), it can be found that results obtained from the proposed method show excellent agreement with those from the COMSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' To have a better clarification, the relative error of the observation point at each time point is outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 3(b), which is defined as RE = (Tp − Tc) /Tc (36) where Tp and Tc denote the results obtained from the proposed method and the COMSOL, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' TABLE II THERMAL PROPERTIES OF DIFFERENT MATERIALS Thermal conductivity (W · m−1 · K−1) Heat capacity (J · kg−1 · K−1) Density (kg · m−3) Copper 400 385 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='7 × 103 Nickel 91 440 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='9 × 103 Al2O3 10 750 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='9 × 103 Silicon 130 700 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='3 × 103 0 2 4 6 Time (s) 300 320 340 360 380 400 Temperature (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='05 Voltage (V) COMSOL Proposed V (a) 0 2 4 6 Time (s) 6 3 0 3 6 8 Relative Error 10-6 Error (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Simulation of the temperature with voltage pulses imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (a) Temperature at P1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='3) (mm) obtained from the proposed scheme and the COMSOL, (b) The relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 1 Real( i) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 1 Imag( i) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='9 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The distribution of the eigenvalues of AT with ∆t = 2 × 10−5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' To test the stability, eigenvalues of AT in this context are analyzed and presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Since all the eigenvalues are located inside the unit circle, the stability in this circumstance can be concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 5, the temperature distribution of the plane z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='3 (mm) of the structure at t = 4 (s) obtained from these two JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 8, AUGUST 2021 7 methods are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Then, the electric potential distribution of the same plane at t = 4 (s) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' For a fair comparison, the structure is split into a similar number of tetrahedrons, and the same time step is adopted for the proposed scheme and the COMSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The CPU time and memory consumption are only about 265 s and 15 MB for the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' By comparison, the figures are 2014 s and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='1 GB for the COMSOL, which implies an efficiency improvement of the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Temperature (K) 389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='0 (a) (b) 389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='1 389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='3 389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='4 y (mm) x (mm) 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 1 y (mm) x (mm) 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Temperature distribution of the plane z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='03 mm at 4 s obtained from (a) the COMSOL, (b) the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (mV) 10 0 (a) (b) 2 4 6 8 y (mm) x (mm) 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 1 y (mm) x (mm) 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 1 12 14 \uf066 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Electric potential distribution of the plane z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='03 mm at 4 s obtained from (a) the COMSOL, (b) The proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 0 2 4 6 Time (s) 300 320 340 360 380 Temperature (K) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='03 Voltage (V) COMSOL Proposed V (a) 0 2 4 6 Time (s) 8 6 4 2 0 2 4 Relative Error 10-6 Error (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Simulation of temperature with a step voltage imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (a) Temperature at P1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='3) (mm) obtained from the proposed scheme and the COMSOL, (b) The relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In addition, a ladder signal is imposed for testing, the results and relative errors are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Similar to the previous situation, results obtained from the proposed scheme agree well with the COMSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Efficiency Improvement by the LTS Method To further demonstrate the stability and efficiency improve- ment of the LTS technique, the structure of two conductors composed of copper and nickel covered by a silicon box is considered, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The thermal properties of the media are listed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In this implementation, a volt- age pulse varying over time is imposed on one face of the cop- per, which can be written as V1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='04 + VGauss sin (300πt) (V), where VGauss is described in (35) with V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='03, τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='01, and t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' For the electrical analysis, only the copper is considered and discretized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' For the thermal analysis, the whole structure is discretized into tetrahedrons and then divided into three groups according to material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The convection boundary condition is applied on six surfaces of the silicon block to represent the thermal transfer between the object and the environment, with h = 1000 W/(m2 · K) and the ambient temperature Ta = 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Group II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='58 Group III 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='12 Nickle Copper Silicon Unit: mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='34 Electrical mesh Heat mesh Mapping 1t \uf044 2t \uf044 3t \uf044 Group I Port Sink Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Geometry of the conductors and silicon box, and illustration of the independent meshes adopted in electrical and thermal analysis in different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='02 Time (s) 300 320 340 360 380 400 Temperature (K) 2 4 6 8 10 Voltage (mV) COMSOL-p1 COMSOL-p2 COMSOL-p3 Proposed-p1 Proposed-p2 Proposed-p3 V (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='02 Time (s) 10 8 6 4 2 0 2 Relative Error 10-4 Error-p1 Error-p2 Error-p3 (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='02 Time (s) 5 0 5 10 Relative Error 10-4 Error-p1 Error-p2 Error-p3 (c) 0 1 2 3 4 5 Number of tetrahedron 105 0 1 2 3 4 Simulation Time (s) 104 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 6 Ratio Proposed-GTS Proposed-LTS Ratio (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Simulation of the temperature with the voltage pulses imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (a) Temperature at P1 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='06, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='03) (mm), P2 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='52, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='03) (mm) and P3 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='29, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='03) (mm) obtained from the proposed-LTS scheme and COMSOL, (b) The relative error between the proposed-GTS scheme and the COMSOL, (c) The relative error between the proposed-LTS scheme and the COMSOL, (d) The comparison of the CPU time for the proposed-GTS scheme and the proposed-LTS scheme with the structures discretized into different number of tetrahedrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 0:06+00 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 00+00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content="50'06+000:5 06+000." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='Q88 380°2 LGbGLLG6 (K)LGGLLG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content="8 380' 380'5 380'3 380°4 (K) 38020." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content="0 o's 04 oe 8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='0 b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='Q88 380°2 LGbGLLG6 (K)JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 8, AUGUST 2021 8 Since the temperature changes more dramatically over time for the Joule heating caused by conducting current in the copper, it is discretized into finer elements and recognized as Group#3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The larger volume of the outer layer composed of silicon is desired to be discretized into coarser elements to save computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Three observing probes are placed at P1 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='06, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='03) (mm), P2 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='52, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='03) (mm), and P3 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='29, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='03) (mm) to record the temperature variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 9(a), the temperature of probes obtained from the proposed-LTS scheme and the COMSOL are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' An excellent agreement can be observed as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The CPU time and memory consumption of the COMSOL, and the proposed scheme with global time stepping, as well as the proposed scheme with local time stepping are compared in Table III, which indicates the efficiency improvement of the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' For the proposed-LTS scheme, time steps adopted in different groups have the relationship ∆t3 = 2∆t2 = 6∆t1, with the minimum time step ∆t1 = 5 × 10−8 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' For the COMSOL and the proposed-GTS scheme, the time step is selected as ∆t = 5 × 10−8 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The relative error of the proposed-GTS scheme and the proposed-LTS scheme in comparison with the COMSOL are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 9(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 9(c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' It can be observed that about four times speed up is achieved with a reasonable accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In this occasion, the eigenvalues of AT are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 11(a), since all the eigenvalues are located inside the unit circle, the stability can be guaranteed theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' When time steps are chosen as ∆t3 = 5∆t2 = 100∆t1 with ∆t1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 × 10−8 s, the eigenvalues distribution is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 11(b), and the stability can still be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Temperature (K) 370 372 374 376 378 380 382 (a) y (mm) x (mm) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 (b) y (mm) x (mm) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Temperature distribution of the plane z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='03 mm at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='01 s obtained from (a) the COMSOL, (b) the proposed-LTS scheme with ∆t3 = 2∆t2 = 6∆t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' TABLE III COMPARISON OF THE COMPUTATIONAL COST BETWEEN THE PROPOSED-GTS SCHEME, THE PROPOSED-LTS SCHEME AND THE COMSOL Tetrahedrons Memory CPU Time (s) COMSOL 1016 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='7 GB 9239 Proposed-GTS 1357 25 MB 698 Proposed-LTS 1357 27 MB 172 When the structure is discretized into finer meshes, the sim- ulation time is compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 9(d), it can be obtained that the speedup ratio, which is defined by the ratio of the runtime of the two schemes, is relatively stable with varied number of tetrahedrons if the time step relationship is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' However, the saved time has seen an upward trend with the number of tetrahedrons growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' As for the memory consumption, when 418,546 tetrahedrons are generated in this context, the memory consumption is about 3 GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Therefore, the capability of the proposed-LTS scheme applied for simulating multiscale problem can be demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 1 Real( i) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 1 Imag( i) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='9 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 (a) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 1 Real( i) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 1 Imag( i) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='9 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The eigenvalues distribution of AT for the proposed-LTS scheme with different time step relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (a) ∆t3 = 2∆t2 = 6∆t1, (b) ∆t3 = 5∆t2 = 100∆t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' CO-SIMULATION EXAMPLES In this section, two representative PDN structures are pre- sented to demonstrate the capability of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' By comparing the results of electrical–thermal coupling sim- ulation with the results of electrical simulation, the effect of temperature variation on potential distribution is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' PDN structure with power planes A simplified PDN structure is considered in this example [36], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 12, which can be recognized as a combination of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The overhead and cross-sectional view of the grid unit are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 13 (a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 13 (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The dimension of the cross section of conductor grid is 10×1 mm2, the radius and height of micro bumps are 10 and 20 mm, respectively, with the figures for connecting conductors between different grid layers are 7 and 30 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The dimension of vias connecting the neighboring two conductor layers is 5×5×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='6 mm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' For simplicity, the conductor layers, and the vias, as well as the micro bumps are all made of copper and the structure is covered by a silicon rectangular with the dimension of 920 × 1040 × 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='4 mm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Similar to previous example, a periodic voltage pulse is imposed on the upper surfaces of micro bumps, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14 (a), while the connecting conductors at the bottom layer are recognized as the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The related parameters of the voltage pulses are shown in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' For the electrical analysis, only the units through which current flows are considered, including micro bumps, power planes and con- necting conductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' For the thermal analysis, all the structures are discretized into tetrahedrons and then divided into three regions, including power region, ground region and silicon region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=" The time steps adopted in different regions have the X 30:0 3150 340 31e:0 318°0 380°0 385°0 LGLG3100 315'0 3140 318'0 380'0 385'0JOURNAL OF LATEX CLASS FILES, VOL." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 8, AUGUST 2021 9 relationship ∆t3 = 2∆t2 = 4∆t1, with the minimum time step ∆t1 = 2 × 10−11 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The convection boundary condition is applied on the six surfaces of silicon block to represent the thermal transfer between the structure and the environment, with h = 20 W/(m2 · K) and the ambient temperature Ta = 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Unit I Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Illustration of the PDN with power planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 10 10 8 (a) 10 10 8 Unit: mm Ground pad Power pad (b) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='6 5 20 20 14 30 Unit: mm bump Via p3 p6 p7 p4 p5 p1 p2 (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The overhead and side view of the PDN structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (a) one type of unit cell, (b) another type of unit cell for the conductor grid, (c) the size of the structure and the location of seven observing probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' There are 25,1889 and 545,915 tetrahedrons generated in electrical and thermal simulation, respectively, which result in 2,282,092 unknowns during an interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In order to compare the temperature rise effect in different sections, seven observe probes are placed to record the temperature variation, with the coordinates of these probes listed in Table VI, which are also marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 13(c): (1) P1, P3 and P5 are on the vias connecting conductor layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (2) P2 and P4 are on the power grids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (3) P6 and P7 are on the connecting conductors between different grid layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The temporary temperature at probes on three different units obtained from the proposed-LTS scheme is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14 (b)–(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Then, the voltage difference distribution between considering thermal effect and without heat impact is con- sidered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Due to the simulation time constraints, the temperature rises are not conspicuous, but it can be anticipated to keep rising at subsequent time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The total number of time steps for the finest group is 16,000, and the simulation costs 41,960 s in total and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='1 GB memory, with 1,365 s spent on pre-processing and 40,595 s on time stepping, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' TABLE IV THE SPATIAL COORDINATES OF SEVEN PROBES (UNITS: mm) P1 P2 P3 P4 P5 P6 P7 x 9 9 9 9 9 9 9 y 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 77 77 72 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 77 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 0 50 100 150 200 Time (ns) 10 20 30 40 50 Voltage (V) (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The imposed voltage pulse and transient temperature at probes on different units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (a) the imposed pulse, (b) temperature on the first unit, (c) the second unit, (d) the third unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' T at Ps T T at P6 T at P- 302 300 0 100 200 Time (ns)1 300306 T at Pi T at P2 T at P3 304 T at n306 P4 () T at Ps 304 _T at P7 302 300 0 100 200 Time (ns)300310 T at Pi T at P2 308 T at P3 TT at Ps 302 _T at P6 T T at P- 301 300 0 100 200 Time (ns)300304 4rTatPi T at P2 TatP3 303 T at nJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 8, AUGUST 2021 10 \uf066 (mV) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Electric potential difference between considering and ignoring thermal effects at t = 300 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' PDN structure with chips To further demonstrate the capability of the proposed scheme, a PDN structure with 4 simplified chips is tested in this section, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 16 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The PDN structure consists of three conductor layers, with the size of each layer identical to the former example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The size information of the chips are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 16 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In view of the scale difference of the structure, it can be divided vertically into three regions including chip region, connecting conductor region and PDN region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The structure is made of copper for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Similar to the former example, for the electrical analysis, a periodic voltage signal is imposed on the upper surface of the chips, which is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 17 (a), while the bottom of the conductor layer is recognized as the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' For the thermal analysis, time steps adopted in different groups have the relationship ∆t3 = 2∆t2 = 6∆t1, with the minimum time step ∆t1 = 8 × 10−11 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The convection boundary condition is applied on the outer surfaces of the structure to represent the thermal transfer with the environment, with h = 20 W/(m2 · K) and the ambient temperature Ta = 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (a) Unit: mm 170 150 30 1 3 Region I Region II Region III (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Illustration of the PDN structure with chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (a) 3-D diagram, (b) overhead overview and size information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 0 2000 4000 6000 Time (ns) 4 6 8 10 12 Voltage (V) (a) 0 2000 4000 6000 Time (ns) 300 302 304 306 308 T (K) T at p1 T at p2 T at p3 T at p4 T at p5 T at p6 T at p7 (b) 0 2000 4000 6000 Time (ns) 300 301 302 303 304 305 T (K) T at p1 T at p2 T at p3 T at p4 T at p5 T at p6 T at p7 (c) 0 2000 4000 6000 Time (ns) 300 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='2 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='4 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='6 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='8 301 T (K) T at p1 T at p2 T at p3 T at p4 T at p5 T at p6 T at p7 (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The imposed voltage pulse and transient temperature at probes on different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (a) the imposed pulse, (b) temperature on the third layer, (c) the second layer, (d) the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' There are 232,172 and 110,682 tetrahedrons generated in the electrical and thermal simulation, respectively, which results in 530,146 unknowns during an interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The total number of time steps for the finest group is 80,000, which costs 59,597 s in total and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='1 GB memory, with 72 s spent on pre-processing and 59,525 s on time stepping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In order to compare the temperature rises in different layers, seven observing probes are placed on each layer to record the temperature variation, with the coordinates of probes on the top layer identical to the former example, as listed in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Temperature profiles of plane y = 85 mm of the PDN structure at four instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (a) 2560 ns, (b) 5120 ns, (c) 6400 ns, (d) colormap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Current density amplitude of plane y = 85 mm of the PDN structure at 6400 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The temporary temperature at the probes on different layers obtained from the proposed-LTS scheme is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 17 (b)–(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 18 shows the temperature profiles at the plane 303 (D) 3:02 301 3001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='8e+09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='5e+9 Amplitude (A/m) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='0e+9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='0e+8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='4e-04I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='1- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content="0 J'O 5'O e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content='sJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 8, AUGUST 2021 11 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Temperature profile of the exterior surface of the PDN structure at t = 6400 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' y = 85 mm at four instances, 2560, 5120, and 6400 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' The current density amplitude distribution of this plane at 6400 ns is also presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' It can be obtained that temperature rise concentrates in small areas for greater current density and spread away from those locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Then, the voltage difference distribution between consid- ering thermal effect and without heat impact is considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Despite the maximum number is quite small for the limited time in this simulation, it is foreseeable that the influence will keep accumulating over time, which indicates the indispensability of taking thermal effect into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' CONCLUSION In this article, a transient electrical–thermal co-simulation scheme has been developed based on the FEM and the DGTD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' In the thermal simulation, an auxiliary variable is introduced to degrade the parabolic equation to a hyperbolic one, which can be solved by DGTD method directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' By adopting different discretized volumes and independent grids for the electrical solver and the thermal solver, redundant computational overhead can be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' On the premise of guaranteeing the stability, a flexible explicit LTS technique based on interpolation method is incorporated into the solver to improve the capability of solving multi-scale problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' With the LTS technique, the sophisticated structure can be divided into groups and different time steps are allowed in separate groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Two numerical examples are provided to demonstrate the validity, flexibility, as well as the efficiency improvement by the LTS technique in comparison with COMSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} +page_content=' Further- more, the electrical–thermal behavior of two multiscale PDN systems is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQfSfeg/content/2301.00088v1.pdf'} 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sha256:b619d392ff909233e6a0c919d6c8d4cf0fb198aa4f23de4b38b7bdb2b7e888c1 +size 1874737 diff --git a/HNAyT4oBgHgl3EQfrvkK/vector_store/index.pkl b/HNAyT4oBgHgl3EQfrvkK/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..adddcf555c0b30321c39c04ae9c2121a31b366f8 --- /dev/null +++ b/HNAyT4oBgHgl3EQfrvkK/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0fb90872cc4e50f8461eb458b75ce0d5bcd32acff546202c0336b68da9969262 +size 166289 diff --git a/HNE0T4oBgHgl3EQfhgGY/content/tmp_files/2301.02433v1.pdf.txt b/HNE0T4oBgHgl3EQfhgGY/content/tmp_files/2301.02433v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8782e0ce6e9f55989780fb64a0141fb3a69d18a9 --- /dev/null +++ b/HNE0T4oBgHgl3EQfhgGY/content/tmp_files/2301.02433v1.pdf.txt @@ -0,0 +1,517 @@ +The Growing Liberality Observed in Primary Animal +and Plant Cultures is Common to the Social Amoeba + + +Norichika Ogata +Nihon BioData Corporation, Kawasaki, Japan +norichik@nbiodata.com + + +Abstract—Tissue culture environment liberates cells from +ordinary laws of multi-cellular organisms. This liberation enables +cells several behaviors, such as proliferation, dedifferentiation, +acquisition of pluripotency, immortalization, and reprogramming. +Recently, the quantitative value of cellular dedifferentiation and +differentiation was defined as “liberality”, which is measurable as +Shannon entropy of numerical transcriptome data and Lempel- +Zip complexity of nucleotide sequence transcriptome data. The +increasing liberality induced by the culture environment had first +been observed in animal cells and had reconfirmed in plant cells. +The phenomena may be common across the kingdom, also in a +social amoeba. We measured the liberality of the social amoeba +which +disaggregated +from +multicellular +aggregates +and +transferred into a liquid medium. + +Keywords—Genomics, +Transcriptome, +Social +Amoeba, +Dictyostelium, Dedifferentiation, Liberality, Primary Culture, +Lempel-Ziv Complexity. +I. +INTRODUCTION +Tissue culture is performed to maintain isolated portions of +multicellular organisms in an artificial milieu that is outside the +individual organism and for considerable periods of time [1]. It +is known over a century that cells derived from cultured explants +are, in general, different from cells of the corresponding tissue +in a living organism [2, 3]. In these tissue cultures, cells are +liberated from stimulations and prohibition which is ordinary in +multi-cellular organisms [4]. This liberation is essential for +growth, +dedifferentiation, +acquisition +of +pluripotency, +immortalization and reprogramming. The proliferations of +cultured cells were also considered to be a result of +dedifferentiation [2]. +In this decade, several studies [5-8] following our research [9] +repeatedly measured the degree of cellular dedifferentiation and +differentiation as a Shannon entropy of numerical transcriptome +data. The Shannon entropy is a kind of alpha diversity in the +ecology [10], and the transcriptome Shannon entropy is simply +transcriptome diversity [11]. It is not incorrect to call it the +(alpha) diversity of the transcriptome, but that would leave its +biological significance undefined, as would each principal +component that came up in the principal component analysis. +We can quantitatively assess, judge, and define that +dedifferentiation is an increase in the Shannon entropy of the +transcriptome, it is more accurate to position the value not as a +mere bioinformatics measure; however, as a number with +obvious biological and bioengineering significance, such as +viable cell rate, cell density, specific growth rate, or pcd +(pg/cell/day) [12]. Here we call the quantitative value of cellular +dedifferentiation and differentiation “liberality,” since a +previous study explained the changes were happening to +cultured cells as “libère” [13]. The study of cellular liberality has +entered a phase in which evidence is being gathered to +strengthen the theory; the degree of cellular dedifferentiation +and differentiation is measurable. +We previously observed increasing liberality in animal and +plant primary cultures [5, 14]. In this study, we measured the +liberality of the social amoeba, Dictyostelium discoideum, +having a transition from single-celled amoebae to a multicellular +organism as a natural part of its life cycle [15]. Dictyostelium +dedifferentiation was initiated by disaggregation of multicellular +aggregates and transfer of the cells into a HL5 liquid medium in +a study [16]. We could not reproduce the mapping used in the + +Fig 1. Scatter plot of culture time vs cellular liberality. +The social amoeba cells disaggregated of multicellular +aggregates and transferred the cells into a HL5 liquid +medium. The Lempel-Ziv complexity of transcriptome were +measured in each culture time. + +Cellular Liberality (Lempel-Ziv complexity of fastq files) +0.149 +0 +0.148 +00 +0 +0 +0 +0.147 +0 +0 +0 +0 +0 +0 +146 +0 +0.1 +0.145 +0 +一 +0 +0.5 +1 +2 +4 +8 +16 +Culture Time (Hoursstudy since the referential genome was not clear in the +manuscript; There is “Reads were mapped to the Dictyostelium +genome (version obtained from Gareth Bloomfield, masking the +duplication on chromosome 2) using Tophat v2.0.9.”. Therefore, +we measured the liberality from the transcriptome data without +the referential genome and the reads mapping process. A recent +study enabled measuring the liberality as Lempel-Ziv +complexity of fastq files [17, 18]. +II. +MATERIALS AND METHODS +Transcriptome data set was obtained from DDBJ SRA +(SRA1039371) [16]; 30 fastq data ware used (SRR11039842- +SRR11039872). In the entry, time course total RNA sampling +during cultivation of the amoeba (0, 0.5, 1, 1.5, 2, 2.5, 3, 4, 5, +6, 8, 10, 12, 14 and 18 hours). Each sample has two biological +replicates. We downloaded sra compressed fastq data. The file +size of decompressed raw textual fastq files and bz2 +compressed files were measured. We measured Lempel-Ziv +complexity of the fastq files as file size rate between bz2 +compressed fastq data and raw textual fastq data. Then we +compared the culture time and the Lampel-Ziv complexity. +III. +RESULTS AND DISCUSSION +Dictyostelium dedifferentiation is initiated by disaggregation +of multicellular aggregates and transfer of the cells into a HL5 +liquid media. The liberality immediately increased and reached +the peak in 3 hours. This result suggested that dedifferentiation +of cells in primary explant culture is a common phenomenon for +diverse multi-cellular organisms including social amoeba. After +4 hours of the amoeba disaggregation, the liberality started +decreasing. The liberality of the amoeba 18 hours after +disaggregation were equal to 0 hours after disaggregation. This +liberality decreasing following the liberality increase was +observed in the previous study of plants, that was thought to be +re-differentiation to construct a new plant. The dedifferentiated +amoeba would immediately start re-differentiation to achieve a +slightly better position in the new environment. Although, in the +Dictyostelium paper the amoeba had been thought to +dedifferentiate for 18 hours, our observations of the liberality +suggest that amoeba dedifferentiation completed in 4 hours. +Indeed, the Dictyostelim paper indicated the importantest +change of amoeba at 2~4 hours after disaggregation; two +transcription factors, bzpS and mybD showed clear peak of +expression (Fig. 3a [16]), the PC1 in a primary component +analysis of transcriptome does not change after 4 hours after +disaggregation (Fig. 3b [16]), and cell motility showed peak at +3 hours after disaggregation (Fig. 5 [16]). +Even plants complete dedifferentiation within 6 hours [14, +19], so there is no reason why the amoeba dedifferentiation +should be slower than that. We recommend that studies +attempting +to +examine +dedifferentiation +incorporate +measurements of cellular liberality to accurately monitor the +state of cellular dedifferentiation [20]. As was evident in the +present study, measurement processes that include mapping of +reads to a reference genome are less reproducible, making the +measurement of liberality based on LZ complexity of nucleotide +sequence files [18] more useful. +REFERENCES +[1] +Margaret Ransone Murray GK. A Bibliography of the Research in Tissue +Culture, 1884-1950. New York: Academic Press; 1953. 1741 p. +[2] +Champy C. Quelques résultats de la méthode de culture de tissus. I. +Généralités. II. Le muscle lisse. . Archives de zoologie expérimentale et +générale. +1913-1914;53:42-51. +https://ia800207.us.archive.org/8/items/archivesdezoolog53cent/archives +dezoolog53cent.pdf +[3] +Carleton HM. Tissue culture: A critical summary. Journal of +Experimental Biology. 1923;1:131-51. +[4] Canguilhem G. La connaissance de la vie. Paris: Librairie Philosophique +J Vrin; 1965. +[5] Guo M, Bao EL, Wagner M, Whitsett JA, and Xu Y, SLICE: determining +cell differentiation and lineage based on single cell entropy. 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Biol. +2022; 5(4):116-118. doi: 10.26502/jbsb.5107039. +[15] Devreotes P, Dictyostelium discoideum: a model system for cell-cell +interactions in development, Science. 1989; 245(4922):1054-8. +[16] Nichols JME, Antolović V, Reich JD, Brameyer S, Paschke P, Chubb JR, +Cell and molecular transitions during efficient dedifferentiation. eLife. +2020; 9:e55435. +[17] Lempel A, Ziv J, On the Complexity of Finite Sequences, IEEE +Transactions onf Information Theory. 1976; IT-22(1). +[18] Ogata N, Hosaka A, Cellular liberality is measurable as Lempel-Ziv +complexity of fastq files, in 2022 IEEE 22nd International Conference on +Bioinformatics and Bioengineering (BIBE), Taichung, Taiwan, 2022 pp. +321-326. doi: 10.1109/BIBE55377.2022.00072 +[19] Nishiyama T, Miyawaki K, Ohshima M, Thompson K, Nagashima A, +Hasebe M, et al. Digital gene expression profiling by 5'-end sequencing +of cDNAs during reprogramming in the moss Physcomitrella patens. +PLoS One. 2012;7(5):e36471. doi: 10.1371/journal.pone.0036471. +PubMed PMID: 22574165; PubMed Central PMCID: PMCPMC3344888. +[20] Ogata N, Kozaki T, Yokoyama T, Hata T, Iwabuchi K. Comparison +between the Amount of Environmental Change and the Amount of +Transcriptome Change. PLoS One. 2015;10(12):e0144822. Epub +2015/12/15. doi: 10.1371/journal.pone.0144822. PubMed PMID: +26657512; PubMed Central PMCID: PMC4678807. + + + +Appendix. Codes obtaining data used in this study. + +# GSM4300090: Media 0h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691764/SRR11039842/SRR11039842.sra ./ + +# GSM4300091: Media 0.5h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691765/SRR11039843/SRR11039843.sra ./ + +# GSM4300092: Media 1h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691766/SRR11039844/SRR11039844.sra ./ + +# GSM4300093: Media 1.5h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691767/SRR11039846/SRR11039846.sra ./ + +# GSM4300094: Media 2h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691768/SRR11039847/SRR11039847.sra ./ + +# GSM4300095: Media 2.5h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691770/SRR11039848/SRR11039848.sra ./ + +# GSM4300096: Media 3h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691771/SRR11039849/SRR11039849.sra ./ + +# GSM4300097: Media 4h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691772/SRR11039850/SRR11039850.sra ./ + +# GSM4300098: Media 5h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691773/SRR11039851/SRR11039851.sra ./ + +# GSM4300099: Media 6h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691774/SRR11039852/SRR11039852.sra ./ + +# GSM4300100: Media 8h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691775/SRR11039853/SRR11039853.sra ./ + +# GSM4300101: Media 10h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691776/SRR11039854/SRR11039854.sra ./ + +# GSM4300102: Media 12h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691777/SRR11039855/SRR11039855.sra ./ + +# GSM4300103: Media 14h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691778/SRR11039856/SRR11039856.sra ./ + +# GSM4300104: Media 18h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691779/SRR11039857/SRR11039857.sra ./ + +# GSM4300105: Media 0h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691780/SRR11039858/SRR11039858.sra ./ + +# GSM4300106: Media 0.5h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691781/SRR11039859/SRR11039859.sra ./ + +# GSM4300107: Media 1h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691782/SRR11039860/SRR11039860.sra ./ + +# GSM4300108: Media 1.5h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691783/SRR11039861/SRR11039861.sra ./ + +# GSM4300109: Media 2h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691784/SRR11039862/SRR11039862.sra ./ + +# GSM4300110: Media 2.5h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691785/SRR11039863/SRR11039863.sra ./ + +# GSM4300111: Media 3h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691786/SRR11039864/SRR11039864.sra ./ + +# GSM4300112: Media 4h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691787/SRR11039865/SRR11039865.sra ./ + + +# GSM4300113: Media 5h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691788/SRR11039866/SRR11039866.sra ./ + +# GSM4300114: Media 6h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691801/SRR11039867/SRR11039867.sra ./ + +# GSM4300115: Media 8h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691802/SRR11039868/SRR11039868.sra ./ + +# GSM4300116: Media 10h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691803/SRR11039869/SRR11039869.sra ./ + +# GSM4300117: Media 12h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691804/SRR11039870/SRR11039870.sra ./ + +# GSM4300118: Media 14h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691805/SRR11039871/SRR11039871.sra ./ + +# GSM4300119: Media 18h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691806/SRR11039872/SRR11039872.sra ./ + +# GSM4300120: Undifferentiated rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691807/SRR11039873/SRR11039873.sra ./ + +# GSM4300121: Undifferentiated rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691808/SRR11039874/SRR11039874.sra ./ + +# GSM4300122: Bacteria/Buffer 0h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691809/SRR11039875/SRR11039875.sra ./ + +# GSM4300123: Bacteria 0.5h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691810/SRR11039876/SRR11039876.sra ./ + +# GSM4300124: Bacteria 1h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691811/SRR11039877/SRR11039877.sra ./ + +# GSM4300125: Bacteria 1.5h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691812/SRR11039878/SRR11039878.sra ./ + +# GSM4300126: Bacteria 2h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691814/SRR11039879/SRR11039879.sra ./ + +# GSM4300127: Bacteria 3h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691815/SRR11039880/SRR11039880.sra ./ + +# GSM4300128: Bacteria 4h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691816/SRR11039881/SRR11039881.sra ./ + +# GSM4300129: Bacteria 5h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691817/SRR11039882/SRR11039882.sra ./ + +# GSM4300130: Bacteria 6h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691818/SRR11039883/SRR11039883.sra ./ + +# GSM4300131: Bacteria 8h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691819/SRR11039884/SRR11039884.sra ./ + +# GSM4300132: Bacteria 12h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691820/SRR11039885/SRR11039885.sra ./ + +# GSM4300133: Bacteria 24h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691821/SRR11039886/SRR11039886.sra ./ + +# GSM4300134: Buffer 0.5h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691822/SRR11039887/SRR11039887.sra ./ + +# GSM4300135: Buffer 1h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691823/SRR11039888/SRR11039888.sra ./ + +# GSM4300136: Buffer 2h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691824/SRR11039889/SRR11039889.sra ./ + + +# GSM4300137: Buffer 3h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691825/SRR11039890/SRR11039890.sra ./ + +# GSM4300138: Buffer 4h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691826/SRR11039891/SRR11039891.sra ./ + +# GSM4300139: Buffer 6h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691827/SRR11039892/SRR11039892.sra ./ + +# GSM4300140: Bacteria/Buffer 0h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691828/SRR11039893/SRR11039893.sra ./ + +# GSM4300141: Bacteria 0.5h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691829/SRR11039894/SRR11039894.sra ./ + +# GSM4300142: Bacteria 1h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691830/SRR11039895/SRR11039895.sra ./ + +# GSM4300143: Bacteria 1.5h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691831/SRR11039896/SRR11039896.sra ./ + +# GSM4300144: Bacteria 2h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691832/SRR11039897/SRR11039897.sra ./ + +# GSM4300145: Bacteria 3h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691833/SRR11039898/SRR11039898.sra ./ + +# GSM4300146: Bacteria 4h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691834/SRR11039899/SRR11039899.sra ./ + +# GSM4300147: Bacteria 5h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691835/SRR11039900/SRR11039900.sra ./ + +# GSM4300148: Bacteria 6h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691836/SRR11039901/SRR11039901.sra ./ + +# GSM4300149: Bacteria 8h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691837/SRR11039902/SRR11039902.sra ./ + +# GSM4300150: Bacteria 12h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691838/SRR11039903/SRR11039903.sra ./ + +# GSM4300151: Bacteria 24h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691839/SRR11039904/SRR11039904.sra ./ + +# GSM4300152: Buffer 0.5h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691840/SRR11039906/SRR11039906.sra ./ + +# GSM4300153: Buffer 1h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691841/SRR11039907/SRR11039907.sra ./ + +# GSM4300154: Buffer 2h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691842/SRR11039908/SRR11039908.sra ./ + +# GSM4300155: Buffer 3h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691843/SRR11039909/SRR11039909.sra ./ + +# GSM4300156: Buffer 4h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691844/SRR11039910/SRR11039910.sra ./ + +# GSM4300157: Buffer 6h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691846/SRR11039911/SRR11039911.sra ./ + +# GSM4300158: Development 0h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691847/SRR11039912/SRR11039912.sra ./ + +# GSM4300159: Development 2h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691848/SRR11039913/SRR11039913.sra ./ + +# GSM4300160: Development 4h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691849/SRR11039914/SRR11039914.sra ./ + + +# GSM4300161: Development 6h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691850/SRR11039915/SRR11039915.sra ./ + +# GSM4300162: Development 8h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691789/SRR11039916/SRR11039916.sra ./ + +# GSM4300163: Development 10h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691790/SRR11039917/SRR11039917.sra ./ + +# GSM4300164: Development 12h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691791/SRR11039918/SRR11039918.sra ./ + +# GSM4300165: Development 14h rep1; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691792/SRR11039919/SRR11039919.sra ./ + +# GSM4300166: Development 0h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691793/SRR11039920/SRR11039920.sra ./ + +# GSM4300167: Development 2h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691794/SRR11039921/SRR11039921.sra ./ + +# GSM4300168: Development 4h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691795/SRR11039922/SRR11039922.sra ./ + +# GSM4300169: Development 6h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691796/SRR11039923/SRR11039923.sra ./ + +# GSM4300170: Development 8h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691797/SRR11039924/SRR11039924.sra ./ + +# GSM4300171: Development 10h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691798/SRR11039925/SRR11039925.sra ./ + +# GSM4300172: Development 12h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691799/SRR11039926/SRR11039926.sra ./ + +# GSM4300173: Development 14h rep2; Dictyostelium discoideum; RNA-Seq +wget ftp://ftp.ddbj.nig.ac.jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691800/SRR11039927/SRR11039927.sra ./ + diff --git a/HNE0T4oBgHgl3EQfhgGY/content/tmp_files/load_file.txt b/HNE0T4oBgHgl3EQfhgGY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c5feea5a90c41e427b8861cdaad82f169b00f78 --- /dev/null +++ b/HNE0T4oBgHgl3EQfhgGY/content/tmp_files/load_file.txt @@ -0,0 +1,884 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf,len=883 +page_content='The Growing Liberality Observed in Primary Animal and Plant Cultures is Common to the Social Amoeba Norichika Ogata Nihon BioData Corporation, Kawasaki, Japan norichik@nbiodata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='com Abstract—Tissue culture environment liberates cells from ordinary laws of multi-cellular organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' This liberation enables cells several behaviors, such as proliferation, dedifferentiation, acquisition of pluripotency, immortalization, and reprogramming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Recently, the quantitative value of cellular dedifferentiation and differentiation was defined as “liberality”, which is measurable as Shannon entropy of numerical transcriptome data and Lempel- Zip complexity of nucleotide sequence transcriptome data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' The increasing liberality induced by the culture environment had first been observed in animal cells and had reconfirmed in plant cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' The phenomena may be common across the kingdom, also in a social amoeba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' We measured the liberality of the social amoeba which disaggregated from multicellular aggregates and transferred into a liquid medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Keywords—Genomics, Transcriptome, Social Amoeba, Dictyostelium, Dedifferentiation, Liberality, Primary Culture, Lempel-Ziv Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' INTRODUCTION Tissue culture is performed to maintain isolated portions of multicellular organisms in an artificial milieu that is outside the individual organism and for considerable periods of time [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' It is known over a century that cells derived from cultured explants are, in general, different from cells of the corresponding tissue in a living organism [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' In these tissue cultures, cells are liberated from stimulations and prohibition which is ordinary in multi-cellular organisms [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' This liberation is essential for growth, dedifferentiation, acquisition of pluripotency, immortalization and reprogramming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' The proliferations of cultured cells were also considered to be a result of dedifferentiation [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' In this decade, several studies [5-8] following our research [9] repeatedly measured the degree of cellular dedifferentiation and differentiation as a Shannon entropy of numerical transcriptome data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' The Shannon entropy is a kind of alpha diversity in the ecology [10], and the transcriptome Shannon entropy is simply transcriptome diversity [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' It is not incorrect to call it the (alpha) diversity of the transcriptome, but that would leave its biological significance undefined, as would each principal component that came up in the principal component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' We can quantitatively assess, judge, and define that dedifferentiation is an increase in the Shannon entropy of the transcriptome, it is more accurate to position the value not as a mere bioinformatics measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' however, as a number with obvious biological and bioengineering significance, such as viable cell rate, cell density, specific growth rate, or pcd (pg/cell/day) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Here we call the quantitative value of cellular dedifferentiation and differentiation “liberality,” since a previous study explained the changes were happening to cultured cells as “libère” [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' The study of cellular liberality has entered a phase in which evidence is being gathered to strengthen the theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' the degree of cellular dedifferentiation and differentiation is measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' We previously observed increasing liberality in animal and plant primary cultures [5, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' In this study, we measured the liberality of the social amoeba, Dictyostelium discoideum, having a transition from single-celled amoebae to a multicellular organism as a natural part of its life cycle [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium dedifferentiation was initiated by disaggregation of multicellular aggregates and transfer of the cells into a HL5 liquid medium in a study [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' We could not reproduce the mapping used in the Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Scatter plot of culture time vs cellular liberality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' The social amoeba cells disaggregated of multicellular aggregates and transferred the cells into a HL5 liquid medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' The Lempel-Ziv complexity of transcriptome were measured in each culture time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Cellular Liberality (Lempel-Ziv complexity of fastq files) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='149 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='148 00 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='147 0 0 0 0 0 0 146 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='145 0 一 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5 1 2 4 8 16 Culture Time (Hoursstudy since the referential genome was not clear in the manuscript;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' There is “Reads were mapped to the Dictyostelium genome (version obtained from Gareth Bloomfield, masking the duplication on chromosome 2) using Tophat v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='9.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Therefore, we measured the liberality from the transcriptome data without the referential genome and the reads mapping process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' A recent study enabled measuring the liberality as Lempel-Ziv complexity of fastq files [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' MATERIALS AND METHODS Transcriptome data set was obtained from DDBJ SRA (SRA1039371) [16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' 30 fastq data ware used (SRR11039842- SRR11039872).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' In the entry, time course total RNA sampling during cultivation of the amoeba (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5, 2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5, 3, 4, 5, 6, 8, 10, 12, 14 and 18 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Each sample has two biological replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' We downloaded sra compressed fastq data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' The file size of decompressed raw textual fastq files and bz2 compressed files were measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' We measured Lempel-Ziv complexity of the fastq files as file size rate between bz2 compressed fastq data and raw textual fastq data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Then we compared the culture time and the Lampel-Ziv complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RESULTS AND DISCUSSION Dictyostelium dedifferentiation is initiated by disaggregation of multicellular aggregates and transfer of the cells into a HL5 liquid media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' The liberality immediately increased and reached the peak in 3 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' This result suggested that dedifferentiation of cells in primary explant culture is a common phenomenon for diverse multi-cellular organisms including social amoeba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' After 4 hours of the amoeba disaggregation, the liberality started decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' The liberality of the amoeba 18 hours after disaggregation were equal to 0 hours after disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' This liberality decreasing following the liberality increase was observed in the previous study of plants, that was thought to be re-differentiation to construct a new plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' The dedifferentiated amoeba would immediately start re-differentiation to achieve a slightly better position in the new environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Although, in the Dictyostelium paper the amoeba had been thought to dedifferentiate for 18 hours, our observations of the liberality suggest that amoeba dedifferentiation completed in 4 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Indeed, the Dictyostelim paper indicated the importantest change of amoeba at 2~4 hours after disaggregation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' two transcription factors, bzpS and mybD showed clear peak of expression (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' 3a [16]), the PC1 in a primary component analysis of transcriptome does not change after 4 hours after disaggregation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' 3b [16]), and cell motility showed peak at 3 hours after disaggregation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' 5 [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Even plants complete dedifferentiation within 6 hours [14, 19], so there is no reason why the amoeba dedifferentiation should be slower than that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' We recommend that studies attempting to examine dedifferentiation incorporate measurements of cellular liberality to accurately monitor the state of cellular dedifferentiation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' As was evident in the present 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[15] Devreotes P, Dictyostelium discoideum: a model system for cell-cell interactions in development, Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' 245(4922):1054-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' [16] Nichols JME, Antolović V, Reich JD, Brameyer S, Paschke P, Chubb JR, Cell and molecular transitions during efficient dedifferentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' eLife.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' 9:e55435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' [17] Lempel A, Ziv J, On the Complexity of Finite Sequences, IEEE Transactions onf Information Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' IT-22(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' [18] Ogata N, Hosaka A, Cellular liberality is measurable as Lempel-Ziv complexity of fastq files, in 2022 IEEE 22nd International Conference on Bioinformatics and Bioengineering (BIBE), Taichung, Taiwan, 2022 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' 321-326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='1109/BIBE55377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='00072 [19] Nishiyama T, Miyawaki K, Ohshima M, Thompson K, Nagashima A, Hasebe M, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=" Digital gene expression profiling by 5'-end sequencing of cDNAs during reprogramming in the moss Physcomitrella patens." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' PLoS One.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='7(5):e36471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='1371/journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='pone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='0036471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' PubMed PMID: 22574165;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' PubMed Central PMCID: PMCPMC3344888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' [20] Ogata N, Kozaki T, Yokoyama T, Hata T, Iwabuchi K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Comparison between the Amount of Environmental Change and the Amount of Transcriptome Change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' PLoS One.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='10(12):e0144822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Epub 2015/12/15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='1371/journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='pone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='0144822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' PubMed PMID: 26657512;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' PubMed Central PMCID: PMC4678807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Codes obtaining data used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' # GSM4300090: Media 0h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691764/SRR11039842/SRR11039842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300091: Media 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691765/SRR11039843/SRR11039843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300092: Media 1h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691766/SRR11039844/SRR11039844.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300093: Media 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691767/SRR11039846/SRR11039846.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300094: Media 2h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691768/SRR11039847/SRR11039847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300095: Media 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691770/SRR11039848/SRR11039848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300096: Media 3h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691771/SRR11039849/SRR11039849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300097: Media 4h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691772/SRR11039850/SRR11039850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300098: Media 5h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691773/SRR11039851/SRR11039851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300099: Media 6h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691774/SRR11039852/SRR11039852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300100: Media 8h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691775/SRR11039853/SRR11039853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300101: Media 10h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691776/SRR11039854/SRR11039854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300102: Media 12h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691777/SRR11039855/SRR11039855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300103: Media 14h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691778/SRR11039856/SRR11039856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300104: Media 18h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691779/SRR11039857/SRR11039857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300105: Media 0h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691780/SRR11039858/SRR11039858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300106: Media 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691781/SRR11039859/SRR11039859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300107: Media 1h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691782/SRR11039860/SRR11039860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300108: Media 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691783/SRR11039861/SRR11039861.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300109: Media 2h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691784/SRR11039862/SRR11039862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300110: Media 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691785/SRR11039863/SRR11039863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300111: Media 3h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691786/SRR11039864/SRR11039864.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300112: Media 4h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691787/SRR11039865/SRR11039865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300113: Media 5h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691788/SRR11039866/SRR11039866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300114: Media 6h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691801/SRR11039867/SRR11039867.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300115: Media 8h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691802/SRR11039868/SRR11039868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300116: Media 10h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691803/SRR11039869/SRR11039869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300117: Media 12h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691804/SRR11039870/SRR11039870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300118: Media 14h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691805/SRR11039871/SRR11039871.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300119: Media 18h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691806/SRR11039872/SRR11039872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300120: Undifferentiated rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691807/SRR11039873/SRR11039873.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300121: Undifferentiated rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691808/SRR11039874/SRR11039874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300122: Bacteria/Buffer 0h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691809/SRR11039875/SRR11039875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300123: Bacteria 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691810/SRR11039876/SRR11039876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300124: Bacteria 1h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691811/SRR11039877/SRR11039877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300125: Bacteria 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691812/SRR11039878/SRR11039878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300126: Bacteria 2h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691814/SRR11039879/SRR11039879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300127: Bacteria 3h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691815/SRR11039880/SRR11039880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300128: Bacteria 4h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691816/SRR11039881/SRR11039881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300129: Bacteria 5h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691817/SRR11039882/SRR11039882.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300130: Bacteria 6h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691818/SRR11039883/SRR11039883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300131: Bacteria 8h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691819/SRR11039884/SRR11039884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300132: Bacteria 12h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691820/SRR11039885/SRR11039885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300133: Bacteria 24h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691821/SRR11039886/SRR11039886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300134: Buffer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691822/SRR11039887/SRR11039887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300135: Buffer 1h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691823/SRR11039888/SRR11039888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300136: Buffer 2h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691824/SRR11039889/SRR11039889.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300137: Buffer 3h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691825/SRR11039890/SRR11039890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300138: Buffer 4h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691826/SRR11039891/SRR11039891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300139: Buffer 6h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691827/SRR11039892/SRR11039892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300140: Bacteria/Buffer 0h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691828/SRR11039893/SRR11039893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300141: Bacteria 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691829/SRR11039894/SRR11039894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300142: Bacteria 1h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691830/SRR11039895/SRR11039895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300143: Bacteria 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691831/SRR11039896/SRR11039896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300144: Bacteria 2h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691832/SRR11039897/SRR11039897.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300145: Bacteria 3h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691833/SRR11039898/SRR11039898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300146: Bacteria 4h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691834/SRR11039899/SRR11039899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300147: Bacteria 5h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691835/SRR11039900/SRR11039900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300148: Bacteria 6h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691836/SRR11039901/SRR11039901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300149: Bacteria 8h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691837/SRR11039902/SRR11039902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300150: Bacteria 12h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691838/SRR11039903/SRR11039903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300151: Bacteria 24h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691839/SRR11039904/SRR11039904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300152: Buffer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='5h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691840/SRR11039906/SRR11039906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300153: Buffer 1h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691841/SRR11039907/SRR11039907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300154: Buffer 2h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691842/SRR11039908/SRR11039908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300155: Buffer 3h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691843/SRR11039909/SRR11039909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300156: Buffer 4h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691844/SRR11039910/SRR11039910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300157: Buffer 6h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691846/SRR11039911/SRR11039911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300158: Development 0h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691847/SRR11039912/SRR11039912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300159: Development 2h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691848/SRR11039913/SRR11039913.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300160: Development 4h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691849/SRR11039914/SRR11039914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300161: Development 6h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691850/SRR11039915/SRR11039915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300162: Development 8h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691789/SRR11039916/SRR11039916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300163: Development 10h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691790/SRR11039917/SRR11039917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300164: Development 12h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691791/SRR11039918/SRR11039918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300165: Development 14h rep1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691792/SRR11039919/SRR11039919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300166: Development 0h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691793/SRR11039920/SRR11039920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300167: Development 2h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691794/SRR11039921/SRR11039921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300168: Development 4h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691795/SRR11039922/SRR11039922.' metadata={'source': 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+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691796/SRR11039923/SRR11039923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300170: Development 8h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691797/SRR11039924/SRR11039924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300171: Development 10h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691798/SRR11039925/SRR11039925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300172: Development 12h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691799/SRR11039926/SRR11039926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/ # GSM4300173: Development 14h rep2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' Dictyostelium discoideum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content=' RNA-Seq wget ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ddbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='nig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='jp/ddbj_database/dra/sralite/ByExp/litesra/SRX/SRX769/SRX7691800/SRR11039927/SRR11039927.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='sra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} +page_content='/' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE0T4oBgHgl3EQfhgGY/content/2301.02433v1.pdf'} diff --git a/I9E2T4oBgHgl3EQfAAZV/content/tmp_files/2301.03586v1.pdf.txt b/I9E2T4oBgHgl3EQfAAZV/content/tmp_files/2301.03586v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f02157bd0764067a87deb723929532b7e9410ae --- /dev/null +++ b/I9E2T4oBgHgl3EQfAAZV/content/tmp_files/2301.03586v1.pdf.txt @@ -0,0 +1,965 @@ +arXiv:2301.03586v1 [math.NT] 7 Jan 2023 +THE PRIME NUMBER THEOREM AND PRIMORIAL +NUMBERS +JONATAN GOMEZ +JGOMEZPE@UNAL.EDU.CO +UNIVERSIDAD NACIONAL DE COLOMBIA +Abstract. Counting the number of prime numbers up to a certain natural +number and describing the asymptotic behavior of such a counting function has +been studied by famous mathematicians like Gauss, Legendre, Dirichlet, and +Euler. The prime number theorem determines that such asymptotic behavior +is similar to the asymptotic behavior of the number divided by its natural +logarithm. +In this paper, we take advantage of a multiplicative representa- +tion of a number and the properties of the logarithm function to express the +prime number theorem in terms of primorial numbers, and n-primorial tota- +tive numbers. A primorial number is the multiplication of the first n prime +numbers while n-primorial totatives are the numbers that are coprime to the +n-th primorial number. By doing this we can define several different functions +that can be used to approximate the behavior of the prime counting function +asymptotically. +1. Introduction +1.1. Prime Numbers. A prime number is a natural number having exactly two +factors, 1 and itself [Nar00]. The natural number 1 is not considered a prime number +since it has only one factor (1). From now on, pn will denote the nth prime number, +here n ≥ 1. The first eleven prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. +1.2. Prime Numbers Theorem. [Nar00]. Let π(x) be the prime number-counting +function, i.e., the function that determines the number of primes less than or +equal to a natural number x. The prime number theorem proved independently +by Hadamard [Had96] and de la Vallée Poussin [Val96], describes the asymptotic +behavior of π(x) as follows: +(1.1) +lim +x→∞ +π(x) +� +x +log(x) +� = 1 +Using asymptotic notation, we can rewrite the prime number theorem as follows: +(1.2) +π(x) ∼ +x +log(x) +Table 1.2 shows the asymptotic behavior of π(x) and +x +log(x). Results for π(x) are +taken from https://en.wikipedia.org/wiki/Prime_number_theorem and results for +x +log(x) are obtained with the Python program (PNTprimorials.py) freely available +at professor Jonatan Gomez github repository [Gom]. +1 + +THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS +2 +x +π(x) +π(x)/ +x +log(x) +101 +4 +0.921 +102 +25 +1.151 +103 +168 +1.161 +104 +1229 +1.132 +105 +9592 +1.104 +106 +78498 +1.084 +107 +664579 +1.071 +108 +5761455 +1.061 +109 +50847534 +1.054 +1010 +455052511 +1.048 +1011 +4118054813 +1.043 +1012 +37607912018 +1.039 +1013 +346065536839 +1.036 +1014 +3204941750802 +1.033 +1015 +29844570422669 +1.031 +1016 +279238341033925 +1.029 +1017 +2623557157654233 +1.027 +1018 +24739954287740860 +1.025 +1019 +234047667276344607 +1.024 +1020 +2220819602560918840 +1.023 +1021 +21127269486018731928 +1.022 +1022 +201467286689315906290 +1.021 +1023 +1925320391606803968923 +1.020 +1024 +18435599767349200867866 +1.019 +1025 +176846309399143769411680 +1.018 +Table +1. Asymptotic +behavior +of +π(x) +and +x +log(x). +Re- +sults for +x +log(x) are obtained with the Python program (PNT- +primorials.py) +freely +available +at +professor +Jonatan +Gomez +github repository [Gom]. +Results for π(x) are taken from +https://en.wikipedia.org/wiki/Prime_number_theorem and +Now, we write down some Theorems about prime numbers that have been proven +previously in the literature, and we derive some technical properties from such +Theorems. +Theorem 1. (Bertrand’s postulate) For any positive natural number n we have +that pn+1 ≤ 2pn − 1. +Proof. Chebyshev proved this Theorem in [Tch52]. +□ +Corollary 2. For any positive natural number n we have that log +� +pn+2 +pn +� +< 2 +Proof. By applying Theorem 1 twice we have that pn+2 < 2pn+1 < 4pn for all +positive natural number n. Clearly, pn+2 +pn +< 4 so, log +� +pn+2 +pn +� +< log(4) < 2. +□ +Theorem 3. For any positive natural number m there exists a positive natural +number N such that pm +n+1 < �n +i=1 pi for all n ≥ N. + +THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS +3 +Proof. Suzuki proved this Theorem in [Suz13]. +□ +Corollary 4. For any positive natural number m there exists a positive natural +number N such that +log(pn+1) +log( +�n +i=1 pi) < 1 +m for all n ≥ N. +Proof. Consider inequality in Theorem 4, apply log function on both sides of the +inequality, divide both sides by the right side, apply property log(xm) = m ∗ log(x) +on the left side, and divide both sides of the inequality by m. +□ +Theorem 5. Functions f(x) = � +p≤x +p +p−1 and h(x) = eγ ∗ log(x) have the same +asymptotic behavior. Here γ is the Euler’s constant. +Proof. This Theorem can be proven as a Corollary of the Theorem proved by +Mertens in [Mer74]. +□ +Corollary 6. Functions f •(x) = � +p≤x p and h•(x) = eγ ∗ log(x)∗ � +p≤x p− 1 have +the same asymptotic behavior. Here γ is the Euler’s constant. +1.3. Primorial. Many interesting operations can be defined over prime numbers, +for example, we can define the primorial of the nth prime number as the product +of the first n ∈ N+ prime numbers [Dub87], i.e., #(n) = pn# = �n +i=1 pk. This +definition is similar to the definition of the factorial function, so it is possible to +define the primorial as a recursive function #(0) = 1 and #(n) = pn#(n − 1) for +n ≥ 1. The first five primorials are 2, 6, 30, 210, 2310. +1.4. Coprime or Relative-prime Numbers. Two natural numbers a, b ∈ N are +coprime or relative-prime numbers iff their greatest common divisor is 1 (gcd(a, b) = +1). According to this definition, i) 1 is relative-prime with any other positive natural +number, ii) 0 is not relative-prime with any natural number, and iii) any prime +number is relative-prime with any other prime number. Notice that we can define +prime numbers in terms of coprime numbers: A natural number p > 1 is a prime +number iff p is relative-prime to q for all natural numbers 1 ≤ q < p. +1.5. Natural numbers less than n (Zn). Let n ∈ N, the set of natural numbers +less than n, is Zn = {0, 1, . . ., n−1}. For example, Z2 = {0, 1} and Z#(2) = Z2∗3 = +Z6 = {0, 1, 3, 4, 5}. Amount the properties hold by Zn, we are especially interested +in the structure of the subset of natural numbers that are relative-prime to #(n) +(sometimes called as totative numbers of #(n)). For instance, consider Z#(2) = Z6, +the subset of Z6 defined by the natural numbers that are relative-prime to 6 is {1, 5}. +It is clear that any prime number q such pn < q < #(n) will be a relative-prime +number to #(n). +1.6. Euler’s totient function (ϕ(n)). Euler’s totient function [Eul63] counts the +natural numbers, in Zn, which are relative-prime to n. Take for example Z#(3) = +Z30, the subset of natural numbers relative-prime to 30 is {1, 7, 11, 13, 17, 19, 23, 29}, +therefore ϕ(30) = 8. +Euler’s totient function is a multiplicative function, i.e., +ϕ(ab) = ϕ(a)ϕ(b) for two coprime numbers a and b. Moreover, for a prime number +p, we have that Zp − {0} is the set of totative numbers of p, therefore ϕ(p) = p − 1. +Using these two properties of Euler’s totient function, we can easily compute it on +primorial numbers: ϕ(#(n)) = �n +i=1(pk − 1) = �n +i=1 ϕ(pk). + +THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS +4 +1.7. Primorial sets. We define a n-primorial set by ’replacing’ numbers 0 and 1 +in Z#(n), with numbers #(n) and #(n) + 1, respectively [Gom23]. Let n ∈ N, the +n-primorial set is defined as Z# +n = {2, 3, . . ., #(n), #(n) + 1}. The following is the +list of the first five n-primorial sets: +(1) Z# +1 = {2, 3} +(2) Z# +2 = {2, 3, 4, 5, 6, 7} +(3) Z# +3 = {2, 3, . . ., 30, 31} +(4) Z# +4 = {2, 3, . . ., 210, 211} +(5) Z# +5 = {2, 3, . . ., 2310, 2311} +1.8. Primorial totatives. We can extend the notion of totatives of #(n) to set +Z# +n by considering the subset of natural numbers in Z# +n being relative-prime to +#(n). The following is the list of the first three n-totative sets: +(1) The 1-totative set is tot(1) = {3} +(2) The 2-totative set is tot(2) = {5, 7} +(3) The 3-totative set is tot(3) = {7, 11, 13, 17, 19, 23, 29, 31} +Notice that any prime number p ∈ Z# +n must be either pi for some i = 1, 2, . . . , n +or a n-totative number, i.e., we can express the Prime Number Theorem in terms +of totative numbers. We explore this relationship in Section 4. +Lemma 7. The number of n-totatives is tot(n) = �n +i=1(pk − 1). +Proof. #(n) and (#(n) + 1) are coprime numbers, therefore the number of n- +totatives is equal to the totatives of #(n), i.e, the number of n-totatives is ϕ(#(n)) = +�n +i=1 ϕ(pk) = �n +i=1(pk − 1) = tot(n). +□ +2. The log Function and Multiplicative Number Representations +We can establish relationships between the prime number theorem and primorial +numbers if we represent numbers greater or equal to one in terms of multiplicative +fractions of prime or primorial numbers. Before that, we develop a technical Lemma +for analyzing asymptotic behavior. +Lemma 8. Let f, g, and h be real number functions such that 0 < f(x) ≤ g(x) ≤ +h(x) for all x greater than certain number N. If f and h have the same asymptotic +behavior then f, g, and h have the same asymptotic behavior. +Proof. Obvious, 1 = f(x) +f(x) ≤ g(x) +f(x) ≤ h(x) +f(x), by the Squeeze or sandwich Theorem we +have 1 = limx→∞ +f(x) +f(x) ≤ limx→∞ +g(x) +f(x) ≤ limx→∞ +h(x) +f(x) = 1. +□ +Definition 9. Let A = {ai}i∈N+ be a monotonic increasing succession such that +a1 ≥ 1. For any number x ≥ a1, any positive natural number n, and 0 ≤ r < 1: +(1) n(x) as the positive natural number such that an(x) ≤ x < an(x)+1. +(2) r(x) = +x−an(x) +an(x)+1−an(x) , +(3) s(n, r) = 1 + +� +an+1 +an +− 1 +� +r, +(4) s(x) = s(n(x), r(x)), and +(5) y(n, r) = an ∗ s(n, r). +Notice that s(n, r) = 1 + +� +an+1 +an +− 1 +� +r = 1 + +� +an+1−an +an +� +r = an+1r+an(1−r) +an +is +kind of the r-th multiplicative fraction of an+1 respect to succession A. + +THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS +5 +Lemma 10. Let A = {ai}i∈N+ be a monotonic increasing succession such that +a1 ≥ 1. For any number x ≥ a1, any positive natural number n, and 0 ≤ r < 1 +(1) an ≤ y(n, r) < an+1, +(2) x = y(n(x), r(x)) = an(x) ∗ s(x) +(3) m = n(y(m, t)), and t = r(y(m, t)) +Proof. (1) y(n, r) = an ∗s(n, r) = an +� +an+1r+an(1−r) +an +� += an+1r+an(1−r). Clearly, +y(n, r) is a convex combination between an+1 and an, then an ≤ y(n, r) < an+1. +(2) an(x) ∗ s(x) = an(x) +� +1 + +� an(x)+1−an(x) +an(x) +� � +x−an(x) +an(x)+1−an(x) +�� += an(x) ∗ s(x), i.e., +an(x) ∗ s(x) = an(x) +� +1 + +x−an(x) +an(x) +� += an(x) +� +x +an(x) +� += x. (3) Follows from (1) and +(2). +□ +We can define, in particular, two different representations of a number x ≥ 2 if +we consider Lemma 10: +(1) The prime representation of x ≥ 2 uses the succession of prime numbers +(A = {pn}n∈N+). Therefore, x = pn⋆(x)∗s⋆(x) with n⋆(x) such that pn⋆(x) ≤ +x < pn⋆(x)+1, s⋆(x) = 1+ +� pn⋆(x)+1 +pn⋆(x) +− 1 +� +r⋆(x), and r⋆(x) = +x−pn⋆(x) +pn⋆(x)+1−pn⋆(x) . +(2) The primorial representation of x ≥ 2 uses the succession of primorial +numbers (A = {#(n)}n∈N+). Therefore, x = #(n′(x)) ∗ s′(x) with n′(x) +such that #(n′(x)) ≤ x < #(n′(x) + 1), s′(x) = 1 + +� +pn′(x)+1 − 1 +� +r′(x), +and r′(x) = +x−#(n′(x)) +#(n′(x)+1)−#(n′(x)). +We can take advantage of the prime representation of a number for computing +the log(x) function, see Equation 2.1. +(2.1) +log(x) = log +� +pn⋆(x) ∗ s⋆(x) +� += log(pn⋆(x)) + log(s⋆(x)) +We can manipulate Equation 2.1 to produce functions having the asymptotic +behavior of log(x), see Equations 2.2 and 2.3. +(2.2) +log−(x) = log +� +pn⋆(x)−1 +� +(2.3) +log+(x) = log +� +pn⋆(x)+1 +� +Theorem 11. Functions log−(x) and log+(x) have the same asymptotic behavior, +i.e., +lim +x→∞ +log−(x) +log+(x) = 1 +Proof. Given a real number ǫ > 0, we can take n as the smallest natural number +such that +2 +log(pn) < ǫ. Now, we consider M = pn. If x > M then n⋆(x) > n. We +have log+(x) − log−(x) = log(pn⋆(x)+1) − log(pn•(x)−1) = log +� pn⋆(x)+1 +pn⋆(x)−1 +� +. By Corol- +lary 2 we have log+(x) − log−(x) < 2. Clearly, +��� log+(x) +log−(x) − 1 +��� = log+(x)−log−(x) +log−(x) +< +2 +log−(x) < ǫ. +□ +Corollary 12. Functions: +(1) log−(x) + +THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS +6 +(2) loga(x) = log−(x) + log(a(x)) with 1 ≤ a(x) ≤ +pn⋆(x)+1 +pn⋆(x)−1 +(3) log(x) +(4) log⋆(x) = log(pn⋆(x)) +(5) log∗(x) = log−(x) + log +� +1 + +� pn⋆(x) +pn⋆(x)−1 − 1 +� +r⋆(x) +� +, and +(6) log+(x) +Have the same asymptotic behavior. +Proof. Since every function log?(x) defined in (2)-(6) satisfies log−(x) ≤ log?(x) ≤ +log+(x), then by Theorem 11 and Lemma 8 the proof is completed. +□ +Now, we take advantage of the primorial representation of a number for com- +puting the log(x) function, see Equation 2.4. +(2.4) +log(x) = log (#(n′(x)) ∗ s′(x)) = log(s′(x))+log + + +n′(x) +� +i=1 +pi + + = log(s′(x))+ +n′(x) +� +i=1 +log(pi) +We can manipulate Equation 2.4 to produce functions with the asymptotic be- +havior of log(x), see Equations 2.5 and 2.6. +(2.5) +log−(x) = log (#(n′(x) − 1)) = log + + +n′(x)−1 +� +i=1 +pi + + +(2.6) +log+(x) = log (#(n′(x) + 1)) = log + + +n′(x)+1 +� +i=1 +pi + + +Theorem 13. Functions log−(x) and log+(x) have the same asymptotic behavior, +i.e., +lim +x→∞ +log−(x) +log+(x) = 1 +Proof. Given a real number ǫ > 0, we can take m as the smallest natural number +such that 2 +m < ǫ. Now, by Theorem 3 we can take N as the smallest natural number +such that pm +n+1 < #(n) for all n ≥ N, and by Corollary 4, we have log(pn+1) +log(#(n)) < 1 +m for +all n ≥ N (⋆). If x > M = #(N) then n′(x) ≥ N and +log(pn′(x)+1) +log(#(n′(x))) < 1 +m. Notice that +log(#(n′(x) + 1)) − log(#(n′(x) − 1)) = log +� +#(n′(x)+1) +#(n′(x)−1) +� += log(pn′(x) ∗ pn′(x)+1) < +log(p2 +n′(x)+1). Clearly, +��� +log−(x) +log+(x) − 1 +��� = +log+(x)−log−(x) +log+(x) +< +2 log(pn′(x)+1) +log(#(n′(x)+1)) < +2 log(pn′(x)+1) +log(#(n′(x))) . +Finally, using result (⋆) we have +��� log∗(x) +log(x) − 1 +��� < 2 +m < ǫ. +□ +Corollary 14. Functions: +(1) log−(x) +(2) loga(x)(x) = log−(x) + log(a(x)) with 1 ≤ a(x) ≤ pn′(x) ∗ pn′(x)+1 +(3) log(x) +(4) log#(x) = log(#(n′(x))) +(5) log∗(x) = log−(x) + log(1 + (pn′(x) − 1)r′(x)) + +THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS +7 +(6) log⋄(x) = log−(x) + log(n′(x) + r′(x)), and +(7) log+(x) +Have the same asymptotic behavior. +Proof. Since every function log?(x) defined in (2)-(6) satisfies log−(x) ≤ log?(x) ≤ +log+(x), then by Theorem 13 and Lemma 8 the proof is completed. +□ +3. Primorials and n-Totative Numbers +Notice that if we consider x = pn, Theorem 16 and Corollary 17 suggest a +relationship between the primorial number #(n) and the quantity of n-totatives +numbers when n → ∞. Upon this suggestion, we propose a relationship between +any positive number x ≥ 2 and n-totative numbers by considering the primorial +representation of a number and defining a function that approximates the quantity +of n-totative numbers up to x, see Equation 3.1. +(3.1) +tot∗(x) = tot(n′(x)) ∗ t∗(x) = t∗(x) ∗ +n′(x) +� +i=1 +pi − 1 +Here, t∗(x) = 1+(pn′(x)+1 −2)∗r′(x), with n′(x) and r′(x) as defined in Section +2, and tot(n) as defined in Section 1.8. Before establishing the relationship, we need +the following technical lemma. +Lemma 15. 1 ≤ 1+(a−1)r +1+(a−2)r < +a +a−1 for all a > 1 and all 0 ≤ r < 1. +Proof. Consider r = 0: clearly, 1+(a−1)r +1+(a−2)r = 1 +1 = 1. Consider 0 < r < 1: clearly +a − 2 < a − 1 then 1 + (a − 2)r < 1 + (a − 1)r, so 1 < 1+(a−1)r +1+(a−2)r. Now r < 1, i.e., +−1 + r < 0. By adding a + a2r − 2ar on both sides of the inequality and organizing +terms we have a − 1 + a2r − 2ar + r < a + a2r − 2ar. Grouping terms we have +(a − 1)(1 + (a − 1)r) < a(1 + (a − 2)r) and dividing both sides of inequality by +(a − 1)(1 + (a − 2)r) we have 1+(a−1)r +1+(a−2)r < +a +a−1. +□ +Now, we follow a similar scheme as the one we used in the previous section to +establish the relationship. +Theorem 16. Functions f(x) = �n′(x) +i=1 +pi +pi−1 and g(x) = �n′(x)+1 +i=1 +pi +pi−1 have the +same asymptotic behavior. +Proof. Obvious, limx→∞ +g(x) +f(x) = limx→∞ +pn′(x)+1 +pn′(x)+1−1 = 1+limx→∞ +1 +pn′(x)+1−1 = 1 +□ +Corollary 17. Function f ◦(x) = +x +tot∗(x) has the same asymptotic behavior of func- +tions f(x) = �n′(x) +i=1 +pi +pi−1 and g(x) = �n′(x)+1 +i=1 +pi +pi−1 . +Proof. Clearly, +x +tot∗(x) = +s′(x)∗#(n′(x)) +t∗(x)∗�n′(x) +i=1 +pi−1 = f(x)∗ s′(x) +t∗(x) = f(x)∗ +� 1+(pn′(x)+1−1)r′(x) +1+(pn′(x)+1−2)r′(x) +� +by Lemma 15 we have f(x) ≤ +x +tot∗(x) < f(x) ∗ +pn′(x)+1 +pn′(x)+1−1 = g(x). Then by Theorem +16 and Lemma 8 the proof is completed. +□ +Finally, we use Theorem 16 to establish the desired relationship. + +THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS +8 +Theorem 18. Functions f(x) = �n′(x) +i=1 +pi +pi−1 and g◦(x) = eγ ∗ log(y(x)) with γ +the Euler’s constant and y(x) = pn′(x) ∗ +� +1 + +� pn′(x)+1 +pn′(x) +− 1 +� +r′(x) +� +have the same +asymptotic behavior. +Proof. Given a real number ǫ > 0, we consider Theorem 16 and take M such +that +��� +� +p≤z +p +p−1 +eγ∗log(z) − 1 +��� < ǫ for all z > y(M). +If x > M then y(x) > y(M) and +��� +� +p≤y(x) +p +p−1 +eγ∗log(y(x)) − 1 +��� < ǫ. +We can see that �n′(x) +i=1 +pi +pi−1 = � +p≤y(x) +p +p−1 therefore, +���� +�n′(x) +i=1 +pi +pi−1 +eγ∗log(y(x)) − 1 +���� < ǫ. +□ +Corollary 19. Functions f ◦(x) = +x +tot∗(x) and g◦(x) = eγ ∗ log(y(x)) with γ the +Euler’s constant and y(x) = pn′(x) ∗ +� +1 + +� pn′(x)+1 +pn′(x) +− 1 +� +r′(x) +� +have the same as- +ymptotic behavior. +Proof. Follows from Corollary 17 and Theorem 18. +□ +Corollary 20. Functions id(x) = x and h◦(x) = tot∗(x) ∗ eγ ∗ log(y(x)) with γ +the Euler’s constant and y(x) = pn′(x) ∗ +� +1 + +� pn′(x)+1 +pn′(x) +− 1 +� +r′(x) +� +have the same +asymptotic behavior. +Proof. Follows from Corollary 19. +□ +4. Putting All Together +Notice that functions having the same asymptotic behavior as functions id(x) and +log(x) can replace them in the prime number theorem. So, we can express the Prime +Number Theorem in several ways, one of them using n-totative numbers. Table +4 shows the asymptotic behavior of π(x) vs +x +log(x), +x +log∗(x), +x +log⋄(x), +h◦(x) +log(h◦(x)), and +x +log(h◦(x)). These results are obtained with the Python program (PNTprimorials.py) +freely available at Professor Jonatan Gomez github repository [Gom]. +5. Conclusions and Future Work +We have developed several functions having the same asymptotic behavior of +π(x) the prime number counting function. We do this by representing any posi- +tive positive number in terms of primorial numbers and multiplicative fractions of +prime numbers. We were also able to define a function with the same asymptotic +behavior of π(x) but defined in terms of primorial n-totative numbers. We study +this relationship since any prime number lower or equal than #(n) + 1 is a prime +number pi with i = 1, 2, . . . , n or a primorial n-totative number, i.e., π(x) can be +expressed in terms of tot(n). +Our future work will concentrate on defining the class of functions with the same +asymptotic behavior of π(x), defining a function with a smaller convergence ratio +to π(x) for small values of x. We will in-depth study such functions but defined +in terms of primorial n-totative numbers and define some function approximations +for counting twin, cousin, sexy primes, and a constellation of prime numbers in the +same way we did for π(x) in terms of n-totative numbers. + +REFERENCES +9 +π(x)/ +x +x +log(x) +x +log∗(x) +x +log⋄(x) +h◦(x) +log(h◦(x)) +x +log(h◦(x)) +101 +0.921 +0.392 +0.702 +1.101 +0.787 +102 +1.151 +0.683 +0.988 +1.327 +1.106 +103 +1.161 +0.770 +1.018 +1.305 +1.137 +104 +1.132 +0.820 +1.025 +1.236 +1.119 +105 +1.104 +0.840 +1.014 +1.227 +1.093 +106 +1.084 +0.858 +1.011 +1.180 +1.077 +107 +1.071 +0.875 +1.013 +1.186 +1.064 +108 +1.061 +0.881 +1.004 +1.145 +1.057 +109 +1.054 +0.885 +0.997 +1.137 +1.050 +1010 +1.048 +0.893 +0.998 +1.103 +1.045 +1011 +1.043 +0.902 +0.998 +1.089 +1.041 +1012 +1.039 +0.905 +0.995 +1.102 +1.037 +1013 +1.036 +0.910 +0.996 +1.081 +1.034 +1014 +1.033 +0.914 +0.995 +1.069 +1.032 +1015 +1.031 +0.919 +0.996 +1.072 +1.030 +1016 +1.029 +0.924 +0.997 +1.061 +1.028 +1017 +1.027 +0.926 +0.996 +1.077 +1.026 +1018 +1.025 +0.929 +0.996 +1.076 +1.024 +1019 +1.024 +0.931 +0.995 +1.065 +1.023 +1020 +1.023 +0.933 +0.995 +1.061 +1.022 +1021 +1.022 +0.935 +0.995 +1.046 +1.021 +1022 +1.021 +0.938 +0.995 +1.049 +1.020 +1023 +1.020 +0.940 +0.996 +1.042 +1.019 +1024 +1.019 +0.941 +0.995 +1.054 +1.018 +1025 +1.018 +0.943 +0.996 +1.048 +1.017 +Table 2. Asymptotic behavior of π(x), +x +log(x), +x +log∗(x), +x +log⋄(x), +h◦(x) +log(h◦(x)), and +x +log(h◦(x)). +Results are obtained with the Python +program (PNTprimorials.py) freely available at Professor Jonatan +Gomez github repository [Gom]. +References +[Eul63] +Leonhard Euler. “Theoremata arithmetica nova methodo demonstrata”. +In: Novi commentarii academiae scientiarum imperialis Petropolitanae +8 (1763), pp. 74–104. +[Tch52] +P. Tchebychev. “Mémoire sur les nombres premiers”. In: mathématiques +pures et appliquées, Série 1 (1852), pp. 371–382. url: http://sites.mathdoc.fr/JMPA/PDF/JMPA_1852_1_17_A19_0.pdf. +[Mer74] +Franz Mertens. “Ein Beitrag zur analytischen Zahlentheorie.” ger. In: +Journal für die reine und angewandte Mathematik 78 (1874), pp. 46–62. +url: http://eudml.org/doc/148244. +[Had96] +Jacques Hadamard. “Sur la distribution des zéros de la fonction ζ(s) et +ses conséquences arithmétiques”. In: Bulletin de la Société Mathématique +de France 24 (1896). + +REFERENCES +10 +[Val96] +Charles-Jean de la Vallée Poussin. “Recherches analytiques sur la théorie +des nombres premiers”. In: Annales de la Société scientifique de Bruxelles +(1896). +[Suz13] +T. Suzuki. “A Theorem on the Series of Prime Numbers”. In: Tôhuku +Math 3 (1913), pp. 83–86. url: https://www.jstage.jst.go.jp/article/tmj1911/3/0/3_0_83/_pdf/-char/ja. +[Dub87] +Harvey Dubner. “Factorial and primorial primes”. In: Recreational Math +21(4) (1987). +[Nar00] +Władysław Narkiewicz. The Development of Prime Number Theory: From +Euclid to Hardy and Littlewood. Monographs on mathematics. Springer, +2000. isbn: 9783642085574. +[Gom23] +Jonatan Gomez. On Primorial Numbers. 2023. +[Gom] +Jonatan Gomez. Prime Numbers Programs. url: https://github.com/jgomezpe/primenumbers. +(accessed: 5.1.2023). + diff --git a/I9E2T4oBgHgl3EQfAAZV/content/tmp_files/load_file.txt b/I9E2T4oBgHgl3EQfAAZV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..76f80b5139ce871dc8a7dbfab257af2494e1d71b --- /dev/null +++ b/I9E2T4oBgHgl3EQfAAZV/content/tmp_files/load_file.txt @@ -0,0 +1,476 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf,len=475 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='03586v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='NT] 7 Jan 2023 THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS JONATAN GOMEZ JGOMEZPE@UNAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='EDU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='CO UNIVERSIDAD NACIONAL DE COLOMBIA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Counting the number of prime numbers up to a certain natural number and describing the asymptotic behavior of such a counting function has been studied by famous mathematicians like Gauss, Legendre, Dirichlet, and Euler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' The prime number theorem determines that such asymptotic behavior is similar to the asymptotic behavior of the number divided by its natural logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' In this paper, we take advantage of a multiplicative representa- tion of a number and the properties of the logarithm function to express the prime number theorem in terms of primorial numbers, and n-primorial tota- tive numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' A primorial number is the multiplication of the first n prime numbers while n-primorial totatives are the numbers that are coprime to the n-th primorial number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' By doing this we can define several different functions that can be used to approximate the behavior of the prime counting function asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Prime Numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' A prime number is a natural number having exactly two factors, 1 and itself [Nar00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' The natural number 1 is not considered a prime number since it has only one factor (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' From now on, pn will denote the nth prime number, here n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' The first eleven prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Prime Numbers Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' [Nar00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Let π(x) be the prime number-counting function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', the function that determines the number of primes less than or equal to a natural number x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' The prime number theorem proved independently by Hadamard [Had96] and de la Vallée Poussin [Val96], describes the asymptotic behavior of π(x) as follows: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='1) lim x→∞ π(x) � x log(x) � = 1 Using asymptotic notation, we can rewrite the prime number theorem as follows: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='2) π(x) ∼ x log(x) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='2 shows the asymptotic behavior of π(x) and x log(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Results for π(x) are taken from https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='org/wiki/Prime_number_theorem and results for x log(x) are obtained with the Python program (PNTprimorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='py) freely available at professor Jonatan Gomez github repository [Gom].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 1 THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS 2 x π(x) π(x)/ x log(x) 101 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='921 102 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='151 103 168 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='161 104 1229 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='132 105 9592 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='104 106 78498 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='084 107 664579 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='071 108 5761455 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='061 109 50847534 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='054 1010 455052511 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='048 1011 4118054813 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='043 1012 37607912018 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='039 1013 346065536839 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='036 1014 3204941750802 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='033 1015 29844570422669 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='031 1016 279238341033925 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='029 1017 2623557157654233 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='027 1018 24739954287740860 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='025 1019 234047667276344607 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='024 1020 2220819602560918840 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='023 1021 21127269486018731928 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='022 1022 201467286689315906290 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='021 1023 1925320391606803968923 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='020 1024 18435599767349200867866 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='019 1025 176846309399143769411680 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='018 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Asymptotic behavior of π(x) and x log(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Re- sults for x log(x) are obtained with the Python program (PNT- primorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='py) freely available at professor Jonatan Gomez github repository [Gom].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Results for π(x) are taken from https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='org/wiki/Prime_number_theorem and Now, we write down some Theorems about prime numbers that have been proven previously in the literature, and we derive some technical properties from such Theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (Bertrand’s postulate) For any positive natural number n we have that pn+1 ≤ 2pn − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Chebyshev proved this Theorem in [Tch52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' For any positive natural number n we have that log � pn+2 pn � < 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' By applying Theorem 1 twice we have that pn+2 < 2pn+1 < 4pn for all positive natural number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Clearly, pn+2 pn < 4 so, log � pn+2 pn � < log(4) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' For any positive natural number m there exists a positive natural number N such that pm n+1 < �n i=1 pi for all n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Suzuki proved this Theorem in [Suz13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' For any positive natural number m there exists a positive natural number N such that log(pn+1) log( �n i=1 pi) < 1 m for all n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Consider inequality in Theorem 4, apply log function on both sides of the inequality, divide both sides by the right side, apply property log(xm) = m ∗ log(x) on the left side, and divide both sides of the inequality by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Functions f(x) = � p≤x p p−1 and h(x) = eγ ∗ log(x) have the same asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Here γ is the Euler’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' This Theorem can be proven as a Corollary of the Theorem proved by Mertens in [Mer74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Functions f •(x) = � p≤x p and h•(x) = eγ ∗ log(x)∗ � p≤x p− 1 have the same asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Here γ is the Euler’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Primorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Many interesting operations can be defined over prime numbers, for example, we can define the primorial of the nth prime number as the product of the first n ∈ N+ prime numbers [Dub87], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', #(n) = pn# = �n i=1 pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' This definition is similar to the definition of the factorial function, so it is possible to define the primorial as a recursive function #(0) = 1 and #(n) = pn#(n − 1) for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' The first five primorials are 2, 6, 30, 210, 2310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Coprime or Relative-prime Numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Two natural numbers a, b ∈ N are coprime or relative-prime numbers iff their greatest common divisor is 1 (gcd(a, b) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' According to this definition, i) 1 is relative-prime with any other positive natural number, ii) 0 is not relative-prime with any natural number, and iii) any prime number is relative-prime with any other prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Notice that we can define prime numbers in terms of coprime numbers: A natural number p > 1 is a prime number iff p is relative-prime to q for all natural numbers 1 ≤ q < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Natural numbers less than n (Zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Let n ∈ N, the set of natural numbers less than n, is Zn = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', n−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' For example, Z2 = {0, 1} and Z#(2) = Z2∗3 = Z6 = {0, 1, 3, 4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Amount the properties hold by Zn, we are especially interested in the structure of the subset of natural numbers that are relative-prime to #(n) (sometimes called as totative numbers of #(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' For instance, consider Z#(2) = Z6, the subset of Z6 defined by the natural numbers that are relative-prime to 6 is {1, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' It is clear that any prime number q such pn < q < #(n) will be a relative-prime number to #(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Euler’s totient function (ϕ(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Euler’s totient function [Eul63] counts the natural numbers, in Zn, which are relative-prime to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Take for example Z#(3) = Z30, the subset of natural numbers relative-prime to 30 is {1, 7, 11, 13, 17, 19, 23, 29}, therefore ϕ(30) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Euler’s totient function is a multiplicative function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', ϕ(ab) = ϕ(a)ϕ(b) for two coprime numbers a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Moreover, for a prime number p, we have that Zp − {0} is the set of totative numbers of p, therefore ϕ(p) = p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Using these two properties of Euler’s totient function, we can easily compute it on primorial numbers: ϕ(#(n)) = �n i=1(pk − 1) = �n i=1 ϕ(pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Primorial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' We define a n-primorial set by ’replacing’ numbers 0 and 1 in Z#(n), with numbers #(n) and #(n) + 1, respectively [Gom23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Let n ∈ N, the n-primorial set is defined as Z# n = {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', #(n), #(n) + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' The following is the list of the first five n-primorial sets: (1) Z# 1 = {2, 3} (2) Z# 2 = {2, 3, 4, 5, 6, 7} (3) Z# 3 = {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', 30, 31} (4) Z# 4 = {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', 210, 211} (5) Z# 5 = {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', 2310, 2311} 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Primorial totatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' We can extend the notion of totatives of #(n) to set Z# n by considering the subset of natural numbers in Z# n being relative-prime to #(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' The following is the list of the first three n-totative sets: (1) The 1-totative set is tot(1) = {3} (2) The 2-totative set is tot(2) = {5, 7} (3) The 3-totative set is tot(3) = {7, 11, 13, 17, 19, 23, 29, 31} Notice that any prime number p ∈ Z# n must be either pi for some i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' , n or a n-totative number, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', we can express the Prime Number Theorem in terms of totative numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' We explore this relationship in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' The number of n-totatives is tot(n) = �n i=1(pk − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' #(n) and (#(n) + 1) are coprime numbers, therefore the number of n- totatives is equal to the totatives of #(n), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='e, the number of n-totatives is ϕ(#(n)) = �n i=1 ϕ(pk) = �n i=1(pk − 1) = tot(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' The log Function and Multiplicative Number Representations We can establish relationships between the prime number theorem and primorial numbers if we represent numbers greater or equal to one in terms of multiplicative fractions of prime or primorial numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Before that, we develop a technical Lemma for analyzing asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Let f, g, and h be real number functions such that 0 < f(x) ≤ g(x) ≤ h(x) for all x greater than certain number N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' If f and h have the same asymptotic behavior then f, g, and h have the same asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Obvious, 1 = f(x) f(x) ≤ g(x) f(x) ≤ h(x) f(x), by the Squeeze or sandwich Theorem we have 1 = limx→∞ f(x) f(x) ≤ limx→∞ g(x) f(x) ≤ limx→∞ h(x) f(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Let A = {ai}i∈N+ be a monotonic increasing succession such that a1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' For any number x ≥ a1, any positive natural number n, and 0 ≤ r < 1: (1) n(x) as the positive natural number such that an(x) ≤ x < an(x)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (2) r(x) = x−an(x) an(x)+1−an(x) , (3) s(n, r) = 1 + � an+1 an − 1 � r, (4) s(x) = s(n(x), r(x)), and (5) y(n, r) = an ∗ s(n, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Notice that s(n, r) = 1 + � an+1 an − 1 � r = 1 + � an+1−an an � r = an+1r+an(1−r) an is kind of the r-th multiplicative fraction of an+1 respect to succession A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS 5 Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Let A = {ai}i∈N+ be a monotonic increasing succession such that a1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' For any number x ≥ a1, any positive natural number n, and 0 ≤ r < 1 (1) an ≤ y(n, r) < an+1, (2) x = y(n(x), r(x)) = an(x) ∗ s(x) (3) m = n(y(m, t)), and t = r(y(m, t)) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (1) y(n, r) = an ∗s(n, r) = an � an+1r+an(1−r) an � = an+1r+an(1−r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Clearly, y(n, r) is a convex combination between an+1 and an, then an ≤ y(n, r) < an+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (2) an(x) ∗ s(x) = an(x) � 1 + � an(x)+1−an(x) an(x) � � x−an(x) an(x)+1−an(x) �� = an(x) ∗ s(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', an(x) ∗ s(x) = an(x) � 1 + x−an(x) an(x) � = an(x) � x an(x) � = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (3) Follows from (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ We can define, in particular, two different representations of a number x ≥ 2 if we consider Lemma 10: (1) The prime representation of x ≥ 2 uses the succession of prime numbers (A = {pn}n∈N+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Therefore, x = pn⋆(x)∗s⋆(x) with n⋆(x) such that pn⋆(x) ≤ x < pn⋆(x)+1, s⋆(x) = 1+ � pn⋆(x)+1 pn⋆(x) − 1 � r⋆(x), and r⋆(x) = x−pn⋆(x) pn⋆(x)+1−pn⋆(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (2) The primorial representation of x ≥ 2 uses the succession of primorial numbers (A = {#(n)}n∈N+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Therefore, x = #(n′(x)) ∗ s′(x) with n′(x) such that #(n′(x)) ≤ x < #(n′(x) + 1), s′(x) = 1 + � pn′(x)+1 − 1 � r′(x), and r′(x) = x−#(n′(x)) #(n′(x)+1)−#(n′(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' We can take advantage of the prime representation of a number for computing the log(x) function, see Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='1) log(x) = log � pn⋆(x) ∗ s⋆(x) � = log(pn⋆(x)) + log(s⋆(x)) We can manipulate Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='1 to produce functions having the asymptotic behavior of log(x), see Equations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='2) log−(x) = log � pn⋆(x)−1 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='3) log+(x) = log � pn⋆(x)+1 � Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Functions log−(x) and log+(x) have the same asymptotic behavior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', lim x→∞ log−(x) log+(x) = 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Given a real number ǫ > 0, we can take n as the smallest natural number such that 2 log(pn) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Now, we consider M = pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' If x > M then n⋆(x) > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' We have log+(x) − log−(x) = log(pn⋆(x)+1) − log(pn•(x)−1) = log � pn⋆(x)+1 pn⋆(x)−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' By Corol- lary 2 we have log+(x) − log−(x) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Clearly, ��� log+(x) log−(x) − 1 ��� = log+(x)−log−(x) log−(x) < 2 log−(x) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Functions: (1) log−(x) THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS 6 (2) loga(x) = log−(x) + log(a(x)) with 1 ≤ a(x) ≤ pn⋆(x)+1 pn⋆(x)−1 (3) log(x) (4) log⋆(x) = log(pn⋆(x)) (5) log∗(x) = log−(x) + log � 1 + � pn⋆(x) pn⋆(x)−1 − 1 � r⋆(x) � , and (6) log+(x) Have the same asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Since every function log?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (x) defined in (2)-(6) satisfies log−(x) ≤ log?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (x) ≤ log+(x), then by Theorem 11 and Lemma 8 the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ Now, we take advantage of the primorial representation of a number for com- puting the log(x) function, see Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='4) log(x) = log (#(n′(x)) ∗ s′(x)) = log(s′(x))+log \uf8eb \uf8ed n′(x) � i=1 pi \uf8f6 \uf8f8 = log(s′(x))+ n′(x) � i=1 log(pi) We can manipulate Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='4 to produce functions with the asymptotic be- havior of log(x), see Equations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='5) log−(x) = log (#(n′(x) − 1)) = log \uf8eb \uf8ed n′(x)−1 � i=1 pi \uf8f6 \uf8f8 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='6) log+(x) = log (#(n′(x) + 1)) = log \uf8eb \uf8ed n′(x)+1 � i=1 pi \uf8f6 \uf8f8 Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Functions log−(x) and log+(x) have the same asymptotic behavior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', lim x→∞ log−(x) log+(x) = 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Given a real number ǫ > 0, we can take m as the smallest natural number such that 2 m < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Now, by Theorem 3 we can take N as the smallest natural number such that pm n+1 < #(n) for all n ≥ N, and by Corollary 4, we have log(pn+1) log(#(n)) < 1 m for all n ≥ N (⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' If x > M = #(N) then n′(x) ≥ N and log(pn′(x)+1) log(#(n′(x))) < 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Notice that log(#(n′(x) + 1)) − log(#(n′(x) − 1)) = log � #(n′(x)+1) #(n′(x)−1) � = log(pn′(x) ∗ pn′(x)+1) < log(p2 n′(x)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Clearly, ��� log−(x) log+(x) − 1 ��� = log+(x)−log−(x) log+(x) < 2 log(pn′(x)+1) log(#(n′(x)+1)) < 2 log(pn′(x)+1) log(#(n′(x))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Finally, using result (⋆) we have ��� log∗(x) log(x) − 1 ��� < 2 m < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Functions: (1) log−(x) (2) loga(x)(x) = log−(x) + log(a(x)) with 1 ≤ a(x) ≤ pn′(x) ∗ pn′(x)+1 (3) log(x) (4) log#(x) = log(#(n′(x))) (5) log∗(x) = log−(x) + log(1 + (pn′(x) − 1)r′(x)) THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS 7 (6) log⋄(x) = log−(x) + log(n′(x) + r′(x)), and (7) log+(x) Have the same asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Since every function log?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (x) defined in (2)-(6) satisfies log−(x) ≤ log?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (x) ≤ log+(x), then by Theorem 13 and Lemma 8 the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Primorials and n-Totative Numbers Notice that if we consider x = pn, Theorem 16 and Corollary 17 suggest a relationship between the primorial number #(n) and the quantity of n-totatives numbers when n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Upon this suggestion, we propose a relationship between any positive number x ≥ 2 and n-totative numbers by considering the primorial representation of a number and defining a function that approximates the quantity of n-totative numbers up to x, see Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='1) tot∗(x) = tot(n′(x)) ∗ t∗(x) = t∗(x) ∗ n′(x) � i=1 pi − 1 Here, t∗(x) = 1+(pn′(x)+1 −2)∗r′(x), with n′(x) and r′(x) as defined in Section 2, and tot(n) as defined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Before establishing the relationship, we need the following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 1 ≤ 1+(a−1)r 1+(a−2)r < a a−1 for all a > 1 and all 0 ≤ r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Consider r = 0: clearly, 1+(a−1)r 1+(a−2)r = 1 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Consider 0 < r < 1: clearly a − 2 < a − 1 then 1 + (a − 2)r < 1 + (a − 1)r, so 1 < 1+(a−1)r 1+(a−2)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Now r < 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', −1 + r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' By adding a + a2r − 2ar on both sides of the inequality and organizing terms we have a − 1 + a2r − 2ar + r < a + a2r − 2ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Grouping terms we have (a − 1)(1 + (a − 1)r) < a(1 + (a − 2)r) and dividing both sides of inequality by (a − 1)(1 + (a − 2)r) we have 1+(a−1)r 1+(a−2)r < a a−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ Now, we follow a similar scheme as the one we used in the previous section to establish the relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Functions f(x) = �n′(x) i=1 pi pi−1 and g(x) = �n′(x)+1 i=1 pi pi−1 have the same asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Obvious, limx→∞ g(x) f(x) = limx→∞ pn′(x)+1 pn′(x)+1−1 = 1+limx→∞ 1 pn′(x)+1−1 = 1 □ Corollary 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Function f ◦(x) = x tot∗(x) has the same asymptotic behavior of func- tions f(x) = �n′(x) i=1 pi pi−1 and g(x) = �n′(x)+1 i=1 pi pi−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Clearly, x tot∗(x) = s′(x)∗#(n′(x)) t∗(x)∗�n′(x) i=1 pi−1 = f(x)∗ s′(x) t∗(x) = f(x)∗ � 1+(pn′(x)+1−1)r′(x) 1+(pn′(x)+1−2)r′(x) � by Lemma 15 we have f(x) ≤ x tot∗(x) < f(x) ∗ pn′(x)+1 pn′(x)+1−1 = g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Then by Theorem 16 and Lemma 8 the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ Finally, we use Theorem 16 to establish the desired relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' THE PRIME NUMBER THEOREM AND PRIMORIAL NUMBERS 8 Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Functions f(x) = �n′(x) i=1 pi pi−1 and g◦(x) = eγ ∗ log(y(x)) with γ the Euler’s constant and y(x) = pn′(x) ∗ � 1 + � pn′(x)+1 pn′(x) − 1 � r′(x) � have the same asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Given a real number ǫ > 0, we consider Theorem 16 and take M such that ��� � p≤z p p−1 eγ∗log(z) − 1 ��� < ǫ for all z > y(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' If x > M then y(x) > y(M) and ��� � p≤y(x) p p−1 eγ∗log(y(x)) − 1 ��� < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' We can see that �n′(x) i=1 pi pi−1 = � p≤y(x) p p−1 therefore, ���� �n′(x) i=1 pi pi−1 eγ∗log(y(x)) − 1 ���� < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ Corollary 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Functions f ◦(x) = x tot∗(x) and g◦(x) = eγ ∗ log(y(x)) with γ the Euler’s constant and y(x) = pn′(x) ∗ � 1 + � pn′(x)+1 pn′(x) − 1 � r′(x) � have the same as- ymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Follows from Corollary 17 and Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ Corollary 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Functions id(x) = x and h◦(x) = tot∗(x) ∗ eγ ∗ log(y(x)) with γ the Euler’s constant and y(x) = pn′(x) ∗ � 1 + � pn′(x)+1 pn′(x) − 1 � r′(x) � have the same asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Follows from Corollary 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Putting All Together Notice that functions having the same asymptotic behavior as functions id(x) and log(x) can replace them in the prime number theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' So, we can express the Prime Number Theorem in several ways, one of them using n-totative numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Table 4 shows the asymptotic behavior of π(x) vs x log(x), x log∗(x), x log⋄(x), h◦(x) log(h◦(x)), and x log(h◦(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' These results are obtained with the Python program (PNTprimorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='py) freely available at Professor Jonatan Gomez github repository [Gom].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Conclusions and Future Work We have developed several functions having the same asymptotic behavior of π(x) the prime number counting function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' We do this by representing any posi- tive positive number in terms of primorial numbers and multiplicative fractions of prime numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' We were also able to define a function with the same asymptotic behavior of π(x) but defined in terms of primorial n-totative numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' We study this relationship since any prime number lower or equal than #(n) + 1 is a prime number pi with i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' , n or a primorial n-totative number, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=', π(x) can be expressed in terms of tot(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Our future work will concentrate on defining the class of functions with the same asymptotic behavior of π(x), defining a function with a smaller convergence ratio to π(x) for small values of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' We will in-depth study such functions but defined in terms of primorial n-totative numbers and define some function approximations for counting twin, cousin, sexy primes, and a constellation of prime numbers in the same way we did for π(x) in terms of n-totative numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' REFERENCES 9 π(x)/ x x log(x) x log∗(x) x log⋄(x) h◦(x) log(h◦(x)) x log(h◦(x)) 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='392 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='702 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='787 102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='683 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='988 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Results are obtained with the Python program (PNTprimorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='py) freely available at Professor Jonatan Gomez github repository [Gom].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' References [Eul63] Leonhard Euler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' “Theoremata arithmetica nova methodo demonstrata”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' In: Novi commentarii academiae scientiarum imperialis Petropolitanae 8 (1763), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 74–104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' [Tch52] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Tchebychev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' “Mémoire sur les nombres premiers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' In: mathématiques pures et appliquées, Série 1 (1852), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 371–382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' url: http://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='mathdoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='fr/JMPA/PDF/JMPA_1852_1_17_A19_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' [Mer74] Franz Mertens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' “Ein Beitrag zur analytischen Zahlentheorie.” ger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' In: Journal für die reine und angewandte Mathematik 78 (1874), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 46–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' url: http://eudml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='org/doc/148244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' [Had96] Jacques Hadamard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' “Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' In: Bulletin de la Société Mathématique de France 24 (1896).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' REFERENCES 10 [Val96] Charles-Jean de la Vallée Poussin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' “Recherches analytiques sur la théorie des nombres premiers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' In: Annales de la Société scientifique de Bruxelles (1896).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' [Suz13] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Suzuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' “A Theorem on the Series of Prime Numbers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' In: Tôhuku Math 3 (1913), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 83–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='jstage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='jst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='jp/article/tmj1911/3/0/3_0_83/_pdf/-char/ja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' [Dub87] Harvey Dubner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' “Factorial and primorial primes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' In: Recreational Math 21(4) (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' [Nar00] Władysław Narkiewicz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' The Development of Prime Number Theory: From Euclid to Hardy and Littlewood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Monographs on mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Springer, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' isbn: 9783642085574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' [Gom23] Jonatan Gomez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' On Primorial Numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' [Gom] Jonatan Gomez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' Prime Numbers Programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' url: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='com/jgomezpe/primenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content=' (accessed: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} +page_content='2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfAAZV/content/2301.03586v1.pdf'} diff --git a/INE1T4oBgHgl3EQfrwUe/content/tmp_files/2301.03357v1.pdf.txt b/INE1T4oBgHgl3EQfrwUe/content/tmp_files/2301.03357v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7fccc6c09bd44d9df3c3bbb3c3c81234a9b5d0e --- /dev/null +++ b/INE1T4oBgHgl3EQfrwUe/content/tmp_files/2301.03357v1.pdf.txt @@ -0,0 +1,1422 @@ +1 + +d-wave superconductivity as a model for diborides apart MgB2 +Evgeny F. Talantsev1,2,* +1M.N. Miheev Institute of Metal Physics, Ural Branch, Russian Academy of Sciences, +18, S. Kovalevskoy St., Ekaterinburg, 620108, Russia +2NANOTECH Centre, Ural Federal University, 19 Mira St., Ekaterinburg, 620002, +Russia +*corresponding author’s E-mail: evgney.f.talantsev@gmail.ru + +Abstract +Recently, Pei et al. (arXiv2105.13250) reported that ambient pressure 𝛽-MoB2 (space group: +𝑅3̅𝑚) exhibits a phase transition to 𝛼-MoB2 (space group: 𝑃6/𝑚𝑚𝑚) at pressure P ~ 70 GPa +and this high-pressure phase is a high-temperature superconductor exhibited 𝑇𝑐 = 32 𝐾 at +P~110 GPa. Despite 𝛼-MoB2 has the same crystalline structure as ambient pressure MgB2 +and the 𝑇𝑐’s of 𝛼-MoB2 and MgB2 are very close, the first principles calculations showed that +in 𝛼-MoB2 the states near the Fermi level, 𝜀𝐹, are dominated by the d-electrons of Mo atoms, +while in MgB2 the p-orbitals of boron atomic sheets dominantly contribute to the states near +the 𝜀𝐹. More recently, Hire et al. (arXiv2212.14869) reported that the 𝑃6/𝑚𝑚𝑚-phase can +be stabilized at ambient pressure in Nb1-xMoxB2 solid solutions, and these ternary alloys +exhibit 𝑇𝑐~8 𝐾. In addition, Pei et al. (Sci. China-Phys. Mech. Astron. 65, 287412 (2022)) +showed that compressed WB2 exhibits 𝑇𝑐~15 𝐾 at P~121 GPa. Here, we analyzed +experimental data reported for P6/mmm-phases of Nb1-xMoxB2 (x = 0.25; 1.0) and highly- +compressed WB2, and showed that these three phases exhibit d-wave superconductivity. We +deduced +2Δ𝑚(0) +𝑘𝐵𝑇𝑐 = 4.1 ± 0.2 for 𝛼-MoB2, +2Δ𝑚(0) +𝑘𝐵𝑇𝑐 = 5.3 ± 0.1 for Nb0.75Mo0.25B2, and +2Δ𝑚(0) +𝑘𝐵𝑇𝑐 = +4.9 ± 0.2 for WB2. We found that Nb0.75Mo0.25B2 exhibits high strength of nonadiabaticity, +which is quantified by the ratio of +𝑇𝜃 +𝑇𝐹 = 3.5, which is by one order of magnitude exceeds the +ratio in MgB2, -MoB2, WB2, pnictides, cuprates, and highly-compressed hydrides. + + +2 + +d-wave superconductivity as a model for diborides apart MgB2 +I. Introduction. +The discovery of near-room temperature superconductivity in highly compressed sulphur +hydride by Drozdov et al [1] sparked theoretical and experimental studies of a variety of +materials which potentially can exhibit a high-temperature superconductivity to be +compressed at high pressure [2-25]. This research field represents one of the most +fascinating scientific exploration where advanced first principles calculations conjuncts with +top world class of experimental studies [26-39]. +One of the interesting results in this conjunctive exploration has been reported by Pei et al +[40] who found that the stoichiometric compound MoB2 exhibits the phase transition from +the 𝛽-MoB2-phase (space group: 𝑅3̅𝑚) to 𝛼-MoB2-phase (space group: 𝑃6/𝑚𝑚𝑚) at critical +pressure P ~ 70 GPa. This high-pressure phase, 𝛼-MoB2, exhibits the same crystalline +structure as ambient pressure MgB2 and, what is the most intriguing experimental result +reported by Pei et al [40], the 𝛼-MoB2 phase is a high-temperature superconductor with 𝑇𝑐 = +32 𝐾 (at P = 109.7 GPa), which is remarkably close to 𝑇𝑐 = 39 − 42 𝐾 in MgB2 [41,42]. +First principles calculations performed by Pei et al [40] showed that several bands in the +𝛼-MoB2 crossing the Fermi level, 𝜀𝐹, which causes the metallic type of conductivity in this +phase. Pei et al [40] also showed the molybdenum d-orbitals (especially the dz2 orbital) have +larger contributions than the boron p-orbitals near the 𝜀𝐹. In overall, despite 𝛼-MoB2 phase +exhibits the same crystal structure as MgB2 and the superconducting transition temperature +for these compounds are comparable, their electronic structures are different. For instance, +the out-of-plane phonon mode of molybdenum ions are strongly coupled with molybdenum +d-electrons near the 𝜀𝐹 in 𝛼-MoB2 [40], while the in-plane boron-boron stretching mode in +MgB2 interacts intensively with the σ-bond in the boron honeycomb lattice near the 𝜀𝐹 [40]. +Pei et al [40] also calculated the electron-phonon coupling constant, 𝜆𝑒−𝑝ℎ = 1.60, in 𝛼- + +3 + +MoB2 at 𝑃 = 90 𝐺𝑃𝑎. Similar findings, including 𝜆𝑒−𝑝ℎ = 1.60, were reported by Quan et al +[43] who performed first principles calculations for highly-pressurized 𝛼-MoB2 phase. +These results give a ground to expect that the 𝛼-MoB2 phase can exhibit d-wave +superconducting energy gap symmetry (or, at least, s+d-wave gap symmetry with significant +d-wave component), which is different from the two-band s-wave MgB2. +More recently, Hire et al [44] showed that the 𝑃6/𝑚𝑚𝑚-phase can be stabilized at +ambient pressure in Nb1-xMoxB2 (x = 0.25; 0.50; 0.75 and 0.9) solid solutions. Despite the +superconducting transition temperature in Nb1-xMoxB2 (x = 0.25; 0.50; 0.75 and 0.9) were +significantly lower (i.e., 𝑇𝑐 = (6.5 − 8.1) 𝐾 [44]) these values are still high enough to make a +proposal that the same pairing mechanism emerges in ambient pressure superconductors Nb1- +xMoxB2 and highly-pressurized 𝛼-MoB2. +Hire et al [44] also performed first principles calculation, measurements of the +temperature dependent magnetoresistance 𝑅(𝑇, 𝐵), and specific heat measurements from +which several parameters of Nb1-xMoxB2 (x = 0.25; 0.50; 0.75 and 0.9) superconductors (and, +in particular, the Debye temperature, 𝑇𝜃) were determined. +Pei et al [45] and Lim et al [46] extended the range of superconducting diborides by the +discovery of highly-compressed phase of WB2 (𝑇𝑐~15 𝐾 at P~121 GPa) for which Pei et al +[45] proposed space group: P63/mmc (which is distorted P6/mmm), while Lim et al [46] +concluded that this highly-pressurized superconducting phase of WB2 formed by staking +faulted P63/mmc-P6/mmm phase (which can be found to be similar to the stacking faulted +123-124 phases in Y-Ba-Cu-O system [47-49]). +Here, we performed detailed analysis of the magnetoresistance data reported by Pei et al +[40], Hire et al [44], Pei et al [45], and showed that the P6/mmm-phases of Nb1-xMoxB2 (x = +0.25; 1.0) and WB2 (P=121.3 GPa) exhibit the d-wave superconducting gap symmetry. We +also found that ambient pressure Nb1-xMoxB2 (x = 0.25) superconductors characterized by + +4 + +high strength of nonadiabaticity, which can be characterized by the ratio of +𝑇𝜃 +𝑇𝐹 = 3.5 (where +𝑇𝐹 is the Fermi temperature, which is by more than one order of magnitude exceeds the +𝑇𝜃 +𝑇𝐹 +ratio in MgB2, -MoB2, WB2, pnictides, cuprates, and highly-compressed hydrides. + +II. Results +2.1. P6/mmm 𝜶-MoB2 (𝑷 = 𝟏𝟎𝟗. 𝟕 𝑮𝑷𝒂) +2.1.1. Debye temperature and the electron-phonon coupling constant +Debye temperature, 𝑇𝜃, can be deduced from the fit of experimentally measured +temperature dependent resistance curve, 𝑅(𝑇), to the Bloch-Grüneisen (BG) equation +[50,51]. In many reports, classical BG approach is advanced by introducing so-called the +saturation resistance [52-57]: +𝑅(𝑇) = +1 +1 +𝑅𝑠𝑎𝑡+ +1 +𝑅0+𝐴( 𝑇 +𝑇𝜃 +) +5 +∫ +𝑥5 +(𝑒𝑥−1)(1−𝑒−𝑥) +𝑇𝜃 +𝑇 +0 +𝑑𝑥 + + + + + + + +(1) +where 𝑅𝑠𝑎𝑡, 𝑅0, 𝑇𝜃 and 𝐴 are free fitting parameters. From the deduced 𝑇𝜃 and measured 𝑇𝑐 +(which we defined by as strict as practically possible resistance criterion of +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚 → 0, where +𝑅𝑛𝑜𝑟𝑚 is the normal state resistance at the onset of the superconducting transition, see details +in [56]), the electron-phonon coupling constant, 𝜆𝑒−𝑝ℎ, can be calculated as unique root of +advanced McMillan equation [56]: +𝑇𝑐 = ( +1 +1.45) × 𝑇𝜃 × 𝑒 +−( +1.04(1+𝜆𝑒−𝑝ℎ) +𝜆𝑒−𝑝ℎ−𝜇∗(1+0.62𝜆𝑒−𝑝ℎ)) +× 𝑓1 × 𝑓2 +∗ + + + +(2) +where +𝑓1 = (1 + ( +𝜆𝑒−𝑝ℎ +2.46(1+3.8𝜇∗)) +3 2 +⁄ +) +1 3 +⁄ + + + + + + + +(3) +𝑓2 +∗ = 1 + (0.0241 − 0.0735 × 𝜇∗) × 𝜆𝑒−𝑝ℎ +2 +. + + + + + +(4) + +5 + +where 𝜇∗ is the Coulomb pseudopotential parameter, which we assumed (follow the approach +proposed in [40,44,46]) to be 𝜇∗ = 0.13 for Nb1-xMoxB2 (x = 0.25; 1.0) and WB2. +The fits of 𝑅(𝑇) datasets measured for 𝛼-MoB2 phase at 𝑃 = 91.4 𝑎𝑛𝑑 109.7 𝐺𝑃𝑎 [40] +to Eq. 1 together with deduced 𝑅𝑠𝑎𝑡, 𝑇𝜃, and 𝜆𝑒−𝑝ℎ, are shown in Fig. 1 (where we utilized +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.10 criterion to define 𝑇𝑐, because the same criterion was used by Pei et al [40] +to define the upper critical field in the same 𝛼-MoB2 sample). Deduced 𝜆𝑒−𝑝ℎ(91.4 𝐺𝑃𝑎) = +1.42 is in a good agreement with the value calculated by first principles calculations +𝜆𝑒−𝑝ℎ(90 𝐺𝑃𝑎) = 1.60 [40,43]. + +Figure 1. R(T) data for highly compressed 𝛼-MoB2 (P = 109.7 GPa) and data fit to Eq. 1 (raw data +reported by Pei et al [40]). Green balls indicate the bounds for which R(T) data was used for the fit to +Eq. 1. (a) Deduced 𝑇𝜃 = 301 ± 1 𝐾, 𝑇𝑐,0.10 = 26.6 𝐾, 𝜆𝑒−𝑝ℎ = 1.42, 𝑅𝑠𝑎𝑡 = 0.61 ± 0.02 Ω, fit +quality is 0.9998. (b) Deduced 𝑇𝜃 = 321 ± 1 𝐾, 𝑇𝑐,0.10 = 28.2 𝐾, 𝜆𝑒−𝑝ℎ = 1.41, 𝑅𝑠𝑎𝑡 = 0.50 ± +0.01 Ω, fit quality is 0.9998. 95% confidence bands are shown by pink shadow areas. +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0 +50 +100 +150 +200 +250 +300 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +a +‒MoB2 (P = 91.4 GPa) + raw R(T) + fit to BG model + fitted R(T) range + Tc,0.10 = 26.6 K +resistance () +Rsat = 0.61 ± 0.02  +T = 301 ± 1 K +e-ph = 1.42 +b +‒MoB2 (P = 109.7 GPa) + raw R(T) + fit to BG model + fitted R(T) range + Tc,0.10 = 28.2 K +resistance () +temperature (K) +Rsat = 0.50 ± 0.01  +T = 321 ± 1 K +e-ph = 1.41 + +6 + +2.1.2. Temperature dependent upper critical field +Pei et al [40] in their Figure 2,d utilized several resistance criteria +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚(𝑇) = +0.10, 0.50, 0.90 to derive the upper critical field, 𝐵𝑐2(𝑇), from measured 𝑅(𝑇, 𝐵, 𝑃 = +109.7 𝐺𝑃𝑎) curves. By following general logic [56,58,59] that as low as possible resistance +criterion should be in use, here we utilized the same criterion of +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.10, as the one +which was used to define the 𝑇𝑐 in Fig. 1 and by the lowest criterion to defined 𝐵𝑐2(𝑇) by Pei +et al [40]. +In Fig. 2,a the 𝐵𝑐2(𝑇) dataset is fitted to the equation for temperature dependent upper +critical field for s-wave superconductors [58-60]: +𝐵𝑐2(𝑇) = +𝜙0 +2∙𝜋∙𝜉2(0) ( +1.77−0.43( 𝑇 +𝑇𝑐) +2 ++0.07( 𝑇 +𝑇𝑐) +4 +1.77 +) +2 +× [1 − +1 +2𝑘𝐵𝑇 ∫ +𝑑𝜀 +𝑐𝑜𝑠ℎ2( +√𝜀2+Δ2(𝑇) +2𝑘𝐵𝑇 +) +∞ +0 +] +(5) +where the amplitude of temperature dependent superconducting gap, (T), is given by +[61,62]: +Δ(𝑇) = Δ(0) × tanh [ +𝜋𝑘𝐵𝑇𝑐 +Δ(0) √𝜂 +Δ𝐶 +𝛾𝑇𝑐 ( +𝑇𝑐 +𝑇 − 1)] + + + + + +(6) +where  = 2/3 for s-wave superconductors,  is Sommerfeld constant, and 𝑘𝐵 is the +Boltzmann constant. However, the deduced +2Δ(0) +𝑘𝐵𝑇𝑐 = 2.3 ± 0.1 (Fig. 2,a) is too low to be +attributed to s-wave superconductivity, for which the weak-coupling limit is +2Δ(0) +𝑘𝐵𝑇𝑐 = 3.53 +[63,64]. And also, the fit quality is low R =0.8267. +Then, we fitted the temperature dependent upper critical field data to the d-wave gap +symmetry model. The fitting function can be constructed similarly to its counterparts for s- +wave [58-60] and p-wave [59,60,65]: +𝐵𝑐2(𝑇) = +𝜙0 +2∙𝜋∙𝜉2(0) ( +1.77−0.43( 𝑇 +𝑇𝑐) +2 ++0.07( 𝑇 +𝑇𝑐) +4 +1.77 +) +2 +[1 − +1 +2∙𝑘𝐵∙𝑇 ∙ ∫ +𝑐𝑜𝑠2(𝜃) ∙ +2𝜋 +0 +(∫ +𝑑𝜀 +𝑐𝑜𝑠ℎ2(√𝜀2+Δ2(𝑇,𝜃) +2∙𝑘𝐵∙𝑇 +) +∞ +0 +) ∙ 𝑑𝜃] +(7) + +7 + +where the superconducting energy gap, (T,), is given by [61,62,65]: +Δ(𝑇, 𝜃) = 𝑐𝑜𝑠(2𝜃) × Δ𝑚(0) × tanh [ +𝜋𝑘𝐵𝑇𝑐 +Δ(0) √𝜂 +Δ𝐶 +𝛾𝑇𝑐 ( +𝑇𝑐 +𝑇 − 1)] + + + +(8) +where m(T) is the is the maximum amplitude of the k-dependent d-wave gap,  = 7/5 [65],  +is the angle around the Fermi surface subtended at (, ) in the Brillouin zone (details can be +found elsewhere [61,62]). + +Figure 2. The upper critical field, Bc2(T), data (defined by +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.10 criterion) for 𝛼-MoB2 +(𝑃 = 109.7 𝐺𝑃𝑎) reported by Pei et al [40] and data fits to s-wave (panel a) and d-wave (panel b) +single-band models. Deduced parameters are (for both panels the critical temperature was fixed to the +observed value of Tc = 28.2 K): (a) s-wave fit, (0) = 6.5(2) nm, Δ(0) = 2.8 ± 0.1 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 +⁄ += +1.5 ± 0.8, +2Δ(0) +𝑘𝐵𝑇𝑐 = 2.3 ± 0.2, the goodness of fit is 0.8267; (b) d-wave fit, (0) = 6.2(5) nm, Δ(0) = +5.0 ± 0.2 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 +⁄ += 0.8 ± 0.1, +2Δ(0) +𝑘𝐵𝑇𝑐 = 4.1 ± 0.2, the goodness of fit is 0.9842. + +0 +2 +4 +6 +8 +10 +0 +5 +10 +15 +20 +25 +30 +0 +2 +4 +6 +8 +(T) +Bc2(T) + Bc2(T) data and s-wave fit +Bc2(T) (T) +‒MoB2 (P = 109.7 GPa) +(0) = 2.8 ± 0.1 meV +2(0)/kBTc= 2.3 ± 0.1 +C/Tc = 1.5 ± 0.8 +0 +6 +12 +18 +24 +30 +a + (T) data and s-wave fit +(T) (nm) +(T) +Bc2(T) + Bc2(T) data and d-wave fit +Bc2(T) (T) +temperature (K) +(0) = 5.0 ± 0.2 meV +2(0)/kBTc= 4.1 ± 0.2 +C/Tc = 0.8 ± 0.1 +b +0 +6 +12 +18 +24 + (T) data and d-wave fit +(T) (nm) + +8 + +The fit converged with a better quality (with the goodness of fit is 0.9842) (Fig. 2,b). +Deduced parameters are: ξ(0) = 6.2(5) 𝑛𝑚, Δ(0) = 5.0 ± 0.2 𝑚𝑒𝑉, +2Δ(0) +𝑘𝐵𝑇𝑐 = 4.1 ± 0.2, +Δ𝐶 +𝛾𝑇𝑐 = +0.8 ± 0.1. Considering that the weak coupling limits for d-wave superconductors [61,62,65] +are: +2∙Δ(0) +𝑘𝐵∙𝑇𝑐 = 4.28 and +Δ𝐶 +𝛾𝑇𝑐 = 0.995, we can conclude that deduced parameters in 𝛼-MoB2 +(𝑃 = 109.7 𝐺𝑃𝑎) superconductor is within weak-coupling values for d-wave superconductor. +It should be noted, that the accuracy of the extracted parameters is directly related to the +sampling number of the measurement, and, thus, further increase in the accuracy in the +deduced parameters, can be possible if more raw 𝑅(𝑇, 𝐵) data (especially, measured at low +temperature, down to miliKelvin level) will be available. + +2.1.3. The Fermi temperature and the strength of the nonadiabaticity +The Fermi temperature can be calculated by the equation [58]: +𝑇𝐹 = +𝜋2𝑚𝑒 +8∙𝑘𝐵 × (1 + 𝜆𝑒−𝑝ℎ) × 𝜉2(0) × ( +2Δ𝑚(0) +ℏ +) +2 +, + + + +(9) +where 𝑚𝑒 is bare electron mass, ℏ is the reduced Planck’s constant, and other parameters +have deduced above. In the result, calculated Fermi temperature is 𝑇𝐹 = 1756 ± 25 𝐾. +Calculated 𝑇𝐹 implies that the P6/mmm 𝛼-MoB2 (𝑃 = 109.7 𝐺𝑃𝑎) phase falls in +unconventional superconductors band in the Uemura plot (Fig. 3), because this phase exhibits +typical for many unconventional superconductors (for instance, iron-based, cuprates and +hydrogen-rich superconductors) ratio of +𝑇𝑐 +𝑇𝐹 = 0.016. +Also, we found that the P6/mmm 𝛼-MoB2 (𝑃 = 109.7 𝐺𝑃𝑎) phase exhibits similar +level of the nonadiabaticy ( +𝑇𝜃 +𝑇𝐹 = 0.18 ± 0.02) to iron-based, cuprates and hydrogen-rich +superconductors [66,67] (Figs. 4,5). + + +9 + + + +Figure 3. Uemura plot (Tc vs TF), where the diborides are shown together with other superconducting +families: 2D materials, metals, pnictides, cuprates, and near-room-temperature superconductors. +Reference on original data can be found in Refs. 65,67-73. + + + +Figure 4. Plot of +𝑇𝜃 +𝑇𝐹 vs 𝜆𝑒−𝑝ℎ for several superconducting families and for diborides. This type of +plot proposed by Pietronero et al [66]. References on original data can be found in Refs. 65,67-71. + +10 +100 +1000 +10000 +100000 +0.1 +1 +10 +100 +2H-TaSeS +MgB2 +WB2 +Nb0.75Mo0.25B2 +-MoB2 +LiC6 +IrTe2 +bismuthates +(La,Nd)H10 +CsI +MATBG +intermetallics +Laves +-O2 +noncentrosymmetric +Heusler +A15 +metals +SrTiO3 +LaH10 +H3S +iron-based +H3S (from (0)) +H3S (from (0)) + metals + A15 alloys + Heusler alloys +noncentrosymmetric +Laves phase +intermetallics +SrTiO3 +Ba1-xKxBiO3 +iron-based SCs +cuprates ((0) and Jc(sf,T)) +LaH10 (from (0)) +LaH10 (from (0)) +(La,Nd)H10 (from (0)) +CsI (P=209 GPa) +MATBG (from Jc(sf,T)) +MATBG (from (0)) +IrTe2 (from Jc(sf,T)) +-O2 (P=115 GPa) +LiC6 +2H-TaSeS +Tc/TF = 0.001 +Tc/TF = 0.01 +Tc/TF = 0.05 +-MoB2 (P=110 GPa) +Nb0.75Mo0.25B2 +WB2 (P=121 GPa) +MgB2 +transition temperature, Tc (K) +Fermi temperature, TF (K) +BCS +cuprates +0.25 +0.5 +1 +2 +4 +0.001 +0.01 +0.1 +1 +10 +100 +MgB2 +WB2 +Nb0.75Mo0.25B2 +-MoB2 +LiC6 +metals +A15 +Laves +Heusler +-O2 +CsI +SrTiO3 +noncentrosymmetric +LaH10 +H3S +H3S (from (0)) +H3S (from (0)) + metals + A15 alloys + Heusler alloys +noncentrosymmetric +Laves phase +intermetallics +SrTiO3 +Ba1-xKxBiO3 +iron-based SCs +LaH10 (from (0)) +LaH10 (from (0)) +(La,Nd)H10 (from (0)) +CsI (P=209 GPa) +-O2 (P=115 GPa) +LiC6 +T/TF = 0.025 +T/TF = 0.4 +-MoB2 (P =110 GPa) +Nb0.75Mo0.25B2 +MgB2 +WB2 (P =121 GPa) +T/TF +electron-phonon coupling strength, e-ph + +10 + + + +Figure 5. Plot of +𝑇𝜃 +𝑇𝐹 vs 𝑇𝑐 for several superconducting families and for the diborides. References on +original data can be found in Refs. 65,67-72. + +2.2. Ambient pressure P6/mmm Nd0.75Mo0.25B2 +2.2.1. The electron-phonon coupling constant +Hire et al [20] in their Table 1 reported the Debye temperature for P6/mmm Nb1-xMoxB2 +(𝑥 = 0.25) which was deduced from low-temperature specific heat measurements, 𝑇𝜃 = +625 𝐾. Follow the approach implemented in this study, we processed 𝑅(𝑇, 𝐵 = 0) data +reported by Hire et al [20] by utilizing the resistance criterion of +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.015 and deduced +𝑇𝑐,0.015 = 7.2 𝐾, from which 𝜆𝑒−𝑝ℎ = 0.573 was calculated by Eqs. 2-4. + +2.2.2. Temperature dependent upper critical field +Hire et al [44] in their Figure 6 reported 𝑅(𝑇, 𝐵) data reported, which we processed by +utilizing the resistance criterion of +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.015 and deduced the 𝐵𝑐2(𝑇) dataset. The fits +of this dataset to s-wave (Eqs. 5,6) and d-wave model (Eqs. 7,8) are shown in Fig. 6. +0.1 +1 +10 +100 +0.001 +0.01 +0.1 +1 +10 +100 +2H-TaSeS +WB2 +MgB2 +Nb0.75Mo0.25B2 +-MoB2 +LiC6 +(La,Nd)H10 +Laves +intermetallics +Heusler +CsI +O2 +SrTiO3 +MATBG +A15 +metals +noncentrosymmetric +LaH10 +H3S +cuprates +H3S (from (0)) +H3S (from (0)) + metals + A15 alloys + Heusler alloys +noncentrosymmetric +Laves phase +intermetallics +SrTiO3 +iron-based SCs +cuprates ((0) and Jc(sf,T)) +LaH10 (from (0)) +LaH10 (from (0)) +(La,Nd)H10 (from (0)) +CsI (P=209 GPa) +MATBG (from Jc(sf,T)) +MATBG (from (0)) +2H-TaSeS +-O2 (P=115 GPa) +T/TF = 0.025 +T/TF = 0.4 +Ba1-xKxBiO3 +LiC6 +-MoB2 (P=110 GPa) +Nb0.75Mo0.25B2 +MgB2 +WB2 (P=121 GPa) +T/TF +transition temperature, Tc (K) + +11 + +The deduced parameters for s-wave (Fig. 6,a) contradict to each other, i.e +2Δ(0) +𝑘𝐵𝑇𝑐 = 3.18 ± +0.15 (which is lower than the s-wave weak-coupling limit is +2Δ(0) +𝑘𝐵𝑇𝑐 = 3.53 [63,64]), while +deduced +Δ𝐶 +𝛾𝑇𝑐 = 1.62 ± 0.19 is larger than s-wave weak-coupling limit of +Δ𝐶 +𝛾𝑇𝑐 = 1.43. And the +fit quality R =0.9534 is not high. + +Figure 6. The upper critical field, Bc2(T), data (defined by +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.015 criterion) for P6/mmm +Nb0.75Mo0.25B2 reported by Hire et al [44] and data fits to s-wave (panel a) and d-wave (panel b) +single-band models. Deduced parameters are (for both panels the critical temperature was fixed to the +observed value of Tc = 7.2 K): (a) s-wave fit, (0) = 8.0(7) nm, Δ(0) = 0.987 ± 0.038 𝑚𝑒𝑉, +Δ𝐶 𝛾𝑇𝑐 +⁄ += 1.6 ± 0.2, +2Δ(0) +𝑘𝐵𝑇𝑐 = 3.2 ± 0.1, the goodness of fit is 0.9534; (b) d-wave fit, (0) = 7.5(0) +nm, Δ(0) = 1.65 ± 0.05 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 +⁄ += 1.13 ± 0.03, +2Δ(0) +𝑘𝐵𝑇𝑐 = 5.3 ± 0.1, the goodness of fit is +0.9959. + +0 +1 +2 +3 +4 +5 +6 +0.0 +1.5 +3.0 +4.5 +6.0 +7.5 +0 +1 +2 +3 +4 +5 +6 +(T) +Bc2(T) + Bc2(T) data and s-wave fit +Bc2(T) (T) +Nb0.75Mo0.25B2 +(0) = 987 ± 38 eV +2(0)/kBTc= 3.2 ± 0.1 +C/Tc = 1.6 ± 0.2 +6 +9 +12 +15 +18 +21 +a + (T) data and s-wave fit +(T) (nm) +(T) +Bc2(T) + Bc2(T) data and d-wave fit +Bc2(T) (T) +temperature (K) +(0) = 1.65 ± 0.05 meV +2(0)/kBTc= 5.3 ± 0.1 +C/Tc = 1.13 ± 0.03 +b +6 +9 +12 +15 +18 +21 + (T) data and d-wave fit +(T) (nm) + +12 + +The fit to the d-wave gap symmetry model has a better quality (with the goodness of fit is +0.9959) (Fig. 6,b). Deduced parameters are: ξ(0) = 6.2(5) 𝑛𝑚, Δ(0) = 1.65 ± 0.05 𝑚𝑒𝑉, +2Δ(0) +𝑘𝐵𝑇𝑐 = 5.3 ± 0.1, +Δ𝐶 +𝛾𝑇𝑐 = 1.13 ± 0.03, characterize the material as moderately strong coupled +d-wave superconductor (considering that the weak coupling limits for d-wave +superconductors [61,62,65] are: +2∙Δ(0) +𝑘𝐵∙𝑇𝑐 = 4.28 and +Δ𝐶 +𝛾𝑇𝑐 = 0.995). + +2.2.3. The Fermi temperature and the strength of the nonadiabaticity +The substitution of deduced parameters in Eq. 9 returns the Fermi temperature 𝑇𝐹 = + 180 ± 7 𝐾 in P6/mmm-phase of Nb0.75Mo0.25B2. Calculated 𝑇𝐹 implies that this phase falls +in unconventional superconductors band in the Uemura plot (Fig. 3), because this phase +exhibits typical for many unconventional superconductors ratio of +𝑇𝑐 +𝑇𝐹 = 0.042. +However, what comes from our analysis and reported by Hire et al [44] the Debye +temperature, that P6/mmm-phase of Nb0.75Mo0.25B2 superconductor exhibits strong +nonadiabaticy, because the ratio: +0.4 ≪ +𝑇𝜃 +𝑇𝐹 = 3.5 ± 0.3 + + + + + + + +(10) +is well above typical range for moderate level of nonadiabaticity (0.025 ≤ +𝑇𝜃 +𝑇𝐹 ≤ 0.4) +observed in majority of unconventional superconductors, including iron-based, cuprates and +highly compressed hydrides [67] (Figs. 4,5). + +2.3. P63/mmc WB2 (P = 121.3 GPa) +2.3.1. The Debye temperature and the electron-phonon coupling constant +Pei et al [45] measured 𝑅(𝑇) datasets for WB2 phase at 𝑃 = 121.3 𝐺𝑃𝑎 which we fitted +to Eq. 1 in Fig. 7. The fit converged at 𝑇𝜃 = 440 ± 1 𝐾 and 𝑅𝑠𝑎𝑡 → ∞. From deduced 𝑇𝜃 we + +13 + +found 𝜆𝑒−𝑝ℎ = 0.755, for which we utilized the criterion of +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.18, which is based +on the presence of the inflection of the transition 𝑅(𝑇, 𝐵, 𝑃 = 121.3 𝐺𝑃𝑎) which can be seen +in Fig. 2(b,d) of Ref. 45. + + +Figure 7. R(T) data for highly compressed WB2 (P = 121.3 GPa) and data fit to Eq. 1 (raw data +reported by Pei et al [45]). Green balls indicate the bounds for which R(T) data was used for the fit to +Eq. 1. (a) Deduced 𝑇𝜃 = 440 ± 1 𝐾, 𝑇𝑐,0.18 = 12.5 𝐾, 𝜆𝑒−𝑝ℎ = 0.755, 𝑅𝑠𝑎𝑡 = ∞, fit quality is +0.9997. 95% confidence bands are shown by pink shadow areas. + +2.3.2. Temperature dependent upper critical field +By utilizing the resistance criterion of +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.18 for 𝑅(𝑇, 𝐵) data reported by Pei et +al [45] in their Figure 2,d we deduced the 𝐵𝑐2(𝑇) dataset for WB2 (P = 121.3 GPa). The fit of +the 𝐵𝑐2(𝑇) dataset to s-wave (Eqs. 5,6) and d-wave model (Eqs. 7,8) are shown in Fig. 8. +The deduced parameters for s-wave (Fig. 6,a) contradict to each other, i.e. +2Δ(0) +𝑘𝐵𝑇𝑐 = 2.8 ± +0.1 (which is lower than the s-wave weak-coupling limit of +2Δ(0) +𝑘𝐵𝑇𝑐 = 3.53 [63,64]), while +deduced +Δ𝐶 +𝛾𝑇𝑐 = 1.6 ± 0.4 is slightly larger than s-wave weak-coupling limit of +Δ𝐶 +𝛾𝑇𝑐 = 1.43. +And the fit quality R =0.9019 is not high. +0 +50 +100 +150 +200 +250 +300 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +WB2 (P = 121.3 GPa) +T = 440 ± 1 K +e-ph = 0.755 + raw R(T) + fit ot BG model + fitted R(T) range + Tc,inflection = 12.5 K +resistance () +temperature (K) + +14 + + + +Figure 8. The upper critical field, Bc2(T), data (defined by +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.015 criterion) for P63/mmc +WB2 (P = 121.3 GPa) reported by Pei et al [45] and data fits to s-wave (panel a) and d-wave (panel b) +single-band models. Deduced parameters are: (a) s-wave fit, Tc = 12.45 K (fixed), (0) = 13.8 nm, +Δ(0) = 1.48 ± 0.06 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 +⁄ += 1.6 ± 0.4, +2Δ(0) +𝑘𝐵𝑇𝑐 = 2.8 ± 0.1, the goodness of fit is 0.9019; (b) +d-wave fit, 𝑇𝑐 = 12.2 ± 0.2 𝐾, (0) = 13.0 nm, Δ(0) = 2.58 ± 0.02 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 +⁄ += 1.19 ± 0.07, +2Δ(0) +𝑘𝐵𝑇𝑐 = 4.9 ± 0.1, the goodness of fit is 0.9986. + +The fit to the d-wave gap symmetry model has a better quality (with the goodness of fit is +0.9986) (Fig. 8,b). Deduced parameters are: ξ(0) = 13.0 𝑛𝑚, Δ(0) = 2.58 ± 0.02 𝑚𝑒𝑉, +2Δ(0) +𝑘𝐵𝑇𝑐 = 4.9 ± 0.1, +Δ𝐶 +𝛾𝑇𝑐 = 1.19 ± 0.07, characterize the material as moderately strong coupled +d-wave superconductor (considering that the weak coupling limits for d-wave +superconductors [61,62,65] are: +2∙Δ(0) +𝑘𝐵∙𝑇𝑐 = 4.28 and +Δ𝐶 +𝛾𝑇𝑐 = 0.995). + +0 +1 +2 +0 +2 +4 +6 +8 +10 +12 +14 +0.0 +0.5 +1.0 +1.5 +2.0 +(T) +Bc2(T) + Bc2(T) data and s-wave fit +Bc2(T) (T) +WB2 +(0) = 1.48 ± 0.06 meV +2(0)/kBTc= 2.8 ± 0.1 +C/Tc = 1.6 ± 0.4 +0 +10 +20 +30 +40 +50 +a + (T) data and s-wave fit +(T) (nm) +(T) +Bc2(T) + Bc2(T) data and d-wave fit +Bc2(T) (T) +temperature (K) +Tc = 12.2 ± 0.1 K +(0) = 2.58 ± 0.02 meV +2(0)/kBTc= 4.9 ± 0.1 +C/Tc = 1.19 ± 0.07 +b +0 +10 +20 +30 +40 + (T) data and d-wave fit +(T) (nm) + +15 + +2.3.3. The Fermi temperature and the strength of the nonadiabaticity +The substitution of deduced parameters in Eq. 9 returns the Fermi temperature 𝑇𝐹 = + 1679 ± 68 𝐾 in WB2 (P = 121.3 GPa). Calculated 𝑇𝐹 implies that this phase falls in nearly +conventional superconductors band in the Uemura plot (Fig. 3), because this phase exhibits +reasonably low ratio of +𝑇𝑐 +𝑇𝐹 = 0.0077 ± 0.0003, while typical range for unconventional +superconductors is 0.01 ≤ +𝑇𝑐 +𝑇𝐹 ≤ 0.05. +Also, this superconductor exhibits very moderate strength of nonadiabaticy, because the +ratio: +0.025 < +𝑇𝜃 +𝑇𝐹 = 0.26 ± 0.01 < 0.4 + + + + + +(11) +is typical for majority of high-temperature superconductors, including iron-based, cuprates +and highly compressed hydrides [67] (Figs. 4,5). + +2.4. P6/mmm MgB2 +2.4.1. Temperature dependent upper critical field +To show that our Bc2(T) model (Eqs. 5-8 [58-60,71]) can be considered as an alternative +model to extract primary superconducting parameters from 𝑅(𝑇, 𝐵) datasets (while the Bc2(T) +definition criterion is +𝑅(𝑇) +𝑅𝑛𝑜𝑟𝑚(𝑇) → 0) in addition to widely used Werthamer-Helfand- +Hohenberg model [74,75], in Figure 9 we showed Bc2(T) data reported by Zehetmayer et al +[76] for single crystal MgB2 and datafits to the s-wave (panel a, Eqs. 5,6), d-wave (panel b, +Eqs. 7,8), and so-called two-band -model [77] in assumption of s-wave gap symmetry for +both bands (panel c) [77,78]: +𝐵𝑐2,𝑡𝑜𝑡𝑎𝑙(𝑇) = 𝛼 × 𝐵𝑐2,𝑏𝑎𝑛𝑑1(𝜉𝑡𝑜𝑡𝑎𝑙(0), 𝑇) + (1 − 𝛼) × 𝐵𝑐2,𝑏𝑎𝑛𝑑2(𝜉𝑡𝑜𝑡𝑎𝑙(0), 𝑇), +(12) +where to reduce the number of free-fitting parameters, we implemented the restriction [78]: +𝑇𝑐1 = 𝑇𝑐2, + + + + + + + + +(13) + +16 + + + +Figure 9. The upper critical field, Bc2(T), data for P6/mmm MgB2 reported by Zehetmayer et al [76] +and data fits to single band s-wave (panel a, Eqs. 5,6), single band d-wave (panel b, Eqs. 7,8), and +two-band s-wave [77,78] (panel c, Eqs. 5,6,12-14) models. Deduced parameters are: (a) s-wave fit, +𝑇𝑐 = 36.7 ± 0.4 𝐾, (0) = 10.4 nm, Δ(0) = 5.22 ± 0.09 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 +⁄ += 2.3 ± 0.3, +2Δ(0) +𝑘𝐵𝑇𝑐 = 3.3 ± +0.1, the goodness of fit is 0.9887; (b) d-wave fit, 𝑇𝑐 = 37.8 ± 0.3 𝐾, (0) = 10.0 nm, Δ𝑚(0) = +11.6 ± 0.5 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 +⁄ += 1.15 ± 0.07, +2Δ𝑚(0) +𝑘𝐵𝑇𝑐 = 7.1 ± 0.3, the goodness of fit is 0.9975. (c) two +conditions where used: 𝑇𝑐1 = 𝑇𝑐2 = 37.2 ± 0.2 𝐾 and +Δ𝐶1 +𝛾1𝑇𝑐1 = +Δ𝐶2 +𝛾2𝑇𝑐2 = 1.8 ± 0.1, and other free-fitting +parameters are:total(0) = 10.3 nm, 𝛼 = 0.77 ± 0.06, Δ1(0) = 6.5 ± 0.4 𝑚𝑒𝑉, +2Δ1(0) +𝑘𝐵𝑇𝑐 = 4.1 ± 0.3, +Δ2(0) = 2.7 ± 0.4 𝑚𝑒𝑉, +2Δ2(0) +𝑘𝐵𝑇𝑐 = 1.7 ± 0.2, the goodness of fit is 0.9984. +0 +1 +2 +3 +4 +0 +1 +2 +3 +0 +8 +16 +24 +32 +40 +0 +1 +2 +3 +(T) +Bc2(T) + Bc2(T) data and s-wave fit +Bc2(T) (T) + MgB2 +(single band s-wave) +Tc=36.7±0.4 K +C/Tc=2.3±0.3 +2(0)/kBTc=3.3±0.1 +a +0 +10 +20 +30 +40 +50 +60 + (T) data and s-wave fit +(T) (nm) +(T) +Bc2(T) + Bc2(T) data and d-wave fit +Bc2(T) (T) + MgB2 +(single band d-wave) +Tc=37.8±0.3 K +C/Tc=1.1±0.1 +2(0)/kBTc=7.1±0.3 +b +0 +10 +20 +30 +40 +50 + (T) data and d-wave fit +(T) (nm) +C1/1Tc=C2/2Tc=1.8±0.1 +(T) +Bc2(T) + Bc2(T) data and -model fit +Bc2(T) (T) +temperature (K) +MgB2 (-model) +=0.77±0.06 +21(0)/kBTc=4.1±0.3 +22(0)/kBTc=1.7±0.2 +c +0 +10 +20 +30 +40 +50 + (T) data and -model fit +(T) (nm) + +17 + + +Δ𝐶1 +𝛾1𝑇𝑐1 = +Δ𝐶2 +𝛾2𝑇𝑐2. + + + + + + + + +(14) +The deduced parameters for single band s-wave model (Fig. 9,a) contradict to each other, +i.e. +2Δ(0) +𝑘𝐵𝑇𝑐 = 3.3 ± 0.1 (which is lower than the s-wave weak-coupling limit), while +Δ𝐶 +𝛾𝑇𝑐 = +2.3 ± 0.3 is much larger than the s-wave weak-coupling limit. The deduced ratio of +2Δ𝑚(0) +𝑘𝐵𝑇𝑐 = +7.1 ± 0.3 for d-wave model is nearly twice larger the d-wave weak-coupling limit of +2Δ𝑚(0) +𝑘𝐵𝑇𝑐 = 4.28, which is too large to be realistic value. +However, parameters deduced for two-band -model, 𝛼 = 0.77 ± 0.06, +2Δ1(0) +𝑘𝐵𝑇𝑐 = 4.1 ± 0.3, +2Δ2(0) +𝑘𝐵𝑇𝑐 = 1.7 ± 0.2, are in a good agreement with the values deduced in MgB2 by other +techniques [77], in particular by point contact spectroscopy [79]. + +III. Discussion +Considering that the P6/mmm-phase of Nb0.75Mo0.25B2 exhibits pronounced +nonadiabaticity, +𝑇𝜃 +𝑇𝐹 = 3.5, which is well above an empirical boarder +𝑇𝜃 +𝑇𝐹 ≅ 0.4 below which +the majority of conventional and unconventional superconductors are located (Figs. 4,5), we +can propose that the strength of the nonadiabaticity is a primary reason for relatively low Tc +in this material in comparison with other diboride counterparts. A good support of this +hypnotize can be seen in Fig. 5, where the Tc suppression within four dibories is link with the +increase of the strength of the nonadiabaticity. It can be also seen in Fig. 5, that there are no +materials which simultaneously exhibite 𝑇𝑐 > 10 𝐾 and +𝑇𝜃 +𝑇𝐹 > 0.4. +Another explanation for the relatively low Tc in Nb0.75Mo0.25B2 can be based on the +Abrikosov-Gor’kov [80], Anderson [81], and Openov [82,83] theory of dirty +superconductors. The theory states that uniformly distributed (on the atomic level) impurities + +18 + +exhibited magnetic moment should suppress the superconducting order parameter in s-wave +superconductors, but this kind of impurities should not affect the superconducting order +parameter in d-wave superconductors. However, non-magnetic impurities should cause the +suppression of in d-wave superconductors, but this kind of doping should not affect the s- +wave superconducting state. Considering that the (Nb,Mo)-(0001) planes in P6/mmm-phase +have chemical atomic disorder, because Hire et al [44] did not report any evidence for the +atomic ordering within Nb-Mo atoms in (0001) planes, it is appeared that the Tc suppression +in Nb0.75Mo0.25B2 (and in all materials in the Nb1-xMoxB2 (x = 0.25; 0.50; 0.75 and 0.9) +system) can be interpreted as the Tc suppression in d-wave MoB2 superconductor by +nonmagnetic impurities, i.e. Nb atoms. + +IV. Conclusions +In this work, we deduced primary superconducting parameters in three diborides, i.e. +P6/mmm phases of Nb1-xMoxB2 (x = 0.25; 1.0) and WB2. It was shown that these phases +exhibit d-wave superconducting gap symmetry. We proposed that many fold suppression of +the superconducting transition temperature (down to 𝑇𝑐 = 8 𝐾) in Nb0.75Mo0.25B2, can be +related either to strong nonadiabaticity in this phase (which exhibits the ratio +𝑇𝜃 +𝑇𝐹 = 3.5), either +due the effect of the 𝑇𝑐 suppression in d-wave MoB2 superconductors by nonmagnetic +impurity (which is Nb atoms). + +Acknowledgement +The author thanks financial support provided by the Ministry of Science and Higher +Education of Russia (theme “Pressure” No. АААА-А18-118020190104-3). The research +funding from the Ministry of Science and Higher Education of the Russian Federation (Ural + +19 + +Federal University Program of Development within the Priority-2030 Program) is gratefully +acknowledged. + +Data availability statement +The data that support the findings of this study are available from the corresponding author +upon reasonable request. + +Declaration of interests +The author declares that he has no known competing financial interests or personal +relationships that could have appeared to influence the work reported in this paper. + +References +[1] A.P. Drozdov, M. I. Eremets, I. A. Troyan, V. Ksenofontov, S. I. Shylin, Conventional +superconductivity at 203 kelvin at high pressures in the sulfur hydride system. Nature 525, +73-76 (2015). +[2] A. P. Drozdov, et al. Superconductivity at 250 K in lanthanum hydride under high +pressures Nature 569, 528-531 (2019) +[3] M. Somayazulu, et al. 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Openov, Effect of nonmagnetic and magnetic impurities on the specific heat jump +in anisotropic superconductors Phys Rev B 69, 224516 (2004). + diff --git a/INE1T4oBgHgl3EQfrwUe/content/tmp_files/load_file.txt b/INE1T4oBgHgl3EQfrwUe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..72e29e0b9aae8e1e66adc07e7933a181511d348b --- /dev/null +++ b/INE1T4oBgHgl3EQfrwUe/content/tmp_files/load_file.txt @@ -0,0 +1,1334 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf,len=1333 +page_content='1 d-wave superconductivity as a model for diborides apart MgB2 Evgeny F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Talantsev1,2,* 1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Miheev Institute of Metal Physics, Ural Branch, Russian Academy of Sciences, 18, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Kovalevskoy St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=', Ekaterinburg, 620108, Russia 2NANOTECH Centre, Ural Federal University, 19 Mira St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=', Ekaterinburg, 620002, Russia *corresponding author’s E-mail: evgney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='talantsev@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='ru Abstract Recently, Pei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' (arXiv2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='13250) reported that ambient pressure 𝛽-MoB2 (space group: 𝑅3̅𝑚) exhibits a phase transition to 𝛼-MoB2 (space group: 𝑃6/𝑚𝑚𝑚) at pressure P ~ 70 GPa and this high-pressure phase is a high-temperature superconductor exhibited 𝑇𝑐 = 32 𝐾 at P~110 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Despite 𝛼-MoB2 has the same crystalline structure as ambient pressure MgB2 and the 𝑇𝑐’s of 𝛼-MoB2 and MgB2 are very close, the first principles calculations showed that in 𝛼-MoB2 the states near the Fermi level, 𝜀𝐹, are dominated by the d-electrons of Mo atoms, while in MgB2 the p-orbitals of boron atomic sheets dominantly contribute to the states near the 𝜀𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' More recently, Hire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' (arXiv2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='14869) reported that the 𝑃6/𝑚𝑚𝑚-phase can be stabilized at ambient pressure in Nb1-xMoxB2 solid solutions, and these ternary alloys exhibit 𝑇𝑐~8 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' In addition, Pei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' (Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' China-Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 65, 287412 (2022)) showed that compressed WB2 exhibits 𝑇𝑐~15 𝐾 at P~121 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Here, we analyzed experimental data reported for P6/mmm-phases of Nb1-xMoxB2 (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0) and highly- compressed WB2, and showed that these three phases exhibit d-wave superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' We deduced 2Δ𝑚(0) 𝑘𝐵𝑇𝑐 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 for 𝛼-MoB2, 2Δ𝑚(0) 𝑘𝐵𝑇𝑐 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 for Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2, and 2Δ𝑚(0) 𝑘𝐵𝑇𝑐 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 for WB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' We found that Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 exhibits high strength of nonadiabaticity, which is quantified by the ratio of 𝑇𝜃 𝑇𝐹 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5, which is by one order of magnitude exceeds the ratio in MgB2, \uf061-MoB2, WB2, pnictides, cuprates, and highly-compressed hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2 d-wave superconductivity as a model for diborides apart MgB2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The discovery of near-room temperature superconductivity in highly compressed sulphur hydride by Drozdov et al [1] sparked theoretical and experimental studies of a variety of materials which potentially can exhibit a high-temperature superconductivity to be compressed at high pressure [2-25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' This research field represents one of the most fascinating scientific exploration where advanced first principles calculations conjuncts with top world class of experimental studies [26-39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' One of the interesting results in this conjunctive exploration has been reported by Pei et al [40] who found that the stoichiometric compound MoB2 exhibits the phase transition from the 𝛽-MoB2-phase (space group: 𝑅3̅𝑚) to 𝛼-MoB2-phase (space group: 𝑃6/𝑚𝑚𝑚) at critical pressure P ~ 70 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' This high-pressure phase, 𝛼-MoB2, exhibits the same crystalline structure as ambient pressure MgB2 and, what is the most intriguing experimental result reported by Pei et al [40], the 𝛼-MoB2 phase is a high-temperature superconductor with 𝑇𝑐 = 32 𝐾 (at P = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 GPa), which is remarkably close to 𝑇𝑐 = 39 − 42 𝐾 in MgB2 [41,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' First principles calculations performed by Pei et al [40] showed that several bands in the 𝛼-MoB2 crossing the Fermi level, 𝜀𝐹, which causes the metallic type of conductivity in this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Pei et al [40] also showed the molybdenum d-orbitals (especially the dz2 orbital) have larger contributions than the boron p-orbitals near the 𝜀𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' In overall, despite 𝛼-MoB2 phase exhibits the same crystal structure as MgB2 and the superconducting transition temperature for these compounds are comparable, their electronic structures are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' For instance, the out-of-plane phonon mode of molybdenum ions are strongly coupled with molybdenum d-electrons near the 𝜀𝐹 in 𝛼-MoB2 [40], while the in-plane boron-boron stretching mode in MgB2 interacts intensively with the σ-bond in the boron honeycomb lattice near the 𝜀𝐹 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Pei et al [40] also calculated the electron-phonon coupling constant, 𝜆𝑒−𝑝ℎ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='60, in 𝛼- 3 MoB2 at 𝑃 = 90 𝐺𝑃𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Similar findings, including 𝜆𝑒−𝑝ℎ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='60, were reported by Quan et al [43] who performed first principles calculations for highly-pressurized 𝛼-MoB2 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' These results give a ground to expect that the 𝛼-MoB2 phase can exhibit d-wave superconducting energy gap symmetry (or, at least, s+d-wave gap symmetry with significant d-wave component), which is different from the two-band s-wave MgB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' More recently, Hire et al [44] showed that the 𝑃6/𝑚𝑚𝑚-phase can be stabilized at ambient pressure in Nb1-xMoxB2 (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9) solid solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Despite the superconducting transition temperature in Nb1-xMoxB2 (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9) were significantly lower (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=', 𝑇𝑐 = (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1) 𝐾 [44]) these values are still high enough to make a proposal that the same pairing mechanism emerges in ambient pressure superconductors Nb1- xMoxB2 and highly-pressurized 𝛼-MoB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Hire et al [44] also performed first principles calculation, measurements of the temperature dependent magnetoresistance 𝑅(𝑇, 𝐵), and specific heat measurements from which several parameters of Nb1-xMoxB2 (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9) superconductors (and, in particular, the Debye temperature, 𝑇𝜃) were determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Pei et al [45] and Lim et al [46] extended the range of superconducting diborides by the discovery of highly-compressed phase of WB2 (𝑇𝑐~15 𝐾 at P~121 GPa) for which Pei et al [45] proposed space group: P63/mmc (which is distorted P6/mmm), while Lim et al [46] concluded that this highly-pressurized superconducting phase of WB2 formed by staking faulted P63/mmc-P6/mmm phase (which can be found to be similar to the stacking faulted 123-124 phases in Y-Ba-Cu-O system [47-49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Here, we performed detailed analysis of the magnetoresistance data reported by Pei et al [40], Hire et al [44], Pei et al [45], and showed that the P6/mmm-phases of Nb1-xMoxB2 (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0) and WB2 (P=121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 GPa) exhibit the d-wave superconducting gap symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' We also found that ambient pressure Nb1-xMoxB2 (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25) superconductors characterized by 4 high strength of nonadiabaticity, which can be characterized by the ratio of 𝑇𝜃 𝑇𝐹 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 (where 𝑇𝐹 is the Fermi temperature, which is by more than one order of magnitude exceeds the 𝑇𝜃 𝑇𝐹 ratio in MgB2, \uf061-MoB2, WB2, pnictides, cuprates, and highly-compressed hydrides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' P6/mmm 𝜶-MoB2 (𝑷 = 𝟏𝟎𝟗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 𝟕 𝑮𝑷𝒂) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Debye temperature and the electron-phonon coupling constant Debye temperature, 𝑇𝜃, can be deduced from the fit of experimentally measured temperature dependent resistance curve, 𝑅(𝑇), to the Bloch-Grüneisen (BG) equation [50,51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' In many reports, classical BG approach is advanced by introducing so-called the saturation resistance [52-57]: 𝑅(𝑇) = 1 1 𝑅𝑠𝑎𝑡+ 1 𝑅0+𝐴( 𝑇 𝑇𝜃 ) 5 ∫ 𝑥5 (𝑒𝑥−1)(1−𝑒−𝑥) 𝑇𝜃 𝑇 0 𝑑𝑥 (1) where 𝑅𝑠𝑎𝑡, 𝑅0, 𝑇𝜃 and 𝐴 are free fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' From the deduced 𝑇𝜃 and measured 𝑇𝑐 (which we defined by as strict as practically possible resistance criterion of 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚 → 0, where 𝑅𝑛𝑜𝑟𝑚 is the normal state resistance at the onset of the superconducting transition, see details in [56]), the electron-phonon coupling constant, 𝜆𝑒−𝑝ℎ, can be calculated as unique root of advanced McMillan equation [56]: 𝑇𝑐 = ( 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='45) × 𝑇𝜃 × 𝑒 −( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='04(1+𝜆𝑒−𝑝ℎ) 𝜆𝑒−𝑝ℎ−𝜇∗(1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='62𝜆𝑒−𝑝ℎ)) × 𝑓1 × 𝑓2 ∗ (2) where 𝑓1 = (1 + ( 𝜆𝑒−𝑝ℎ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='46(1+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8𝜇∗)) 3 2 ⁄ ) 1 3 ⁄ (3) 𝑓2 ∗ = 1 + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0241 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0735 × 𝜇∗) × 𝜆𝑒−𝑝ℎ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' (4) 5 where 𝜇∗ is the Coulomb pseudopotential parameter, which we assumed (follow the approach proposed in [40,44,46]) to be 𝜇∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='13 for Nb1-xMoxB2 (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0) and WB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The fits of 𝑅(𝑇) datasets measured for 𝛼-MoB2 phase at 𝑃 = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 𝑎𝑛𝑑 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 𝐺𝑃𝑎 [40] to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 1 together with deduced 𝑅𝑠𝑎𝑡, 𝑇𝜃, and 𝜆𝑒−𝑝ℎ, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 1 (where we utilized 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='10 criterion to define 𝑇𝑐, because the same criterion was used by Pei et al [40] to define the upper critical field in the same 𝛼-MoB2 sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Deduced 𝜆𝑒−𝑝ℎ(91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 𝐺𝑃𝑎) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='42 is in a good agreement with the value calculated by first principles calculations 𝜆𝑒−𝑝ℎ(90 𝐺𝑃𝑎) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='60 [40,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' R(T) data for highly compressed 𝛼-MoB2 (P = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 GPa) and data fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 1 (raw data reported by Pei et al [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Green balls indicate the bounds for which R(T) data was used for the fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' (a) Deduced 𝑇𝜃 = 301 ± 1 𝐾, 𝑇𝑐,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='10 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='6 𝐾, 𝜆𝑒−𝑝ℎ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='42, 𝑅𝑠𝑎𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='02 Ω, fit quality is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' (b) Deduced 𝑇𝜃 = 321 ± 1 𝐾, 𝑇𝑐,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='10 = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 𝐾, 𝜆𝑒−𝑝ℎ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='41, 𝑅𝑠𝑎𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='01 Ω, fit quality is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 95% confidence bands are shown by pink shadow areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='08 0 50 100 150 200 250 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='06 a \uf061‒MoB2 (P = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 GPa) raw R(T) fit to BG model fitted R(T) range Tc,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='10 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='6 K resistance (\uf057) Rsat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='02 \uf057 T\uf071 = 301 ± 1 K \uf06ce-ph = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='42 b \uf061‒MoB2 (P = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 GPa) raw R(T) fit to BG model fitted R(T) range Tc,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='10 = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 K resistance (\uf057) temperature (K) Rsat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='01 \uf057 T\uf071 = 321 ± 1 K \uf06ce-ph = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='41 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Temperature dependent upper critical field Pei et al [40] in their Figure 2,d utilized several resistance criteria 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='90 to derive the upper critical field, 𝐵𝑐2(𝑇), from measured 𝑅(𝑇, 𝐵, 𝑃 = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 𝐺𝑃𝑎) curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' By following general logic [56,58,59] that as low as possible resistance criterion should be in use, here we utilized the same criterion of 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='10, as the one which was used to define the 𝑇𝑐 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 1 and by the lowest criterion to defined 𝐵𝑐2(𝑇) by Pei et al [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2,a the 𝐵𝑐2(𝑇) dataset is fitted to the equation for temperature dependent upper critical field for s-wave superconductors [58-60]: 𝐵𝑐2(𝑇) = 𝜙0 2∙𝜋∙𝜉2(0) ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='77−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='43( 𝑇 𝑇𝑐) 2 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='07( 𝑇 𝑇𝑐) 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='77 ) 2 × [1 − 1 2𝑘𝐵𝑇 ∫ 𝑑𝜀 𝑐𝑜𝑠ℎ2( √𝜀2+Δ2(𝑇) 2𝑘𝐵𝑇 ) ∞ 0 ] (5) where the amplitude of temperature dependent superconducting gap, \uf044(T), is given by [61,62]: Δ(𝑇) = Δ(0) × tanh [ 𝜋𝑘𝐵𝑇𝑐 Δ(0) √𝜂 Δ𝐶 𝛾𝑇𝑐 ( 𝑇𝑐 𝑇 − 1)] (6) where \uf068 = 2/3 for s-wave superconductors, \uf067 is Sommerfeld constant, and 𝑘𝐵 is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' However, the deduced 2Δ(0) 𝑘𝐵𝑇𝑐 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2,a) is too low to be attributed to s-wave superconductivity, for which the weak-coupling limit is 2Δ(0) 𝑘𝐵𝑇𝑐 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='53 [63,64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' And also, the fit quality is low R =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Then, we fitted the temperature dependent upper critical field data to the d-wave gap symmetry model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The fitting function can be constructed similarly to its counterparts for s- wave [58-60] and p-wave [59,60,65]: 𝐵𝑐2(𝑇) = 𝜙0 2∙𝜋∙𝜉2(0) ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='77−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='43( 𝑇 𝑇𝑐) 2 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='07( 𝑇 𝑇𝑐) 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='77 ) 2 [1 − 1 2∙𝑘𝐵∙𝑇 ∙ ∫ 𝑐𝑜𝑠2(𝜃) ∙ 2𝜋 0 (∫ 𝑑𝜀 𝑐𝑜𝑠ℎ2(√𝜀2+Δ2(𝑇,𝜃) 2∙𝑘𝐵∙𝑇 ) ∞ 0 ) ∙ 𝑑𝜃] (7) 7 where the superconducting energy gap, \uf044(T,\uf071), is given by [61,62,65]: Δ(𝑇, 𝜃) = 𝑐𝑜𝑠(2𝜃) × Δ𝑚(0) × tanh [ 𝜋𝑘𝐵𝑇𝑐 Δ(0) √𝜂 Δ𝐶 𝛾𝑇𝑐 ( 𝑇𝑐 𝑇 − 1)] (8) where \uf044m(T) is the is the maximum amplitude of the k-dependent d-wave gap, \uf068 = 7/5 [65], \uf071 is the angle around the Fermi surface subtended at (\uf070, \uf070) in the Brillouin zone (details can be found elsewhere [61,62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The upper critical field, Bc2(T), data (defined by 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='10 criterion) for 𝛼-MoB2 (𝑃 = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 𝐺𝑃𝑎) reported by Pei et al [40] and data fits to s-wave (panel a) and d-wave (panel b) single-band models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Deduced parameters are (for both panels the critical temperature was fixed to the observed value of Tc = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 K): (a) s-wave fit, \uf078(0) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5(2) nm, Δ(0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8, 2Δ(0) 𝑘𝐵𝑇𝑐 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2, the goodness of fit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8267;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' (b) d-wave fit, \uf078(0) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2(5) nm, Δ(0) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 ⁄ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1, 2Δ(0) 𝑘𝐵𝑇𝑐 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2, the goodness of fit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0 2 4 6 8 10 0 5 10 15 20 25 30 0 2 4 6 8 \uf078(T) Bc2(T) Bc2(T) data and s-wave fit Bc2(T) (T) \uf061‒MoB2 (P = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 GPa) \uf044(0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 meV 2\uf044(0)/kBTc= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 \uf044C/\uf067Tc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8 0 6 12 18 24 30 a \uf078(T) data and s-wave fit \uf078(T) (nm) \uf078(T) Bc2(T) Bc2(T) data and d-wave fit Bc2(T) (T) temperature (K) \uf044(0) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 meV 2\uf044(0)/kBTc= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 \uf044C/\uf067Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 b 0 6 12 18 24 \uf078(T) data and d-wave fit \uf078(T) (nm) 8 The fit converged with a better quality (with the goodness of fit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9842) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Deduced parameters are: ξ(0) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2(5) 𝑛𝑚, Δ(0) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 𝑚𝑒𝑉, 2Δ(0) 𝑘𝐵𝑇𝑐 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2, Δ𝐶 𝛾𝑇𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Considering that the weak coupling limits for d-wave superconductors [61,62,65] are: 2∙Δ(0) 𝑘𝐵∙𝑇𝑐 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='28 and Δ𝐶 𝛾𝑇𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='995, we can conclude that deduced parameters in 𝛼-MoB2 (𝑃 = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 𝐺𝑃𝑎) superconductor is within weak-coupling values for d-wave superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' It should be noted, that the accuracy of the extracted parameters is directly related to the sampling number of the measurement, and, thus, further increase in the accuracy in the deduced parameters, can be possible if more raw 𝑅(𝑇, 𝐵) data (especially, measured at low temperature, down to miliKelvin level) will be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The Fermi temperature and the strength of the nonadiabaticity The Fermi temperature can be calculated by the equation [58]: 𝑇𝐹 = 𝜋2𝑚𝑒 8∙𝑘𝐵 × (1 + 𝜆𝑒−𝑝ℎ) × 𝜉2(0) × ( 2Δ𝑚(0) ℏ ) 2 , (9) where 𝑚𝑒 is bare electron mass, ℏ is the reduced Planck’s constant, and other parameters have deduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' In the result, calculated Fermi temperature is 𝑇𝐹 = 1756 ± 25 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Calculated 𝑇𝐹 implies that the P6/mmm 𝛼-MoB2 (𝑃 = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 𝐺𝑃𝑎) phase falls in unconventional superconductors band in the Uemura plot (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 3), because this phase exhibits typical for many unconventional superconductors (for instance, iron-based, cuprates and hydrogen-rich superconductors) ratio of 𝑇𝑐 𝑇𝐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Also, we found that the P6/mmm 𝛼-MoB2 (𝑃 = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 𝐺𝑃𝑎) phase exhibits similar level of the nonadiabaticy ( 𝑇𝜃 𝑇𝐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='02) to iron-based, cuprates and hydrogen-rich superconductors [66,67] (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 4,5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 9 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Uemura plot (Tc vs TF), where the diborides are shown together with other superconducting families: 2D materials, metals, pnictides, cuprates, and near-room-temperature superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Reference on original data can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 65,67-73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Plot of 𝑇𝜃 𝑇𝐹 vs 𝜆𝑒−𝑝ℎ for several superconducting families and for diborides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' This type of plot proposed by Pietronero et al [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' References on original data can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 65,67-71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 10 100 1000 10000 100000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 1 10 100 2H TaSeS MgB2 WB2 Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 \uf061 MoB2 LiC6 IrTe2 bismuthates (La,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='Nd)H10 CsI MATBG intermetallics Laves \uf07a O2 noncentrosymmetric Heusler A15 metals SrTiO3 LaH10 H3S iron based H3S (from \uf078(0)) H3S (from \uf06c(0)) metals A15 alloys Heusler alloys noncentrosymmetric Laves phase intermetallics SrTiO3 Ba1 xKxBiO3 iron based SCs cuprates (\uf06c(0) and Jc(sf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='T)) LaH10 (from \uf078(0)) LaH10 (from \uf06c(0)) (La,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='Nd)H10 (from \uf078(0)) CsI (P=209 GPa) MATBG (from Jc(sf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='T)) MATBG (from \uf078(0)) IrTe2 (from Jc(sf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='T)) \uf07a O2 (P=115 GPa) LiC6 2H TaSeS Tc/TF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='001 Tc/TF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='01 Tc/TF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='05 \uf061 MoB2 (P=110 GPa) Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 WB2 (P=121 GPa) MgB2 transition temperature, Tc (K) Fermi temperature, TF (K) BCS cuprates 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 1 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 1 10 100 MgB2 WB2 Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 \uf061 MoB2 LiC6 metals A15 Laves Heusler \uf07a O2 CsI SrTiO3 noncentrosymmetric LaH10 H3S H3S (from \uf078(0)) H3S (from \uf06c(0)) metals A15 alloys Heusler alloys noncentrosymmetric Laves phase intermetallics SrTiO3 Ba1 xKxBiO3 iron based SCs LaH10 (from \uf078(0)) LaH10 (from \uf06c(0)) (La,Nd)H10 (from \uf078(0)) CsI (P=209 GPa) \uf07a O2 (P=115 GPa) LiC6 T\uf071/TF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='025 T\uf071/TF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 \uf061 MoB2 (P =110 GPa) Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 MgB2 WB2 (P =121 GPa) T\uf071/TF electron phonon coupling strength, \uf06ce ph 10 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Plot of 𝑇𝜃 𝑇𝐹 vs 𝑇𝑐 for several superconducting families and for the diborides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' References on original data can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 65,67-72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Ambient pressure P6/mmm Nd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The electron-phonon coupling constant Hire et al [20] in their Table 1 reported the Debye temperature for P6/mmm Nb1-xMoxB2 (𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25) which was deduced from low-temperature specific heat measurements, 𝑇𝜃 = 625 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Follow the approach implemented in this study, we processed 𝑅(𝑇, 𝐵 = 0) data reported by Hire et al [20] by utilizing the resistance criterion of 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='015 and deduced 𝑇𝑐,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='015 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 𝐾, from which 𝜆𝑒−𝑝ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='573 was calculated by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Temperature dependent upper critical field Hire et al [44] in their Figure 6 reported 𝑅(𝑇, 𝐵) data reported, which we processed by utilizing the resistance criterion of 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='015 and deduced the 𝐵𝑐2(𝑇) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The fits of this dataset to s-wave (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 5,6) and d-wave model (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 7,8) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 1 10 100 2H-TaSeS WB2 MgB2 Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 \uf061-MoB2 LiC6 (La,Nd)H10 Laves intermetallics Heusler CsI \uf07a\uf02dO2 SrTiO3 MATBG A15 metals noncentrosymmetric LaH10 H3S cuprates H3S (from \uf078(0)) H3S (from \uf06c(0)) metals A15 alloys Heusler alloys noncentrosymmetric Laves phase intermetallics SrTiO3 iron-based SCs cuprates (\uf06c(0) and Jc(sf,T)) LaH10 (from \uf078(0)) LaH10 (from \uf06c(0)) (La,Nd)H10 (from \uf078(0)) CsI (P=209 GPa) MATBG (from Jc(sf,T)) MATBG (from \uf078(0)) 2H-TaSeS \uf07a-O2 (P=115 GPa) T\uf071/TF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='025 T\uf071/TF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 Ba1-xKxBiO3 LiC6 \uf061-MoB2 (P=110 GPa) Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 MgB2 WB2 (P=121 GPa) T\uf071/TF transition temperature, Tc (K) 11 The deduced parameters for s-wave (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 6,a) contradict to each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='e 2Δ(0) 𝑘𝐵𝑇𝑐 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='15 (which is lower than the s-wave weak-coupling limit is 2Δ(0) 𝑘𝐵𝑇𝑐 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='53 [63,64]), while deduced Δ𝐶 𝛾𝑇𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='19 is larger than s-wave weak-coupling limit of Δ𝐶 𝛾𝑇𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' And the fit quality R =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9534 is not high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The upper critical field, Bc2(T), data (defined by 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='015 criterion) for P6/mmm Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 reported by Hire et al [44] and data fits to s-wave (panel a) and d-wave (panel b) single-band models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Deduced parameters are (for both panels the critical temperature was fixed to the observed value of Tc = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 K): (a) s-wave fit, \uf078(0) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0(7) nm, Δ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='987 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='038 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2, 2Δ(0) 𝑘𝐵𝑇𝑐 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1, the goodness of fit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9534;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' (b) d-wave fit, \uf078(0) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5(0) nm, Δ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='05 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='03, 2Δ(0) 𝑘𝐵𝑇𝑐 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1, the goodness of fit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 0 1 2 3 4 5 6 \uf078(T) Bc2(T) Bc2(T) data and s-wave fit Bc2(T) (T) Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 \uf044(0) = 987 ± 38 \uf06deV 2\uf044(0)/kBTc= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 \uf044C/\uf067Tc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 6 9 12 15 18 21 a \uf078(T) data and s-wave fit \uf078(T) (nm) \uf078(T) Bc2(T) Bc2(T) data and d-wave fit Bc2(T) (T) temperature (K) \uf044(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='05 meV 2\uf044(0)/kBTc= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 \uf044C/\uf067Tc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='03 b 6 9 12 15 18 21 \uf078(T) data and d-wave fit \uf078(T) (nm) 12 The fit to the d-wave gap symmetry model has a better quality (with the goodness of fit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9959) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 6,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Deduced parameters are: ξ(0) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2(5) 𝑛𝑚, Δ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='05 𝑚𝑒𝑉, 2Δ(0) 𝑘𝐵𝑇𝑐 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1, Δ𝐶 𝛾𝑇𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='03, characterize the material as moderately strong coupled d-wave superconductor (considering that the weak coupling limits for d-wave superconductors [61,62,65] are: 2∙Δ(0) 𝑘𝐵∙𝑇𝑐 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='28 and Δ𝐶 𝛾𝑇𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The Fermi temperature and the strength of the nonadiabaticity The substitution of deduced parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 9 returns the Fermi temperature 𝑇𝐹 = 180 ± 7 𝐾 in P6/mmm-phase of Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Calculated 𝑇𝐹 implies that this phase falls in unconventional superconductors band in the Uemura plot (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 3), because this phase exhibits typical for many unconventional superconductors ratio of 𝑇𝑐 𝑇𝐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' However, what comes from our analysis and reported by Hire et al [44] the Debye temperature, that P6/mmm-phase of Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 superconductor exhibits strong nonadiabaticy, because the ratio: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 ≪ 𝑇𝜃 𝑇𝐹 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 (10) is well above typical range for moderate level of nonadiabaticity (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='025 ≤ 𝑇𝜃 𝑇𝐹 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4) observed in majority of unconventional superconductors, including iron-based, cuprates and highly compressed hydrides [67] (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 4,5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' P63/mmc WB2 (P = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 GPa) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The Debye temperature and the electron-phonon coupling constant Pei et al [45] measured 𝑅(𝑇) datasets for WB2 phase at 𝑃 = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 𝐺𝑃𝑎 which we fitted to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The fit converged at 𝑇𝜃 = 440 ± 1 𝐾 and 𝑅𝑠𝑎𝑡 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' From deduced 𝑇𝜃 we 13 found 𝜆𝑒−𝑝ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='755, for which we utilized the criterion of 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='18, which is based on the presence of the inflection of the transition 𝑅(𝑇, 𝐵, 𝑃 = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 𝐺𝑃𝑎) which can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2(b,d) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' R(T) data for highly compressed WB2 (P = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 GPa) and data fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 1 (raw data reported by Pei et al [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Green balls indicate the bounds for which R(T) data was used for the fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' (a) Deduced 𝑇𝜃 = 440 ± 1 𝐾, 𝑇𝑐,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='18 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 𝐾, 𝜆𝑒−𝑝ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='755, 𝑅𝑠𝑎𝑡 = ∞, fit quality is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 95% confidence bands are shown by pink shadow areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Temperature dependent upper critical field By utilizing the resistance criterion of 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='18 for 𝑅(𝑇, 𝐵) data reported by Pei et al [45] in their Figure 2,d we deduced the 𝐵𝑐2(𝑇) dataset for WB2 (P = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 GPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The fit of the 𝐵𝑐2(𝑇) dataset to s-wave (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 5,6) and d-wave model (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 7,8) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The deduced parameters for s-wave (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 6,a) contradict to each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2Δ(0) 𝑘𝐵𝑇𝑐 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 (which is lower than the s-wave weak-coupling limit of 2Δ(0) 𝑘𝐵𝑇𝑐 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='53 [63,64]), while deduced Δ𝐶 𝛾𝑇𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 is slightly larger than s-wave weak-coupling limit of Δ𝐶 𝛾𝑇𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' And the fit quality R =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9019 is not high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0 50 100 150 200 250 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='06 WB2 (P = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 GPa) T\uf071 = 440 ± 1 K \uf06ce-ph = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='755 raw R(T) fit ot BG model fitted R(T) range Tc,inflection = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 K resistance (\uf057) temperature (K) 14 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The upper critical field, Bc2(T), data (defined by 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚(𝑇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='015 criterion) for P63/mmc WB2 (P = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 GPa) reported by Pei et al [45] and data fits to s-wave (panel a) and d-wave (panel b) single-band models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Deduced parameters are: (a) s-wave fit, Tc = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='45 K (fixed), \uf078(0) = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8 nm, Δ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='06 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4, 2Δ(0) 𝑘𝐵𝑇𝑐 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1, the goodness of fit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' (b) d-wave fit, 𝑇𝑐 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 𝐾, \uf078(0) = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0 nm, Δ(0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='02 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='07, 2Δ(0) 𝑘𝐵𝑇𝑐 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1, the goodness of fit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The fit to the d-wave gap symmetry model has a better quality (with the goodness of fit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9986) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 8,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Deduced parameters are: ξ(0) = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0 𝑛𝑚, Δ(0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='02 𝑚𝑒𝑉, 2Δ(0) 𝑘𝐵𝑇𝑐 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1, Δ𝐶 𝛾𝑇𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='07, characterize the material as moderately strong coupled d-wave superconductor (considering that the weak coupling limits for d-wave superconductors [61,62,65] are: 2∙Δ(0) 𝑘𝐵∙𝑇𝑐 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='28 and Δ𝐶 𝛾𝑇𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0 1 2 0 2 4 6 8 10 12 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0 \uf078(T) Bc2(T) Bc2(T) data and s-wave fit Bc2(T) (T) WB2 \uf044(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='06 meV 2\uf044(0)/kBTc= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 \uf044C/\uf067Tc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 0 10 20 30 40 50 a \uf078(T) data and s-wave fit \uf078(T) (nm) \uf078(T) Bc2(T) Bc2(T) data and d-wave fit Bc2(T) (T) temperature (K) Tc = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 K \uf044(0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='02 meV 2\uf044(0)/kBTc= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 \uf044C/\uf067Tc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='07 b 0 10 20 30 40 \uf078(T) data and d-wave fit \uf078(T) (nm) 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The Fermi temperature and the strength of the nonadiabaticity The substitution of deduced parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 9 returns the Fermi temperature 𝑇𝐹 = 1679 ± 68 𝐾 in WB2 (P = 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 GPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Calculated 𝑇𝐹 implies that this phase falls in nearly conventional superconductors band in the Uemura plot (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 3), because this phase exhibits reasonably low ratio of 𝑇𝑐 𝑇𝐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0077 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0003, while typical range for unconventional superconductors is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='01 ≤ 𝑇𝑐 𝑇𝐹 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Also, this superconductor exhibits very moderate strength of nonadiabaticy, because the ratio: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='025 < 𝑇𝜃 𝑇𝐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='01 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 (11) is typical for majority of high-temperature superconductors, including iron-based, cuprates and highly compressed hydrides [67] (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 4,5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' P6/mmm MgB2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Temperature dependent upper critical field To show that our Bc2(T) model (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 5-8 [58-60,71]) can be considered as an alternative model to extract primary superconducting parameters from 𝑅(𝑇, 𝐵) datasets (while the Bc2(T) definition criterion is 𝑅(𝑇) 𝑅𝑛𝑜𝑟𝑚(𝑇) → 0) in addition to widely used Werthamer-Helfand- Hohenberg model [74,75], in Figure 9 we showed Bc2(T) data reported by Zehetmayer et al [76] for single crystal MgB2 and datafits to the s-wave (panel a, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 5,6), d-wave (panel b, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 7,8), and so-called two-band \uf061-model [77] in assumption of s-wave gap symmetry for both bands (panel c) [77,78]: 𝐵𝑐2,𝑡𝑜𝑡𝑎𝑙(𝑇) = 𝛼 × 𝐵𝑐2,𝑏𝑎𝑛𝑑1(𝜉𝑡𝑜𝑡𝑎𝑙(0), 𝑇) + (1 − 𝛼) × 𝐵𝑐2,𝑏𝑎𝑛𝑑2(𝜉𝑡𝑜𝑡𝑎𝑙(0), 𝑇), (12) where to reduce the number of free-fitting parameters, we implemented the restriction [78]: 𝑇𝑐1 = 𝑇𝑐2, (13) 16 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The upper critical field, Bc2(T), data for P6/mmm MgB2 reported by Zehetmayer et al [76] and data fits to single band s-wave (panel a, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 5,6), single band d-wave (panel b, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 7,8), and two-band s-wave [77,78] (panel c, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 5,6,12-14) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Deduced parameters are: (a) s-wave fit, 𝑇𝑐 = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 𝐾, \uf078(0) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 nm, Δ(0) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='09 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 ⁄ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3, 2Δ(0) 𝑘𝐵𝑇𝑐 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1, the goodness of fit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9887;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' (b) d-wave fit, 𝑇𝑐 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 𝐾, \uf078(0) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0 nm, Δ𝑚(0) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 𝑚𝑒𝑉, Δ𝐶 𝛾𝑇𝑐 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='07, 2Δ𝑚(0) 𝑘𝐵𝑇𝑐 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3, the goodness of fit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' (c) two conditions where used: 𝑇𝑐1 = 𝑇𝑐2 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 𝐾 and Δ𝐶1 𝛾1𝑇𝑐1 = Δ𝐶2 𝛾2𝑇𝑐2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1, and other free-fitting parameters are:\uf020\uf078total(0) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 nm, 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='06, Δ1(0) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 𝑚𝑒𝑉, 2Δ1(0) 𝑘𝐵𝑇𝑐 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3, Δ2(0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 𝑚𝑒𝑉, 2Δ2(0) 𝑘𝐵𝑇𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2, the goodness of fit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0 1 2 3 4 0 1 2 3 0 8 16 24 32 40 0 1 2 3 \uf078(T) Bc2(T) Bc2(T) data and s-wave fit Bc2(T) (T) MgB2 (single band s-wave) Tc=36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 K \uf044C/\uf067Tc=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 2\uf044(0)/kBTc=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 a 0 10 20 30 40 50 60 \uf078(T) data and s-wave fit \uf078(T) (nm) \uf078(T) Bc2(T) Bc2(T) data and d-wave fit Bc2(T) (T) MgB2 (single band d-wave) Tc=37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 K \uf044C/\uf067Tc=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 2\uf044(0)/kBTc=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 b 0 10 20 30 40 50 \uf078(T) data and d-wave fit \uf078(T) (nm) \uf044C1/\uf0671Tc=\uf044C2/\uf0672Tc=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 \uf078(T) Bc2(T) Bc2(T) data and \uf061-model fit Bc2(T) (T) temperature (K) MgB2 (\uf061-model) \uf061=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='06 2\uf0441(0)/kBTc=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 2\uf0442(0)/kBTc=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2 c 0 10 20 30 40 50 \uf078(T) data and \uf061-model fit \uf078(T) (nm) 17 Δ𝐶1 𝛾1𝑇𝑐1 = Δ𝐶2 𝛾2𝑇𝑐2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' (14) The deduced parameters for single band s-wave model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 9,a) contradict to each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 2Δ(0) 𝑘𝐵𝑇𝑐 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 (which is lower than the s-wave weak-coupling limit), while Δ𝐶 𝛾𝑇𝑐 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 is much larger than the s-wave weak-coupling limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The deduced ratio of 2Δ𝑚(0) 𝑘𝐵𝑇𝑐 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3 for d-wave model is nearly twice larger the d-wave weak-coupling limit of 2Δ𝑚(0) 𝑘𝐵𝑇𝑐 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='28, which is too large to be realistic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' However, parameters deduced for two-band \uf061-model, 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='06, 2Δ1(0) 𝑘𝐵𝑇𝑐 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='3, 2Δ2(0) 𝑘𝐵𝑇𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='2, are in a good agreement with the values deduced in MgB2 by other techniques [77], in particular by point contact spectroscopy [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Discussion Considering that the P6/mmm-phase of Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 exhibits pronounced nonadiabaticity, 𝑇𝜃 𝑇𝐹 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5, which is well above an empirical boarder 𝑇𝜃 𝑇𝐹 ≅ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4 below which the majority of conventional and unconventional superconductors are located (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 4,5), we can propose that the strength of the nonadiabaticity is a primary reason for relatively low Tc in this material in comparison with other diboride counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' A good support of this hypnotize can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 5, where the Tc suppression within four dibories is link with the increase of the strength of the nonadiabaticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' It can be also seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 5, that there are no materials which simultaneously exhibite 𝑇𝑐 > 10 𝐾 and 𝑇𝜃 𝑇𝐹 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Another explanation for the relatively low Tc in Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 can be based on the Abrikosov-Gor’kov [80], Anderson [81], and Openov [82,83] theory of dirty superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The theory states that uniformly distributed (on the atomic level) impurities 18 exhibited magnetic moment should suppress the superconducting order parameter in s-wave superconductors, but this kind of impurities should not affect the superconducting order parameter in d-wave superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' However, non-magnetic impurities should cause the suppression of in d-wave superconductors, but this kind of doping should not affect the s- wave superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Considering that the (Nb,Mo)-(0001) planes in P6/mmm-phase have chemical atomic disorder, because Hire et al [44] did not report any evidence for the atomic ordering within Nb-Mo atoms in (0001) planes, it is appeared that the Tc suppression in Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2 (and in all materials in the Nb1-xMoxB2 (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='9) system) can be interpreted as the Tc suppression in d-wave MoB2 superconductor by nonmagnetic impurities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Nb atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Conclusions In this work, we deduced primary superconducting parameters in three diborides, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' P6/mmm phases of Nb1-xMoxB2 (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='0) and WB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' It was shown that these phases exhibit d-wave superconducting gap symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' We proposed that many fold suppression of the superconducting transition temperature (down to 𝑇𝑐 = 8 𝐾) in Nb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='75Mo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='25B2, can be related either to strong nonadiabaticity in this phase (which exhibits the ratio 𝑇𝜃 𝑇𝐹 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='5), either due the effect of the 𝑇𝑐 suppression in d-wave MoB2 superconductors by nonmagnetic impurity (which is Nb atoms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Acknowledgement The author thanks financial support provided by the Ministry of Science and Higher Education of Russia (theme “Pressure” No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' АААА-А18-118020190104-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' The research funding from the Ministry of Science and Higher Education of the Russian Federation (Ural 19 Federal University Program of Development within the Priority-2030 Program) is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Data availability statement The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Declaration of interests The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE1T4oBgHgl3EQfrwUe/content/2301.03357v1.pdf'} +page_content=' Drozdov, M.' metadata={'source': 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a/JdFOT4oBgHgl3EQfxzTs/content/tmp_files/2301.12926v1.pdf.txt b/JdFOT4oBgHgl3EQfxzTs/content/tmp_files/2301.12926v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..74ff4ee5d58a2862a35147561a69759be612dd4f --- /dev/null +++ b/JdFOT4oBgHgl3EQfxzTs/content/tmp_files/2301.12926v1.pdf.txt @@ -0,0 +1,1191 @@ +Asymmetry and condition number of an elliptic-parabolic +system for biological network formation +Clarissa Astuto1, Daniele Boffi1,2, Jan Haskovec1, Peter Markowich1,3, and Giovanni +Russo4 +1King Abdullah University of Science and Technology (KAUST), 4700, Thuwal, +Saudi Arabia +2Department of Mathematics ”F. Casorati”, University of Pavia, Pavia, Italy +3Department of Mathematics, University of Vienna, Vienna, Austria +4Department of Mathematics and Computer Science, University of Catania, +Catania, Italy +January 31, 2023 +Abstract +We present results of numerical simulations of the tensor-valued elliptic-parabolic +PDE model for biological network formation. +The numerical method is based on a +non-linear finite difference scheme on a uniform Cartesian grid in a 2D domain. The +focus is on the impact of different discretization methods and choices of regularization +parameters on the symmetry of the numerical solution. In particular, we show that using +the symmetric alternating-direction implicit (ADI) method for time discretization helps +preserve the symmetry of the solution, compared to the (nonsymmetric) ADI method. +Moreover, we study the effect of regularization by isotropic background permeability +r > 0, showing that increased condition number of the elliptic problem due to decreasing +value of r leads to loss of symmetry. Finally, we perform numerical error analysis of our +method in Wasserstein distance. +1 +Introduction +Principles of formation, adaptation and functioning of biological transportation networks +have been a long standing topic of scientific investigation. It has significant applications in +leaf venation in plants [1], vascular pattern formation [2], mammalian circulatory systems +or neural networks that transport electric charge [3, 4]. Typical subjects of investigation +are geometrical and topological properties of optimal networks, their statistical properties +and robustness with respect to damage or varying external conditions. For instance, in +mammalian circulatory systems one aim of study is the relation between the dilation of +arteries and an augmentation of blood flow [5]. Other studies reveal that local gradient +1 +arXiv:2301.12926v1 [math.NA] 30 Jan 2023 + +of pressure can interfere with the diameter of blood vessels as an adaptive response to the +stress [6, 7, 8, 9]. +In plant leafs, the pattern of their venation seems to influence the cells that are engaged +in photosynthesis, and other functionalities of the plant, such as its longevity and the +optimal water distribution. +Modeling of formation and adaptation of leaf venation is a +very challenging task because of the nature of the problem. Every leaf of the same plant +exhibits different venation patterns [10]. This is reflected by the inherent non-uniqueness of +solutions and, even, instabilities in the corresponding mathematical models. Consequently, +it is difficult to validate results of numerical simulations versus experimental observations. +A small change in the parameters of the model or its discretization (such as the resolution +of the numerical grid) can lead to very different solutions. +The modeling framework for biological network formation introduced by Hu and Cai in [10, +11] involves a purely local dynamic adaptation model based on mechanical laws, consisting of +a system of ordinary differential equations (ODE) on graph edges coupled to a linear system +of equations for the material pressure. The biological nature of the model is reflected by a +metabolic cost function that is proportional to a power of the conductance of the edge. Local +conservation of mass is imposed by the Kirchhoff law. The model responds to merely local +information and naturally incorporates fluctuations in flow distributions. In [11] a related +PDE-based continuum model was proposed, which consists of a parabolic reaction-diffusion +equation for the vector-valued network conductivity, coupled to a Poisson equation for the +pressure. The model was subsequently studied in the series of papers [12, 13, 14, 15, 16]. A +more general model with tensor-valued conductivity was derived in [17]. The two modeling +approaches were compared numerically in [18]. +This paper focuses on numerical treatment of the tensor-valued PDE model of [17], where the +permeability tensor appearing in the Poisson equation is regularized by adding a multiple +rI of the identity matrix, with r > 0. We discretize the system in space using a finite- +difference scheme on a two-dimensional Cartesian grid. The time discretization is carried +out using the alternating-direction implicit (ADI) and symmetric-ADI schemes. The time +discretization is crucial since the system is stiff in all its components. The stiffness may +lead to loss of symmetry of the solution in situations when all parameters, initial and +boundary data are symmetric. The main goal of the paper is to investigate the loss of +symmetry and its dependence on the model parameters, in particular on the value of the +regularization parameter r. Moreover, we carry out convergence analysis of the method. +Here we argue that the Wasserstein distance [19, 20, 21] is an appropriate choice of distance +in the convergence analysis. In particular, it is more suitable than the usual Lp-norms to +compute the discretization error. +The paper is organized as follows: In Section 2 we introduce the tensor-valued PDE model. +In Section 3 we describe the finite-difference semi-implicit schemes that we use for its dis- +cretization. In Section 4 we provide the results of the numerical simulations, with focus on +investigation of the symmetry of the solution and its dependence on the time discretization +2 + +and the parameter values. Finally, in Section 5 we summarize the results and draw some +conclusions. +2 +The PDE Model +The PDE model [17] consists of an elliptic equation for the pressure p = p(t, ⃗x) ∈ R +representing the Darcy’s law, and a parabolic reaction-diffusion equation for the tensor- +valued conductivity C = C(t, ⃗x), +−∇ · ((rI + C∇p) = S, +(1) +∂C +∂t − D2∆C − c2∇p ⊗ ∇p + α|C|γ−2C = 0. +(2) +The term S = S(⃗x) denotes the distribution of sources and sinks, which has to be prescribed +as a datum. +The function r : Ω → R+, with r(x) ≥ r0 > 0, describes the isotropic +background permeability of the medium. In (2) the diffusion coefficient D > 0 controls the +random effects in the transportation medium and the activation parameter c2 > 0 describes +the tendency of the network to align with the pressure gradient. The reaction term α|C|γ−2C +models the metabolic cost of maintaining the network structure, with metabolic coefficient +α > 0 and metabolic exponent γ > 0. For blood circulatory systems we choose γ = 1/2, +see [9] for details, while for modeling of leaf venation in plants we have 1/2 ≤ γ ≤ 1, see +[10, 11]. +We pose (1)–(2) on a bounded domain Ω ⊂ R2 with smooth boundary ∂Ω. We choose +homogeneous Neumann boundary conditions for C and p on ∂Ω, +∇C(t, ⃗x) · ν = 0, +∇p(t, ⃗x) · ν = 0, +⃗x ∈ ∂Ω, t ≥ 0 +(3) +where ν is the outer normal vector to ∂Ω and the boundary condition for C is interpreted +elementwise, i.e., ∇Cij(t, ⃗x) · ν = 0 for all i, j = 1, . . . , d. With the homogeneous Neumann +boundary condition for the pressure p, we impose the global mass balance +� +Ω +S(⃗x)d⃗x = 0 +(4) +to ensure solvability of (1). +Finally, we prescribe a positive semidefinite initial condition C0 ≥ 0 for the conductivity C, +C(t = 0, ⃗x) = C0(⃗x) +in Ω. +(5) +A fundamental observation about the system (1)–(2) is that it represents an L2-gradient +flow of the energy +E[C] = +� +Ω +D2 +2 |∇C|2 + c2∇p[C] · (rI + C)∇p[C] + α +γ |C|γ d⃗x, +3 + +where p[C] is the unique (up to an additive constant) solution of (1) subject to the homo- +geneous Neumann boundary condition. In [17] it has been shown that for γ > 1 the energy +functional is coercive and strictly convex. Consequently, it possesses a unique minimizer +that describes the optimal transportation structure for the given distribution of sources and +sinks S. On the other hand, for 0 < γ < 1 the metabolic term renders the energy highly +non-convex with a multitude of critical points. This fact is manifested in the numerical +simulations carried out in this paper, where we shall observe their strong sensitivity with +respect to the choice of the initial datum C0. +3 +Numerical schemes +In this section we briefly describe a fully second order space and time discretization that +we adopt in our numerical simulations. We refer to [18] for further details. +3.1 +Space discretization +For the discretization in space we choose the two-dimensional quadratic domain Ω = [0, 1]× +[0, 1] where we construct a uniform Cartesian mesh with spatial step h := ∆x = ∆y. We de- +note Ωh the discrete computational domain. The discretized conductivity, Cij ≈ C(xi, yj), +and pressure, pij ≈ p(xi, yj), are defined at the center of the cell (i, j), therefore we have +xi = (i − 1/2)h, yj = (j − 1/2)h, (i, j) ∈ {1, . . . , N}2, hN = 1. We choose a cell cen- +tered discretization because it simplifies the implementation of the homogeneous Neumann +boundary condition and guarantees the exact conservation of total mass of the solution. +The space discretization of Eq. (2) written in compact form (see [18] for more details) reads +∂Ccomp +∂t += D2L Ccomp + c2P − αQ(C)Ccomp +(6) +where L is the discrete Laplacian operator and Dx and, resp., Dy are the discrete first- +order derivative operators in the x- and, resp., y-directions. We use the central difference +approximation. Ccomp = [C(1,1), C(1,2), C(2,2)]T is the vector of the unknowns for the con- +ductivity, and P is the matrix of pressure gradients P = [Dxp Dxp, Dxp Dyp, Dyp Dyp]. For +the metabolic terms, we have +Q(C) = |C|γ−2. +(7) +where | · | denotes the Frobenius norm. +The discretized version of the Poisson equation (1) reads +∂x +�� +r + C(1,1)� +∂xp +� ++ ∂x +� +C(1,2)∂yp +� ++ ∂y +� +C(1,2)∂xp +� ++ ∂y +�� +r + C(2,2)� +∂yp +� += −S +(8) +4 + +where we use the symmetry of the conductance tensor C(1,2) = C(2,1). We discretize the +components of the above formula one by one, since we use different discretizations for each. +For simplicity of notation we define C(1,1) = r +C(1,1) and C(2,2) = r +C(2,2). Then we have +∂x +� +C(1,1)∂x p +� +i,j ≈ +1 +2∆x2 +�� +C(1,1) +i+1,j + C(1,1) +i,j +� +pi+1,j + +� +C(1,1) +i−1,j + C(1,1) +i,j +� +pi−1,j − +� +C(1,1) +i+1,j + C(1,1) +i−1,j + 2C(1,1) +i,j +� +pi,j +� +We omit the term with both y-derivatives because it is analogous to the one with x- +derivatives. The term with mixed derivatives is discretized as follows, +∂x +� +C(1,2)∂y p +� +i,j ≈ +1 +8∆x2 +� +C(1,2) +i+1,j + C(1,2) +i,j +� +(pi+1,j+1 − pi+1,j−1) +− +1 +8∆x2 +�� +C(1,2) +i−1,j + C(1,2) +i,j +� +(pi−1,j+1 − pi−1,j−1) + +� +C(1,2) +i+1,j − C(1,2) +i−1,j +� +(pi,j+1 − pi,j−1) +� +and analogously for the term with the y, x-derivatives. +3.2 +Time discretization: symmetric-ADI method and extrapolation tech- +nique +In this section we describe the time discretization that we apply to the model. It is a crucial +point since the Eq. (2) is very stiff in all its components. +We choose the symmetric alternating-direction implicit (ADI) scheme. From its definition, +the classical ADI scheme [? ] is not symmetric since we choose which direction consid- +ering implicit for the first half step and then the second one is automatically chosen. A +symmetrized version of this method will compute the average of the two choices. +Another improvement we can make to the ADI method can be found in [22]. We extrapo- +late the solution of the reaction-diffusion equation to compute the solution of the Poisson +equation for the pressure: Given the conductivity tensor at time tn and tn−1, we extrapolate +the conductivity at time tn+1/2 +Cn+1/2 = 3 +2Cn − 1 +2Cn−1, +(9) +then compute pn+1/2 by solving the Poisson equation +−∇ +�� +Cn+1/2� +∇pn+1/2� += S, +which in the discretized version reads +− L +� +Cn+1/2� +pn+1/2 = S. +(10) +Finally, we apply the symmetric-ADI method to solve (6). +In practise we apply twice the traditional ADI scheme. The first time we start with the +y-direction implicit in the first step, and, with the x-direction implicit in the second one. +5 + +The second time, we exchange the order for the implicit (and automatically also for the +explicit) steps. At the end, we calculate the average of the two computed solutions. A +numerical comparison between the classical ADI [? ] and the symmetric ADI methods is +presented in Table 1 below. +3.2.1 +Comparisons between ADI and symmetric-ADI method +In this section we compare the two versions of the ADI method to see the improvements in +the symmetric version. We adopt a symmetric numerical scheme, and we choose symmetric +initial datum and source function S. Consequently, the exact solution to the problem retains +symmetry at each time step. +In order to check if our scheme is symmetric, we calculate the asymmetry of the solution +with the following formula +asymm(A) = ||A − AT || +||A + AT || +(11) +In Table 1 we see the comparison between the two ADI schemes, for different choices of +the regularization parameter r, representing the background permeability in the elliptic +equation (1). +Table 1: In this table we see the difference between a traditional ADI scheme and the +symmetric version, for different values of the background permeability r. The choice of the +parameters, the source function S and the initial datum are defined in Eqs. (12-16). +r = 10−2 +r = 10−3 +r = 10−4 +ADI +sym-ADI +ADI +sym-ADI +ADI +sym-ADI +asymm(C) +6.107e-04 +4.390e-08 +2.591e-02 +1.673e-03 +1.536e-01 +1.868e-01 +asymm(p) +3.8862e-05 +4.289e-09 +6.137e-02 +2.688e-03 +2.683e-02 +2.808e-02 +Moreover, investigating the reasons why we lose the symmetry of the solution, we notice +that it is related to the computation of the solution of the Poisson equation (1) and in +the choice of the background permeability r. +When this parameters tends to zero, the +condition number of the iteration matrix L increases, up to the order 108. In Fig. 1 we +show the quantity defined in Eq. (11) for the module of the conductivity tensor C and the +pressure p, as functions of time, together with the condition number of the matrix L, for +different values of r. We show for which values of the parameters we lose the symmetry of +the two solutions. +4 +Numerical Results +The numerical results focus on the effect of the regularization parameter r in (1), and how +some properties of the numerical scheme are strictly connected to it. +6 + +Figure 1: Quantity defined in Eqs. (12-16) for the module of the conductivity tensor C and +the pressure p, as function of time, together with the condition number of L, for different +values of r = 10−1, 10−2, 10−3. +In our simulations we define the following initial conditions and source function S, as in +equations (4) and (5): +C(1,1)(t = 0) = 1, +C(1,2)(t = 0) = 0 +C(2,2)(t = 0) = C(1,1) +(12) +C(1,1)(t = 0) = (2 − |x + y|) exp(−10|x − y|), +C(1,2)(t = 0) = 0, +C(2,2)(t = 0) = C(1,1) +(13) +C(1,1)(t = 0) = 0, +C(1,2)(t = 0) = 0 +C(2,2)(t = 0) = C(1,1) +(14) +S(⃗x) = E − ¯E, +E = exp(−σ(⃗x − ⃗x0)2), +σ = 500, +⃗x0 = (0.25, 0.25) +(15) +where ¯E = mean(E). The values of parameters of the system that we used in the simulations +are as follows: +α = 0.75, c = 5, D = 10−2, ε = 10−3, +(16) +with number of points N = 600, time step ∆t = ∆x and final time tfin = 10. The choice of +the parameter ε is justified in the recent paper [18], where we show that the solutions are +qualitatively very close for ε = 10−3 and ε = 10−4. +In Fig. 1 we show how the symmetry of the the solutions strongly depends on the parameter +r. We calculate the asymmetry of the two variables conductivity C and pressure p, defined in +Eq. (11), at each time step, and we compare these quantities with the conditioning number +of the elliptic operator for the Poisson equation L. Since the Eq. (1) is strongly degenerate +for r → 0, it is not possible to consider negligible values for the parameter. +To better understand how strong is the dependency, in Figs. +2-3 we show the plots of +three different features of the solutions, the module at final time (a), the flux at final +time (b) and the eigenvector associated to the greatest eigenvalue in absolute value (c), for +r = 10−2, 10−3, 10−4. We observe that the symmetry of the solution breaks for r = 10−4, +as we expected from Fig. 1, in which asymm(C) ≈ 10−1. +7 + +N = 600, r = 10-1 +X106 +5 +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +time +×10-10 +asymm +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +time +×10-9 +p +asymm +0.5 +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +timeN = 600, r = 10-2 +X107 +L +cond( +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +time +10-8 +asymm( +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +time +d +×10-8 +0.5 +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +timeN = 600, r = 10-3 +X108 +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +time +X10~3 +asymm( +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +time +X10-3 +p +5 +asymm +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +timeIn Fig. 4 we show the comparison between the solutions obtained with two different initial +conditions, and choosing r = 10−3: Eq. (12) for plots (a) and (c), and Eq. (14) for (b) and +(d). Another difference among the plots is the diffusivity. We choose D = 10−2 for plots (a) +and (b), and D = 0 for plots (c) and (d). The main difference between (a) and (b) panels is +the number of branches. Choosing zero initial condition, we see more branches respect to +the one with initial condition constant, equal to one. This is an effect of the permeability +tensor field P[C] = C + rI in the elliptic operator. In panel (a), it is true that C > r = 10−3 +at initial times, thus, even if we decrease r, the C component is the one prevailing. On +the contrary, in panel (b), choosing the initial condition equal to 0, it is r the parameter +that is leading the elliptic operator at initial times, and being smaller than the module of +the other initial condition, we are able to see more features. Regarding the case of zero +diffusivity, we calculated the condition number of the elliptic operators, for the (c) panel +cond(L) = 7.030 × 107 and for the (d) panel cond(L) = 9.845 × 107. This means that, with +zero diffusivity, the stiffness of the system increases, and the solutions are less symmetric +and accurate. +8 + +Figure 2: In this figure we show three different quantities of the same computations, with +the parameters defined in Eq. (12): the module of the variables at final time (a), the flux at +final time (b) and the eigenvectors associated to the greatest eigenvalue in absolute value (c). +The first row is for r = 10−2, the second row for r = 10−3 and the last one for r = 10−4. +The rest of the data are defined in Eqs. (12-16). +9 + +(a): IICll,r = 10-4 +1 +0.4 +0.9 +0.35 +0.8 +0.3 +0.7 +0.6 +0.25 +2. 0.5 +0.2 +0.4 +0.15 +0.3 +0.1 +0.2 +0.05 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(b): fux, r = 10-4 +0.9 +0.8 +0.7 +0.6 +2. 0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(c): eigenvectors, r = 10-4 +0.9 +0.8 +0.7 +0.6 +2. 0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(a): IICll,r = 10-2 +1 +0.4 +0.9 +0.8 +0.35 +0.7 +0.3 +0.6 +0.25 +2. 0.5 +0.2 +0.4 +0.15 +0.3 +0.1 +0.2 +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(b): fux, r = 10-2 +0.9 +0.8 +0.7 +0.6 +2. 0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(c): eigenvectors, r = 10-2 +0.9 +0.8 +0.7 +0.6 +2. 0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(a): IICll,r = 10-3 +1 +0.45 +0.9 +0.4 +0.8 +0.35 +0.7 +0.3 +0.6 +2. 0.5 +0.25 +0.4 +0.2 +0.3 +0.15 +0.2 +0.1 +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(b): fux, r = 10-3 +0.9 +0.8 +0.7 +0.6 +2. 0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(c): eigenvectors, r = 10-3 +0.9 +0.8 +0.7 +0.6 +2. 0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1Figure 3: In this figure we show three different quantities of the same computations, with +the parameters defined in Eq. (13): the module of the variables at final time (a), the flux at +final time (b) and the eigenvectors associated to the greatest eigenvalue in absolute value (c). +The first row is for r = 10−2, the second row for r = 10−3 and the last one for r = 10−4. +The rest of the data are defined in Eqs. (12-16). +10 + +(a): IICll,r = 10-2 +1 +0.4 +0.9 +0.35 +0.8 +0.3 +0.7 +0.6 +0.25 +2. 0.5 +0.2 +0.4 +0.15 +0.3 +0.1 +0.2 +0.05 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(b): fux, r = 10-2 +0.9 +0.8 +0.7 +0.6 +2. 0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(c): eigenvectors, r = 10-2 +0.9 +0.8 +0.7 +0.6 +2. 0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(a): IICll,r = 10-3 +1 +0.9 +0.45 +0.8 +0.4 +0.7 +0.35 +0.6 +0.3 +2. 0.5 +0.25 +0.4 +0.2 +0.3 +0.15 +0.2 +0.1 +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(b): fux, r = 10-3 +0.9 +0.8 +0.7 +0.6 +2. 0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(c): eigenvectors, r = 10-3 +0.9 +0.8 +0.7 +0.6 +2. 0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(a): IICll,r = 10-4 +1 +0.5 +0.9 +0.45 +0.8 +0.4 +0.7 +0.35 +0.6 +0.3 +2. 0.5 +0.25 +0.4 +0.2 +0.3 +0.15 +0.2 +0.1 +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(b): fux, r = 10-4 +0.9 +0.8 +0.7 +0.6 +9. 0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1(c): eigenvectors, r = 10-4 +0.9 +0.8 +0.7 +0.6 +2. 0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1Figure 4: Comparison between the module of the solutions, at final time, with different +initial conditions and diffusion coefficients D. In (a) and (c) the initial condition is defined +in Eq. (12), while in (b) and (d) we have zero initial condition defined in Eq. (14). In plots +(a) and (b) the diffusion coefficient is D = 10−1, while in plots (c) and (d) we have zero +diffusivity, i.e., D = 0. +4.1 +Accuracy tests: Wasserstein distance +One possible way of introducing distance in function or measure spaces is the Wasserstein +distance. This metric comes from the idea of moving a distribution of mass, minimizing the +average of displacement, see Fig. 5 and, e.g., [23, 24, 25]. +11 + +(a): IICll,r = 10-3 +0.45 +0.9 +0.4 +0.8 +0.35 +0.7 +0.3 +0.6 +2. 0.5 +0.25 +0.4 +0.2 +0.3 +0.15 +0.2 +0.1 +0.1 +0.05 +0 +0.2 +0.4 +0.6 +0.8(b): IICll,r = 10-3 +0.4 +0.9 +0.35 +0.8 +0.3 +0.7 +0.6 +0.25 +2. 0.5 +0.2 +0.4 +0.15 +0.3 +0.1 +0.2 +0.1 +0.05 +0 +0.2 +0.4 +0.6 +0.8(c): UCll,r = 10-3,D = 0 +1.4 +0.9 +1.2 +0.8 +0.7 +1 +0.6 +0.8 +2. 0.5 +0.4 +0.6 +0.3 +0.4 +0.2 +0.2 +0.1 +0 +0.2 +0.4 +0.6 +0.8(d): Cll, r = 10-3, D = 0 +1.6 +0.9 +1.4 +0.8 +1.2 +0.7 +0.6 +1 +2. 0.5 +0.8 +0.4 +0.6 +0.3 +0.4 +0.2 +0.2 +0.1 +0 +0.2 +0.4 +0.6 +0.8Figure 5: ’Vertical’ vs ’horizontal’ distances between a pair of functions. The figure demon- +strates that, loosely speaking, Wasserstein distance depends more on the displacement of the +function than its shape [26]. +The Wasserstein distance of order p ∈ [1, +∞) between the probability measures µ, ν from +the metric space (X, d) is defined as +Wp(µ, ν) = +� +inf +π∈Π(µ,ν) +� +X×X +d(x, y)p dπ(x, y) +�1/p +, +(17) +where Π(µ, ν) is the set of all transference plans between µ and ν, i.e., measures on the +product space (X, d)2 with marginals µ and ν, respectively. +In Fig. 6 and Table 2 we give a comparison between the accuracy of our numerical sim- +ulations calculated with L2-norm and the Wasserstein metric1 for r = 10−2, on the left +panel, and r = 10−3, on the right panel. We used the initial datum (12) and the parameter +settings specified in (12)–(16). We observe an improvement of the order of accuracy using +the Wasserstein metric. When considering the L2 norm, the error analysis does not seem +to be affected by the number of points used in the space discretization. +1https://github.com/nklb/wasserstein-distance +12 + +Table 2: Accuracy tests with Wasserstein distance, errorW, and Richardson extrapolation, +errorR, for two different values of r: on the left r = 10−2 and on the right r = 10−3, at +final time t = 3. With the Wasserstein distance, we are able to show the second order of the +scheme, obtaining a result that is drastically better than the ones obtained with the usual +norms. Here, again, we see that for a smaller values of r (right panel), the accuracy of the +method gets worse. The rest of the data are defined in Eqs. (12-16). +N +errorW +errorR +100 +- +- +200 +3.846e-03 +3.358e-01 +400 +9.579e-04 +2.763e-01 +800 +1.8060e-04 +2.199e-01 +N +errorW +errorR +100 +- +- +200 +4.827e-03 +3.273e-02 +400 +1.514e-03 +3.292e-02 +800 +4.455e-04 +2.609e-02 +Figure 6: Comparison between the accuracy calculated with L2-norm and Wasserstein dis- +tance with initial condition defined in Eq. (12), with r = 10−2 (left panel, without extrapo- +lation technique in time) and r = 10−3 (right panel, with extrapolation in time). The rest +of the data are defined in Eqs. (12)–(16). +In Table 2 we present in the first column, the Wasserstein distance, errorW, between one +solution with N and N/2 number of grid points. In the second column we calculate the error +with the Richardson extrapolation technique, errorR. We observe an better rate of conver- +gence when using the Wasserstein distance, which is related to the fact that the Wasserstein +distance depends to a larger extent on the relative displacement of its arguments, i.e., the +topological features of the network, rather than on the local values of the solution at each +grid point. +13 + +0 +WS distance +slope=-2.20 +1.5 +L2-norm +- slope = -0.30 +-3.5 +-4 +2.3 +2.4 +2.5 +2.6 +2.7 +2.8 +2.9 +3 +log 10(N)5 +Conclusions +In this paper we explored the effects of the background permeability parameter r in the +elliptic-parabolic model (1)–(2) which represents a PDE framework describing the formation +of biological networks. We showed that r is the parameter that influences the condition +number of the elliptic operator in (1). When the condition number becomes too larges, +the symmetry of the numerical solution breaks down. As all the parameters of the system, +the initial datum and the source function are symmetric, using a symmetric numerical +scheme, we expect the solution to be symmetric at each time step. We make use of finite +differences scheme to compute the solution of the system, with central differences for the +space discretization and a symmetric-ADI method in time. Because of the non linearity +in the Poisson equation, we make use of a time extrapolation to improve the accuracy +convergence. +Moreover, we demonstrate that using the Wasserstein distance to measure the order of +convergence of the numerical scheme provides better results than the L2 norm, in particular +when considering small values of the background permeability r. In [18] we were able to +show the (expected) second order accuracy only in the case of r = 0.1. Here, using the +Wasserstein distance to measure the error, we see a significant improvement even for much +smaller values of the background permeability, r = 10−2 and r = 10−3. +In a future work we shall improve the numerical method by implementing a monolithic +algorithm to solve the system. In that way, the two unknowns, conductivity tensor C and +pressure p, are solved implicitly at each time step, with the metabolic term that can be +easily linearized. An application of IMEX schemes will ensure a higher order of accuracy. +The monolithic scheme would also improve the stability in time, allowing us to consider +larger time steps. +References +[1] Robert Malinowski. Understanding of leaf development – the science of complexity. +Plants, 2(3):396–415, 2013. +[2] David Sedmera. Function and form in the developing cardiovascular system. Cardio- +vascular research, 91(2):252–259, 2011. +[3] Anne Eichmann, Ferdinand Le Noble, Monica Autiero, and Peter Carmeliet. 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Bulletin of Mathematical Sciences, 7(1):87–154, 2017. +16 + diff --git a/JdFOT4oBgHgl3EQfxzTs/content/tmp_files/load_file.txt b/JdFOT4oBgHgl3EQfxzTs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7cb9678b62fed582034b8aa6e89a21b1e2db6439 --- /dev/null +++ b/JdFOT4oBgHgl3EQfxzTs/content/tmp_files/load_file.txt @@ -0,0 +1,725 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf,len=724 +page_content='Asymmetry and condition number of an elliptic-parabolic system for biological network formation Clarissa Astuto1, Daniele Boffi1,2, Jan Haskovec1, Peter Markowich1,3, and Giovanni Russo4 1King Abdullah University of Science and Technology (KAUST), 4700, Thuwal, Saudi Arabia 2Department of Mathematics ”F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Casorati”, University of Pavia, Pavia, Italy 3Department of Mathematics, University of Vienna, Vienna, Austria 4Department of Mathematics and Computer Science, University of Catania, Catania, Italy January 31, 2023 Abstract We present results of numerical simulations of the tensor-valued elliptic-parabolic PDE model for biological network formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The numerical method is based on a non-linear finite difference scheme on a uniform Cartesian grid in a 2D domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The focus is on the impact of different discretization methods and choices of regularization parameters on the symmetry of the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In particular, we show that using the symmetric alternating-direction implicit (ADI) method for time discretization helps preserve the symmetry of the solution, compared to the (nonsymmetric) ADI method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Moreover, we study the effect of regularization by isotropic background permeability r > 0, showing that increased condition number of the elliptic problem due to decreasing value of r leads to loss of symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Finally, we perform numerical error analysis of our method in Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 1 Introduction Principles of formation, adaptation and functioning of biological transportation networks have been a long standing topic of scientific investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' It has significant applications in leaf venation in plants [1], vascular pattern formation [2], mammalian circulatory systems or neural networks that transport electric charge [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Typical subjects of investigation are geometrical and topological properties of optimal networks, their statistical properties and robustness with respect to damage or varying external conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' For instance, in mammalian circulatory systems one aim of study is the relation between the dilation of arteries and an augmentation of blood flow [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Other studies reveal that local gradient 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='12926v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='NA] 30 Jan 2023 of pressure can interfere with the diameter of blood vessels as an adaptive response to the stress [6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In plant leafs, the pattern of their venation seems to influence the cells that are engaged in photosynthesis, and other functionalities of the plant, such as its longevity and the optimal water distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Modeling of formation and adaptation of leaf venation is a very challenging task because of the nature of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Every leaf of the same plant exhibits different venation patterns [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' This is reflected by the inherent non-uniqueness of solutions and, even, instabilities in the corresponding mathematical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Consequently, it is difficult to validate results of numerical simulations versus experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' A small change in the parameters of the model or its discretization (such as the resolution of the numerical grid) can lead to very different solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The modeling framework for biological network formation introduced by Hu and Cai in [10, 11] involves a purely local dynamic adaptation model based on mechanical laws, consisting of a system of ordinary differential equations (ODE) on graph edges coupled to a linear system of equations for the material pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The biological nature of the model is reflected by a metabolic cost function that is proportional to a power of the conductance of the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Local conservation of mass is imposed by the Kirchhoff law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The model responds to merely local information and naturally incorporates fluctuations in flow distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In [11] a related PDE-based continuum model was proposed, which consists of a parabolic reaction-diffusion equation for the vector-valued network conductivity, coupled to a Poisson equation for the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The model was subsequently studied in the series of papers [12, 13, 14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' A more general model with tensor-valued conductivity was derived in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The two modeling approaches were compared numerically in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' This paper focuses on numerical treatment of the tensor-valued PDE model of [17], where the permeability tensor appearing in the Poisson equation is regularized by adding a multiple rI of the identity matrix, with r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We discretize the system in space using a finite- difference scheme on a two-dimensional Cartesian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The time discretization is carried out using the alternating-direction implicit (ADI) and symmetric-ADI schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The time discretization is crucial since the system is stiff in all its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The stiffness may lead to loss of symmetry of the solution in situations when all parameters, initial and boundary data are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The main goal of the paper is to investigate the loss of symmetry and its dependence on the model parameters, in particular on the value of the regularization parameter r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Moreover, we carry out convergence analysis of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Here we argue that the Wasserstein distance [19, 20, 21] is an appropriate choice of distance in the convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In particular, it is more suitable than the usual Lp-norms to compute the discretization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The paper is organized as follows: In Section 2 we introduce the tensor-valued PDE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In Section 3 we describe the finite-difference semi-implicit schemes that we use for its dis- cretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In Section 4 we provide the results of the numerical simulations, with focus on investigation of the symmetry of the solution and its dependence on the time discretization 2 and the parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Finally, in Section 5 we summarize the results and draw some conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 2 The PDE Model The PDE model [17] consists of an elliptic equation for the pressure p = p(t, ⃗x) ∈ R representing the Darcy’s law, and a parabolic reaction-diffusion equation for the tensor- valued conductivity C = C(t, ⃗x), −∇ · ((rI + C∇p) = S, (1) ∂C ∂t − D2∆C − c2∇p ⊗ ∇p + α|C|γ−2C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (2) The term S = S(⃗x) denotes the distribution of sources and sinks, which has to be prescribed as a datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The function r : Ω → R+, with r(x) ≥ r0 > 0, describes the isotropic background permeability of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In (2) the diffusion coefficient D > 0 controls the random effects in the transportation medium and the activation parameter c2 > 0 describes the tendency of the network to align with the pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The reaction term α|C|γ−2C models the metabolic cost of maintaining the network structure, with metabolic coefficient α > 0 and metabolic exponent γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' For blood circulatory systems we choose γ = 1/2, see [9] for details, while for modeling of leaf venation in plants we have 1/2 ≤ γ ≤ 1, see [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We pose (1)–(2) on a bounded domain Ω ⊂ R2 with smooth boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We choose homogeneous Neumann boundary conditions for C and p on ∂Ω, ∇C(t, ⃗x) · ν = 0, ∇p(t, ⃗x) · ν = 0, ⃗x ∈ ∂Ω, t ≥ 0 (3) where ν is the outer normal vector to ∂Ω and the boundary condition for C is interpreted elementwise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=', ∇Cij(t, ⃗x) · ν = 0 for all i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' With the homogeneous Neumann boundary condition for the pressure p, we impose the global mass balance � Ω S(⃗x)d⃗x = 0 (4) to ensure solvability of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Finally, we prescribe a positive semidefinite initial condition C0 ≥ 0 for the conductivity C, C(t = 0, ⃗x) = C0(⃗x) in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (5) A fundamental observation about the system (1)–(2) is that it represents an L2-gradient flow of the energy E[C] = � Ω D2 2 |∇C|2 + c2∇p[C] · (rI + C)∇p[C] + α γ |C|γ d⃗x, 3 where p[C] is the unique (up to an additive constant) solution of (1) subject to the homo- geneous Neumann boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In [17] it has been shown that for γ > 1 the energy functional is coercive and strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Consequently, it possesses a unique minimizer that describes the optimal transportation structure for the given distribution of sources and sinks S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' On the other hand, for 0 < γ < 1 the metabolic term renders the energy highly non-convex with a multitude of critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' This fact is manifested in the numerical simulations carried out in this paper, where we shall observe their strong sensitivity with respect to the choice of the initial datum C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 3 Numerical schemes In this section we briefly describe a fully second order space and time discretization that we adopt in our numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We refer to [18] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='1 Space discretization For the discretization in space we choose the two-dimensional quadratic domain Ω = [0, 1]× [0, 1] where we construct a uniform Cartesian mesh with spatial step h := ∆x = ∆y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We de- note Ωh the discrete computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The discretized conductivity, Cij ≈ C(xi, yj), and pressure, pij ≈ p(xi, yj), are defined at the center of the cell (i, j), therefore we have xi = (i − 1/2)h, yj = (j − 1/2)h, (i, j) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' , N}2, hN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We choose a cell cen- tered discretization because it simplifies the implementation of the homogeneous Neumann boundary condition and guarantees the exact conservation of total mass of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The space discretization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (2) written in compact form (see [18] for more details) reads ∂Ccomp ∂t = D2L Ccomp + c2P − αQ(C)Ccomp (6) where L is the discrete Laplacian operator and Dx and, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=', Dy are the discrete first- order derivative operators in the x- and, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=', y-directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We use the central difference approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Ccomp = [C(1,1), C(1,2), C(2,2)]T is the vector of the unknowns for the con- ductivity, and P is the matrix of pressure gradients P = [Dxp Dxp, Dxp Dyp, Dyp Dyp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' For the metabolic terms, we have Q(C) = |C|γ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (7) where | · | denotes the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The discretized version of the Poisson equation (1) reads ∂x �� r + C(1,1)� ∂xp � + ∂x � C(1,2)∂yp � + ∂y � C(1,2)∂xp � + ∂y �� r + C(2,2)� ∂yp � = −S (8) 4 where we use the symmetry of the conductance tensor C(1,2) = C(2,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We discretize the components of the above formula one by one, since we use different discretizations for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' For simplicity of notation we define C(1,1) = r +C(1,1) and C(2,2) = r +C(2,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Then we have ∂x � C(1,1)∂x p � i,j ≈ 1 2∆x2 �� C(1,1) i+1,j + C(1,1) i,j � pi+1,j + � C(1,1) i−1,j + C(1,1) i,j � pi−1,j − � C(1,1) i+1,j + C(1,1) i−1,j + 2C(1,1) i,j � pi,j � We omit the term with both y-derivatives because it is analogous to the one with x- derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The term with mixed derivatives is discretized as follows, ∂x � C(1,2)∂y p � i,j ≈ 1 8∆x2 � C(1,2) i+1,j + C(1,2) i,j � (pi+1,j+1 − pi+1,j−1) − 1 8∆x2 �� C(1,2) i−1,j + C(1,2) i,j � (pi−1,j+1 − pi−1,j−1) + � C(1,2) i+1,j − C(1,2) i−1,j � (pi,j+1 − pi,j−1) � and analogously for the term with the y, x-derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='2 Time discretization: symmetric-ADI method and extrapolation tech- nique In this section we describe the time discretization that we apply to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' It is a crucial point since the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (2) is very stiff in all its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We choose the symmetric alternating-direction implicit (ADI) scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' From its definition, the classical ADI scheme [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' ] is not symmetric since we choose which direction consid- ering implicit for the first half step and then the second one is automatically chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' A symmetrized version of this method will compute the average of the two choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Another improvement we can make to the ADI method can be found in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We extrapo- late the solution of the reaction-diffusion equation to compute the solution of the Poisson equation for the pressure: Given the conductivity tensor at time tn and tn−1, we extrapolate the conductivity at time tn+1/2 Cn+1/2 = 3 2Cn − 1 2Cn−1, (9) then compute pn+1/2 by solving the Poisson equation −∇ �� Cn+1/2� ∇pn+1/2� = S, which in the discretized version reads − L � Cn+1/2� pn+1/2 = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (10) Finally, we apply the symmetric-ADI method to solve (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In practise we apply twice the traditional ADI scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The first time we start with the y-direction implicit in the first step, and, with the x-direction implicit in the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 5 The second time, we exchange the order for the implicit (and automatically also for the explicit) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' At the end, we calculate the average of the two computed solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' A numerical comparison between the classical ADI [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' ] and the symmetric ADI methods is presented in Table 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='1 Comparisons between ADI and symmetric-ADI method In this section we compare the two versions of the ADI method to see the improvements in the symmetric version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We adopt a symmetric numerical scheme, and we choose symmetric initial datum and source function S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Consequently, the exact solution to the problem retains symmetry at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In order to check if our scheme is symmetric, we calculate the asymmetry of the solution with the following formula asymm(A) = ||A − AT || ||A + AT || (11) In Table 1 we see the comparison between the two ADI schemes, for different choices of the regularization parameter r, representing the background permeability in the elliptic equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Table 1: In this table we see the difference between a traditional ADI scheme and the symmetric version, for different values of the background permeability r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The choice of the parameters, the source function S and the initial datum are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (12-16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' r = 10−2 r = 10−3 r = 10−4 ADI sym-ADI ADI sym-ADI ADI sym-ADI asymm(C) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='107e-04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='390e-08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='591e-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='673e-03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='536e-01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='868e-01 asymm(p) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='8862e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='289e-09 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='137e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='688e-03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='683e-02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='808e-02 Moreover, investigating the reasons why we lose the symmetry of the solution, we notice that it is related to the computation of the solution of the Poisson equation (1) and in the choice of the background permeability r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' When this parameters tends to zero, the condition number of the iteration matrix L increases, up to the order 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 1 we show the quantity defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (11) for the module of the conductivity tensor C and the pressure p, as functions of time, together with the condition number of the matrix L, for different values of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We show for which values of the parameters we lose the symmetry of the two solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 4 Numerical Results The numerical results focus on the effect of the regularization parameter r in (1), and how some properties of the numerical scheme are strictly connected to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 6 Figure 1: Quantity defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (12-16) for the module of the conductivity tensor C and the pressure p, as function of time, together with the condition number of L, for different values of r = 10−1, 10−2, 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In our simulations we define the following initial conditions and source function S, as in equations (4) and (5): C(1,1)(t = 0) = 1, C(1,2)(t = 0) = 0 C(2,2)(t = 0) = C(1,1) (12) C(1,1)(t = 0) = (2 − |x + y|) exp(−10|x − y|), C(1,2)(t = 0) = 0, C(2,2)(t = 0) = C(1,1) (13) C(1,1)(t = 0) = 0, C(1,2)(t = 0) = 0 C(2,2)(t = 0) = C(1,1) (14) S(⃗x) = E − ¯E, E = exp(−σ(⃗x − ⃗x0)2), σ = 500, ⃗x0 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='25) (15) where ¯E = mean(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The values of parameters of the system that we used in the simulations are as follows: α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='75, c = 5, D = 10−2, ε = 10−3, (16) with number of points N = 600, time step ∆t = ∆x and final time tfin = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The choice of the parameter ε is justified in the recent paper [18], where we show that the solutions are qualitatively very close for ε = 10−3 and ε = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 1 we show how the symmetry of the the solutions strongly depends on the parameter r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We calculate the asymmetry of the two variables conductivity C and pressure p, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (11), at each time step, and we compare these quantities with the conditioning number of the elliptic operator for the Poisson equation L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Since the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (1) is strongly degenerate for r → 0, it is not possible to consider negligible values for the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' To better understand how strong is the dependency, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 2-3 we show the plots of three different features of the solutions, the module at final time (a), the flux at final time (b) and the eigenvector associated to the greatest eigenvalue in absolute value (c), for r = 10−2, 10−3, 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We observe that the symmetry of the solution breaks for r = 10−4, as we expected from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 1, in which asymm(C) ≈ 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 7 N = 600, r = 10-1 X106 5 0 0 1 2 3 4 5 6 7 8 9 10 time ×10-10 asymm 0 0 1 2 3 4 5 6 7 8 9 10 time ×10-9 p asymm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='5 0 0 1 2 3 4 5 6 7 8 9 10 timeN = 600, r = 10-2 X107 L cond( 0 0 1 2 3 4 5 6 7 8 9 10 time 10-8 asymm( 0 0 1 2 3 4 5 6 7 8 9 10 time d ×10-8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='5 0 0 1 2 3 4 5 6 7 8 9 10 timeN = 600, r = 10-3 X108 0 0 1 2 3 4 5 6 7 8 9 10 time X10~3 asymm( 0 0 1 2 3 4 5 6 7 8 9 10 time X10-3 p 5 asymm 0 0 1 2 3 4 5 6 7 8 9 10 timeIn Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 4 we show the comparison between the solutions obtained with two different initial conditions, and choosing r = 10−3: Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (12) for plots (a) and (c), and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (14) for (b) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Another difference among the plots is the diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We choose D = 10−2 for plots (a) and (b), and D = 0 for plots (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The main difference between (a) and (b) panels is the number of branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Choosing zero initial condition, we see more branches respect to the one with initial condition constant, equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' This is an effect of the permeability tensor field P[C] = C + rI in the elliptic operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In panel (a), it is true that C > r = 10−3 at initial times, thus, even if we decrease r, the C component is the one prevailing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' On the contrary, in panel (b), choosing the initial condition equal to 0, it is r the parameter that is leading the elliptic operator at initial times, and being smaller than the module of the other initial condition, we are able to see more features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Regarding the case of zero diffusivity, we calculated the condition number of the elliptic operators, for the (c) panel cond(L) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='030 × 107 and for the (d) panel cond(L) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='845 × 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' This means that, with zero diffusivity, the stiffness of the system increases, and the solutions are less symmetric and accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 8 Figure 2: In this figure we show three different quantities of the same computations, with the parameters defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (12): the module of the variables at final time (a), the flux at final time (b) and the eigenvectors associated to the greatest eigenvalue in absolute value (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The first row is for r = 10−2, the second row for r = 10−3 and the last one for r = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The rest of the data are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (12-16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 9 (a): IICll,r = 10-4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='7 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='1 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='8 1Figure 3: In this figure we show three different quantities of the same computations, with the parameters defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (13): the module of the variables at final time (a), the flux at final time (b) and the eigenvectors associated to the greatest eigenvalue in absolute value (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The first row is for r = 10−2, the second row for r = 10−3 and the last one for r = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The rest of the data are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (12-16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 10 (a): IICll,r = 10-2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='4 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (12), while in (b) and (d) we have zero initial condition defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In plots (a) and (b) the diffusion coefficient is D = 10−1, while in plots (c) and (d) we have zero diffusivity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=', D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='1 Accuracy tests: Wasserstein distance One possible way of introducing distance in function or measure spaces is the Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' This metric comes from the idea of moving a distribution of mass, minimizing the average of displacement, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 5 and, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=', [23, 24, 25].' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='8(d): Cll, r = 10-3, D = 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='8 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='8Figure 5: ’Vertical’ vs ’horizontal’ distances between a pair of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The figure demon- strates that, loosely speaking, Wasserstein distance depends more on the displacement of the function than its shape [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The Wasserstein distance of order p ∈ [1, +∞) between the probability measures µ, ν from the metric space (X, d) is defined as Wp(µ, ν) = � inf π∈Π(µ,ν) � X×X d(x, y)p dπ(x, y) �1/p , (17) where Π(µ, ν) is the set of all transference plans between µ and ν, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=', measures on the product space (X, d)2 with marginals µ and ν, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 6 and Table 2 we give a comparison between the accuracy of our numerical sim- ulations calculated with L2-norm and the Wasserstein metric1 for r = 10−2, on the left panel, and r = 10−3, on the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We used the initial datum (12) and the parameter settings specified in (12)–(16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We observe an improvement of the order of accuracy using the Wasserstein metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' When considering the L2 norm, the error analysis does not seem to be affected by the number of points used in the space discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='com/nklb/wasserstein-distance 12 Table 2: Accuracy tests with Wasserstein distance, errorW, and Richardson extrapolation, errorR, for two different values of r: on the left r = 10−2 and on the right r = 10−3, at final time t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' With the Wasserstein distance, we are able to show the second order of the scheme, obtaining a result that is drastically better than the ones obtained with the usual norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Here, again, we see that for a smaller values of r (right panel), the accuracy of the method gets worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The rest of the data are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (12-16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' N errorW errorR 100 200 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='846e-03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='358e-01 400 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='579e-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='763e-01 800 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='8060e-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='199e-01 N errorW errorR 100 200 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='827e-03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='273e-02 400 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='514e-03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='292e-02 800 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='455e-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='609e-02 Figure 6: Comparison between the accuracy calculated with L2-norm and Wasserstein dis- tance with initial condition defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (12), with r = 10−2 (left panel, without extrapo- lation technique in time) and r = 10−3 (right panel, with extrapolation in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The rest of the data are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' (12)–(16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In Table 2 we present in the first column, the Wasserstein distance, errorW, between one solution with N and N/2 number of grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In the second column we calculate the error with the Richardson extrapolation technique, errorR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We observe an better rate of conver- gence when using the Wasserstein distance, which is related to the fact that the Wasserstein distance depends to a larger extent on the relative displacement of its arguments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=', the topological features of the network, rather than on the local values of the solution at each grid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' 13 0 WS distance slope=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='5 L2-norm slope = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='5 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='9 3 log 10(N)5 Conclusions In this paper we explored the effects of the background permeability parameter r in the elliptic-parabolic model (1)–(2) which represents a PDE framework describing the formation of biological networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We showed that r is the parameter that influences the condition number of the elliptic operator in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' When the condition number becomes too larges, the symmetry of the numerical solution breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' As all the parameters of the system, the initial datum and the source function are symmetric, using a symmetric numerical scheme, we expect the solution to be symmetric at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' We make use of finite differences scheme to compute the solution of the system, with central differences for the space discretization and a symmetric-ADI method in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Because of the non linearity in the Poisson equation, we make use of a time extrapolation to improve the accuracy convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Moreover, we demonstrate that using the Wasserstein distance to measure the order of convergence of the numerical scheme provides better results than the L2 norm, in particular when considering small values of the background permeability r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In [18] we were able to show the (expected) second order accuracy only in the case of r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Here, using the Wasserstein distance to measure the error, we see a significant improvement even for much smaller values of the background permeability, r = 10−2 and r = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In a future work we shall improve the numerical method by implementing a monolithic algorithm to solve the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' In that way, the two unknowns, conductivity tensor C and pressure p, are solved implicitly at each time step, with the metabolic term that can be easily linearized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' An application of IMEX schemes will ensure a higher order of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' The monolithic scheme would also improve the stability in time, allowing us to consider larger time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' References [1] Robert Malinowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Understanding of leaf development – the science of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdFOT4oBgHgl3EQfxzTs/content/2301.12926v1.pdf'} +page_content=' Plants, 2(3):396–415, 2013.' metadata={'source': 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Engineering Electrical and Computer Engineering +University of Ottawa +Cairo, Egypt + ahass202@uottawa.ca + +Ahmed Kamal +dept. biomedical engineering department +Minai university + Minya, Egypt +ahmd654@yahoo.com + + +Alaa Nagy +dept. Engineering Electrical and Computer Engineering +University of Ottawa +Cairo, Egypt +aelba046@uottawa.ca +Daila Farghl +dept. biomedical engineering department +Minai university +Minya, Egypt + dolly.mostafa93@yahoo.com + + +Abstract— In this paper we discuss a new method for +detecting leukemia in microscopic blood smear images using +deep neural networks to diagnose leukemia early in blood. +leukemia is considered one of the most dangerous mortality +causes for a human being, the traditional process of +diagnosis of leukemia in blood is complex, costly, and time- +consuming, so patients could not receive medical treatment on +time; Computer vision classification technique using deep +learning can overcome the problems of traditional analysis of +blood smears, our system for leukemia detection provides +97.3 % accuracy in classifying samples as cancerous or normal +samples by taking a shot of blood smear and passing it as an +input to the system that will check whether it contains cancer +or not. In case of containing cancer cells, then the +hematological expert passes the sample to a more complex +device such as flow cytometry to generate complete +information about the progress of cancer in the blood. +Keywords— Leukemia cells, leukemia detection, deep +neural networks, deep learning. +I. +INTRODUCTION +Leukemia is a type of cancer affecting blood; if it is +detected late, it will result in death. Leukemia develops when +the bone marrow produces an excessive number of aberrant +white blood cells. The normal of the blood system will be +disrupted when aberrant white blood cells are in excess. +Hematologists can identify abnormal blood when they draw +a blood sample and study it[1]. However, hematologists will +inspect microscopic images visually, and the process is time- +consuming and tiring [1 - 3]. Moreover, the process requires +human experts and is prone to errors due to emotional +disturbance and human physical capability, which has its +limitations. +Moreover, it is not easy to get consistent results from +visual inspection [1]. Visual inspection can only give +qualitative results for further research [1]. Studies indicate +that the majority of modern methods. Use all blood-related +data, such as the number of red blood cells, hemoglobin +level, hematocrit level, mean corpuscular volume, and much +more, as the criterion for categorizing disorders like cancer, +thalassemia, Etc. Expensive testing and equipment labs are +required to know all information about blood. An automatic +image processing system is urgently needed and can +overcome related constraints in visual inspection. The system +to be developed will be based on microscopic images to +recognize leukemia cells in blood smears. The early and fast +identification of the leukemia type greatly aids in providing +the appropriate treatment for a particular type of leukemia +[4]. The currently used diagnostic methods rely on analyzing +immuno- phenotyping, fluorescence in situ hybridization, +cytogenetic analysis, and cytochemistry. Sophisticated and +expensive laboratories are required in order to run the +diagnostic methods, and it has been reported to provide a +high ratio of misidentification; with this system, more images +can be processed, reduce analyzing time, exclude the +influence of subjective factors, and increase the accuracy of +identification process at the same time. In machine learning, +the inspection and classification of leukemia will be based on +the texture, shape, size, color, and statistical analysis of white +blood cells. +In contrast, deep learning makes it much more profound +and gets the whole image's exclusive features. This project is +applied +to +increase +efficiency +globally +and +can +simultaneously benefit and be a massive contribution to the +medical and pattern recognition field. The main objective is +to enhance algorithms that can extract data from human +blood where human blood is the primary source to detect +diseases at an earlier stage and can prevent it quickly [5]. +This system should be robust towards diversity among +individuals, sample collection protocols, time, Etc. This +automated system can produce lab results quickly, easily, +and efficiently. +II. DATASET +Images that were used in this project were downloaded +from the internet and are available in ALL IDB[6], ASH +Image Bank Hematology [7], Stock photo, vectors and +Royalty-free +Images[8], +Shutter +stock[9], +Atlas +of +Hematology [10], Atlas of blood smear analysis[11], Blue +Histology and American Society of Hematology [12], This +dataset is composed of 630 images, contains 480 cancer +images and 150 normal images. + +III. METHODOLOGY +A. Data Preprocessing +1) Remove duplication +As the dataset is collected from various resources, +had found that there are some repetitions, some images +contain a watermark, and other contains websites' logo +totally about 43 images, so now the data set has become +587 images. +2) Resizing of images +As the dataset has a different distribution of size, +and for training the CNN model, it was needed to make +all images in the dataset has the same size, so we applied +a resizing technique and make all image 256 x 256 +pixels to reduce the training time. as shown in figure +[5.1] + +3) Filtering images + +Before the processing stage, we need to remove noise +and enhance line structures in images [13], and this is +available by applying a median filter (3 x3) and +sharpening the image (3 x3) ,as shown in Fig[1]. + + +Figure 1:(a) original image,(b) image resized by 256*256 and +filtered by median and sharpen filters + +4) Data augmentation + +Image data augmentation is a method for artificially +increasing the size of a training dataset by producing +altered copies of the dataset's images [14]. The capacity +of fit models to generalize what they have learned to +new pictures may be improved by training deep-learning +neural network models on more data. Additionally, +augmentation techniques can provide variants of the +images. Through the ImageDataGenerator class, the +Keras deep learning neural network framework can fit +models by adding picture data [15]. There are many +different types of augmentation techniques, some of +them as: +a) Flipping +An image flip means reversing the rows or columns +of pixels in the case of a vertical or horizontal flip [9]. +b) Horizontal and Vertical Shift Augmentation +A shift to an image means moving all pixels of the +image in one direction, such as horizontally or +vertically, while keeping the image dimensions the +same; this means that some of the pixels will be +clipped off the image, and there will be a region of the +image where new pixel values will have to be +specified [16]. +c) Random Zoom Augmentation +A zoom augmentation randomly enlarges the image +and either interpolate or adds new pixel values around +the image [16]. +d) Shearing +Shearing will automatically crop the correct area +from the sheared image so that we have an image with +no black space or padding [16]. +e) Interpolation (Nearest) +A technique for creating new data points within the +range of a discrete set of existing data points is +interpolation [17]. Nearest neighbor interpolation is +the most straightforward approach to interpolation. +Rather than calculate an average value by some +weighting criteria or generate an intermediate value +based on complicated rules, this method simply +determines the "nearest" neighboring pixel and +assumes its intensity value of it [13]. And Fig[2] +indicates a sample image with its augmented one. + + + + +(a) (b) + + Fig 2: (a) original image and (b) augmented image. +B. Processing stage +After augmentation processes, our data become 1550 +images for cancer and 1480 for normal. To fit data to +models, we divided it through coding into three data sets: +training set, validation set, and test set by ratios 60%, 20%, +and 20%, respectively. Then the next stage is to train the +model that can be able to classify the images. + +Our optimizing parameters are accuracy and validation +accuracy: to get the best of them as possible, we trained +three networks with different architectures. + + +a) BasicCNN model +In this model, the input images were (RGB) color +images with a resolution of 128x128 pixels. It consists +of 3 convolutional layers with max pooling layers. A +rectified linear unit follows each convolutional layer +(relu). We used a constant filter size (3x3), and the +number of + + +(a) +(b)Filters (128), the stride of ones (equal 1), and fully +connected layers trained for two categories +classification using the sigmoid activation function. +Where we classified the data set into leukemia cells or +normal cells, this architect achieved 90.99% accuracy +and 84.97 % validation accuracy after 17 epochs. + +Fig.3: Indicates the block diagram of the basic CNN model + +b) Alexnet architecture +In this study, we deployed the pre-trained AlexNet to +detect ALL and classify its subtypes. This architecture +was proposed by Krizhevsky et al., nine who +deployed this architecture for the ImageNet Large +Scale Visual Recognition Challenge 2012,20 and won +the challenge in the first place. Input images were +Red, Green, and Blue (RGB) color images with a +resolution of 227 x 227 pixels. It consists of 5 +convolutional layers with three max polling layers. +Each convolutional layer in AlexNet architecture is +followed by a rectified linear unit (ReLU). All the +parameters, including the filter size, the number of +filters, and the stride for each layer, are illustrated in +Fig.4; we replaced the SoftMax layer with a sigmoid +layer as we want to classify the input image into only +two types of this architect achieved 55.35% accuracy +and 49.76 % validation accuracy after 12 epochs. + + +Figure 4: AlexNet architecture for acute lymphoblastic leukemia +subtype classification. Last 2 layers are newly added. + +c) Modification of model used in published paper +This used a retrained model that had been used in a +published paper [20], shown in figure 5, and we +changed the values of the hyperparameter to become +as shown in figure 6; This network contains five +layers. The first three layers perform feature +extraction, and the other two layers (fully connected +and SoftMax) classify the extracted features. The +input image has a size of 128x128x3. In convolution +layer 1, we used a constant filter size of 5x5 and a +total of 16 different filters. The stride is one, and no +zero-padding was applied. The second and third +convolution layers have the same structure as the +first one but a different number of filters, 32 and 64, +respectively. We used a pooling layer with filter size +two and stride 2 to decrease the volume spatially. +During the model we learned, the mini-batch's +chosen size was 128, and ReLu was used as the +activation function. This architect gives: accuracy = +97.73 % validation accuracy = 94.64 % + + + +Fig. 5: The original architecture of CNN in the mentioned paper. + + +Fig. 6: Architecture of CNN after changes in hyperparameter + +IV. EXPREMENTL RESULT +Our experiments were conducted on Python 3.7 with +3030 images, 60% (1818 images) of them for training, 20% +(606 images) for validation, and the remaining 20% (606 +images) for testing our model. In order to evaluate each +model and clarify the best one, we compare them by some +statistically measured parameters: +A. Accuracy + +Train accuracy +For the basic CNN model, train accuracy comes to +90.99% after 17 epochs; our leukemia classifier is doing an +excellent classification, as shown in fig.7.1a. For AlexNet +architecture, the accuracy achieved its maximum accuracy of +56% after 11 epochs; that means our model is terrible on +leukemia classification as shown in fig.6.1 b, but the +Modification of the model used in Thanh et al. paper [18] +achieved the maximum accuracy over all models 97.73 % +after ten epochs as shown in fig.6.1c. 6.1.3 + + + +FC +Max +Max +Max +Conv layer +Conv layer +Conv layer +Input image +pooling +pooling +sigmoid +pooling +128*128*3 +63*63*128 +61*61*128 +30*30*128 +28*28*12814*14*128 +126*126*128 +No padding +No paddingFullyConnected +Layer +Fully Connected +4096 +Layer +Follo +wedbyRelu +1024 +L1 +L2 +L3 +256 +Norn +Convolution..ReLu..MaxPolling..soon +384 +Convolution5 +256 +Convolution2 +ImageSize=13*13 +Image Size = 13*13 +InputImageSize +4 +Convolution +Convolution +Fully Connected +227x227 +96 +-Filtersize=3*3 +-Filter size= 3*3 +4096 +Layer +Image Size=27*27 +-No of filters=384 +Nooffilters=256 +FullyConnected +Foilowedby Softmax +Convolution1 +Convolution +Stride=1 +-Stride=1 +-Filter size= 5*5 +Maxpooling +Layer +Image Size = 55*55 +-No ofilters= 256 +Filtersize=3*3 +-Stride= 1 +-Stride=2 +Convolution +-Filter size=11*11 +Max pooling +-Nooffilters=96 +-Filter size=3*3 +-Stride=2 +-Stride= 4 +MaxpoolingInput +Conv layer 1 +Max-Pooing 1 +Conylayer2 +Max-Pooling 2Comv layer 3 +sdauragayuto +Com2Feane maps +Peo2Featuremap +Cows:Fetmas +Pooit:Feanremaps +FC Sohmax +100x100x3 +96x96x16 +48x48x16 +46x46x32 +23x23x32 +21x21x64FC +sigmoid +Max +Max +OH +Convlayer +Inputimage +Convlayer +pooling +Convlayer +pooling +128x128x3 +62x62x16 +29x29x32 +25x25x64 +64 +58x58x32 +124x124x16 +Stride=2 +Stride=1 +Stride=2 +Stride=1 +Stride=1 +No padding +NopaddingValidation accuracy +Basic CNN Model validation accuracy reaches 85% after +17 epochs, as shown in fig.7.1a. Therefore, we expect our +model to perform with ~85% accuracy on new data. For +AlexNet architecture, the accuracy achieved its maximum +accuracy of 53.6% after 11 epochs; that means our model is +terrible on leukemia classification, as shown in fig.6.1 b; +This means that we expect our model to perform with +~53.6% accuracy on new data. Nevertheless, in Modification +of the model used in Thanh et al. paper [94] achieved the +maximum validation accuracy over all models at 94.3 % +after ten epochs, as shown in fig.6.1c. Therefore, we expect +our model to perform with ~94.3 % accuracy on new data. +We notice that our train metric increases as epochs increase +while the validation accuracy metric decreases. That means +that our model fits the training set better but slightly loses its +ability to predict new data, indicating that our models are +beginning to overfit. + +Fig.7.1a, curve of val acc & train acc for basic CNN model + + +Fig.7.2b, curve of val acc & train acc for AlexNet architecture + + +Fig.7.2c, curve of validation accuracy & train accuracy for +Modification of model used in Thanh et al paper [18] +B. Confusion Matrix + +A classification problem's predicted outcomes are compiled +in a confusion matrix. The count values describe the number +of accurate and inaccurate predictions for each class. +Because it is feasible to see the relationships between the +classifier outputs and the real ones, this is a great alternative +for reporting results in M-class classification issues. For the +basic CNN model, the number of leukemia images that are +predicted as leukemia is 372, the number of leukemia +images that are predicted as normal is 8, the number of +normal images predicted as normal is 269, and the number +of normal images that are predicted as leukemia is 51, as +shown in fig. 7.3 a. These accuracies show that this model is +good at predicting leukemia images but bad at predicting +normal images. For AlexNet architecture, the number of +leukemia images that are predicted as leukemia is 0, the +number of leukemia images that are predicted as normal is +380, the number of normal images predicted as normal is +157, and the number of normal images that are predicted as +leukemia is 163, as shown in fig.7.3 b. These accuracies +show that this model is terrible at predicting normal images. +The number of leukemia images predicted as leukemia for +the modified model used in the published paper [18] is 369, +the number of leukemia images predicted as normal is 11, +the number of normal images predicted as normal is 301, +and the number of normal images predicted as leukemia is +19; as shown in fig.7.3 c. These accuracies show that this +model has done a great job of predicting normal images. + + + +Fig.7.3a, Confusion matrix of basic CNN model + + + +Fig.7.3b, Confusion matrix of AlexNet Architecture + + +trainaccvsval acc +0.9 +0.8 +0.6 +0.5 +0.4 +Tain +2 +8 +numof Epochstrain acc vs val acc +0.552 +0.550 +0.548 +accuracy +0.546 +0.544 +0.542 +0.540 +train +0.538 +val +0.536 +0 +2 +4 +6 +8 +10 +12 +numofEpochs0.98 +trainaccvsval acc +0.96 +0.92 +train +0.90 +val +0 +1 +2 +3 +4 +5 +6 +7 +8 +numofEpochsConfusionmatrix +320 +372 +280 +class o(cancer) +240 +True label +200 +160 +120 +class 1(normal) +51 +269 +80 +40 +Predicted labelConfusion.matrix +320 +0 +380 +280 +class O(cancer) +240 +Truelabel +200 +160 +120 +class.1(normal) +163 +157 +80 +40 +Predicted label + +Fig.7.3c, Confusion matrix for Modification of model used in +Thanh et al paper [18] +C. Percsision +It is calculated as the proportion of accurate positive results +to those that the classifier predicted to be positive. Our CNN +model has medium precision, AlexNet architecture has very +low precision, and the modified version of the model used in +Thanh et al.'s [18] paper has good precision due to its +goodness method, as shown in fig. 7.4a. +D. Recall +It is determined by dividing the total number of pertinent +samples (all samples that should have been labeled as +positive) by the total number of reliable positive results. +As illustrated in fig. 7.4a for our CNN model, fig. 7.4b for +the AlexNet architecture, and fig. 7.4c for the Thanh et al. +paper [18]. The perfect model regarded recall is the third +model. +The first CNN model in class 1 has a high recall but low +precision. This means that most of the positive examples are +correctly recognized (low FN), but there are a lot of false +positives. Nevertheless, in class 0, low recall and high +precision show that we miss a lot of positive examples (high +FN), but those we predict as positive are indeed positive +(low FP). +E. F1 Score +The harmonic mean of recall and accuracy is the F1 score. +The F1 score has a range of [0, 1]. It tells how accurate the +classifier is (how many instances it classifies correctly) and +how robust it is (it recognizes a significant number of +instances). As illustrated in fig. 7.4a for our CNN model, +fig. 7.4b for the AlexNet architecture, and fig. 7.4c for the +Thanh et al. paper [18]. These figures show that the +modification of the model used in Thanh et al.'s paper [18] +is precise and robust. +F. Support +Support is the number of samples accurately representing +the response within that category. +It provides information on the precise numbers of each class +in the test data. +Figures 7.4a and 7.4b for the fundamental CNN model, 7.4b +for the AlexNet architecture, and 6.8c for a modified version +of the model from the Thanh et al. work [18] serve as +examples. + + + +Fig.7.4a, values of precision, recall, f1 score and support for our +CNN model + + + + +Fig.7.4b, values of precision, recall, f1 score and support for +AlexNet architecture + + +Fig.7.4c, values of precision, recall, f1 score and support for +Modification of model used in Thanh et al paper [18] + +V. +DISCUSSION + +Leukemia is a malignancy that affects the body's blood- +forming tissues, including the lymphatic system and bone +marrow. To get the most effective treatment, the patient +needs early Diagnosis, so we deploy three models using the +power of CNN to classify blood smears into normal and +abnormal. +Our dataset had not been taken under the same conditions as +it was collected from various resources, and it needed to be +bigger to use with DL. To overcome this problem, we used +the power of data augmentation; this solution was suitable +for us; our data before augmentation was 260 images, and +after augmentation became 3030 images. +Our optimizing parameters were accuracy and validation +accuracy; by using CNN, we trained the model: +• +the First model consists of 3 convolutional layers +with max pooling layers. Its accuracy was 90% +and 84.97 % validation accuracy. It was terrible +with our dataset due to its few layers, so we +trained another model +• + the Second model was AlexNet; this architecture +proved its efficiency in CNN models, so we +trained it with our data, input is (RGB) color +images with a resolution of 227 x 227 pixels. It +consists of 5 convolutional layers with three max +polling layers. These models achieved 55.35% +accuracy and 49.76 % validation accuracy. We +found that it does not fit our dataset. So we still +have the same problem of low accuracy and keep +looking for another model +• +In the last model, we used a retrained model that +had been used in a published paper [18]; it +contain7 layers. The first five layers perform +feature extraction, and the other two layers (fully +connected and SoftMax) classify the extracted + +Confusionmatrix +320 +369 +11 +280 +class o(cancer) +240 +True label +200 +160 +120 +class 1(normal +19 +301 +80 +40 +cer) +Predicted labelprecision +recall +f1-score +support +class 0(cancerous) +0.98 +0.62 +0.76 +78 +class 1(Normal) +0.84 +0.99 +0.91 +163precision +recall +f1-score +class 1(cancer) +0.00 +0.0 +0.00 +178 +class e(normal) +0.49 +1.00 +0.66 +172precision +recall +f1-score +support +class o(cancer) +0.97 +0.95 +0.96 +373 +class 1(normal) +0.94 +0.97 +0.95 +327features. The input image has a size of 128x128x3. +This architect has an accuracy of 97.73 % +validation accuracy is 94.64 %, finally, we found +that this model fit our data +• + +VI. CONCLUSIONS + +In this system, we investigated the application of deep +CNNs. We deployed a pre-trained model for detecting and +classifying the blood sample into normal and abnormal +samples using microscopic blood sample images and +convolutional neural network classification algorithms. The +system was built by deep learning, which uses all features in +microscopic images, not only examining changes of specific +features as a classifier input. We have performed the pre- +trained model in a largely augmented dataset to confirm the +system's accuracy and reliability. By performing data +augmentation, we can achieve 97.3% accuracy. The system +has high accuracy, and less processing time (show results in +less than 30 seconds) +, minor errors, and early identification of leukemia +successful in giving the patient the proper care. And cheaper +cost. + +The detection system was built in three parts: +1) the acquisition part, which consists of a digital camera +that has been installed at the top of the eyepiece of the +microscope, +2) pre-trained CNN model responsible for the +classification system. +3) a graphical user interface to display the image obtained +from the camera and show the classification results. +VII. FUTURE WORK +Expanding the focus on classifying the subtypes of leukemia +cells such as Acute Myeloid Leukemia or AML, Chronic +Myeloid Leukemia or CML, Acute Lymphoid Leukemia or +ALL, and Chronic Lymphoid Leukemia or CLL not only +separating between cancerous and non-cancerous cells and +developing a convenient environment to construct an +extensive leukemia dataset as this topic of research suffer +from leaks in images. + +VIII. 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Chapter +A Logical Calculus of the Ideas Immanent in Nervous Activity, +page15–27.ISBN:0-262-01097-6.URL: http://dl.acm.org/citation.cfm? +id=65669.104377. +[6] ALL-IDB Acute Lymphoblastic Leukemia Image Database for Image +Processing. +(n.d.). +https://scotti.di.unimi.it/all/. +https://scotti.di.unimi.it/all/ +[7] ImageBank +| +Home +| +Regular +Bank. +(n.d.). +https://imagebank.hematology.org/. +https://imagebank.hematology.org/ +[8] Stock Photos, Vectors and Royalty Free Images from 123RF. (n.d.). +https://www.123rf.com/. https://www.123rf.com/ +[9] Stock Images, Photos, Vectors, Video, and Music | Shutterstock. +(n.d.). https://www.shutterstock.com/. https://www.shutterstock.com/ +[10] ATLAS +OF +HEMATOLOGY. +(n.d.). +http://www.hematologyatlas.com/principalpage.htm. +http://www.hematologyatlas.com/principalpage.htm +[11] Chronolab. +(n.d.). +https://chronolab.com/atlas/hemat. +https://chronolab.com/atlas/hemat +[12] American Society of Hematology - Hematology.org. 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(n.d.). +https://medium.com/nanonets/how-to-use-deep-learning-when-you- +have-limited-data-part-2-data-augmentation-c26971dc8ced. +https://medium.com/nanonets/how-to-use-deep-learning-when-you- +have-limited-data-part-2-data-augmentation-c26971dc8ced +[17] mdbloice/Augmentor: Image augmentation library in Python for +machine learning. (n.d.). https://github.com/mdbloice/Augmentor. +https://github.com/mdbloice/Augmentor +[18] T. T. P. Thanh, Caleb Vununu, Sukhrob Atoev, Suk-Hwan Lee, and +Ki-Ryong Kwon , “Leukemia Blood Cell Image Classification Using +Convolutional Neural Network “ International Journal of Computer +Theory and Engineering, Vol. 10, No. 2, April 2018 + + + + diff --git a/L9AyT4oBgHgl3EQfT_dJ/content/tmp_files/2301.00116v1.pdf.txt b/L9AyT4oBgHgl3EQfT_dJ/content/tmp_files/2301.00116v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a50f5297fd45d4f61737afd37c46cd8744c2860 --- /dev/null +++ b/L9AyT4oBgHgl3EQfT_dJ/content/tmp_files/2301.00116v1.pdf.txt @@ -0,0 +1,438 @@ +Diffusion within pores fully revealed +by magnetic resonance* +Evren ¨Ozarslan,1∗ Cem Yolcu,1 Alfredo Ordinola,1 Deneb Boito,1 +Tom Dela Haije,2 Mathias Højgaard Jensen,2 Magnus Herberthson3 +1Department of Biomedical Engineering, Link¨oping University, Link¨oping, Sweden +2Department of Computer Science, University of Copenhagen, Copenhagen, Denmark +3Department of Mathematics, Link¨oping University, Link¨oping, Sweden +∗To whom correspondence should be addressed; E-mail: evren.ozarslan@liu.se. +Abstract +Probing the transport of fluids within confined domains is important in many areas including +material science, catalysis, food science, and cell biology. The diffusion propagator fully charac- +terizes the diffusion process, which is highly sensitive to the confining boundaries as well as the +structure within enclosed pores. While magnetic resonance has been used extensively to observe +various features of the diffusion process, its full characterization has been elusive. Here, we address +this challenge by employing a special sequence of magnetic field gradient pulses for measuring the +diffusion propagator, which allows for ‘listening to the drum’ and determining not only the pore’s +shape but also diffusive dynamics within it. +The diffusion propagator indicates the probability that a particle located at position x moves to +x′ between two specified times. The diffusion propagator fully describes the diffusive motion in +environments having restricting or semi-permeable walls, spatially varying diffusivity, external forces, +etc. Let us consider a time-invariant diffusion scenario in d dimensions within a closed and connected +domain Ω under the dimensionless potential energy field U(x). The diffusion propagator for a time +interval of duration t, p(x′, t|x), is then the solution to the system of equations +∇ · +� +D(x′)e−U(x′)∇eU(x′)p(x′, t|x) +� += ∂p(x′, t|x) +∂t +(1a) +lim +t→0 p(x′, t|x) = δ(x′ − x) +(1b) +ˆn · D(x′) e−U(x′) ∇eU(x′)p(x′, t|x) = 0, +x′ ∈ ∂Ω , +(1c) +where ∇ is a vector of partial derivatives with respect to the components of x′, and ˆn is the surface +normal at x′. The first of these is the diffusion equation with diffusion tensor D(x′). The initial +condition is given by Eq. (1b), while the last equation is the reflective boundary condition. In this +*For earlier preprints on this technique, the reader is referred to [1--3]. +1 +arXiv:2301.00116v1 [cond-mat.mtrl-sci] 31 Dec 2022 + +Diffusion within pores revealed by MR +¨Ozarslan et al. +example, U(x), D(x) and Ω are quantities describing the fluid properties or a static picture of the +environment all of which give rise to the particular diffusive dynamics, which is captured by the +propagator. If the diffusion propagator is available, the diffusion tensor and the potential landscape +can be determined, respectively, from its short-time and long-time behaviors, while Ω is given by its +support. Clearly, the diffusive process is an indirect yet powerful means of recovering the structure of +the medium, making it relevant to many disciplines. +Magnetic resonance has been the method of choice for many characterization studies due to its +noninvasive nature and exquisite sensitivity to diffusion, which has been realized since its early +days [4, 5]. In a typical MR experiment, the specimen is subjected to a magnetic field Bz whose +direction defines the z-axis by convention. The magnetic moments of the spin-bearing particles exhibit +coherence, synergistically yielding a magnetization vector that develops along the z-axis. By applying +electromagnetic radiation at a specific frequency, magnetization due to the nuclei of the atoms of +interest can be tilted towards the xy-plane, upon which it undergoes Larmor precession at an angular +frequency given by ω = γBz, where γ is the gyromagnetic ratio, which is specific to the particular +atomic nuclei being examined. Such precession leads to changing magnetic flux around it, inducing +a potential difference in a nearby antenna, which is referred to as the MR signal. During the course +of the MR experiment, different particles acquire different phase shifts +� +− +� +ω(x, t) dt +� +due to the +differences in the local magnetic field and experimental manipulations of Bz. +One such manipulation introduced by Stejskal and Tanner in 1965 involves incorporating pulsed +magnetic field gradients (∇Bz) into MR acquisitions for performing diffusion measurements in a +controllable way [6]; gradient pulses have also been the building blocks of MR imaging [7]. Stejskal +and Tanner’s experiment (see Figure 1a) featuring two gradient pulses of equal duration is still the most +widely employed diffusion encoding method. Here, qa denotes the integral of the gradient vector over +its duration, multiplied by γ. A spin bearing particle, whose average positions during the application of +the first and second pulses denoted by x and x′, suffers phase shifts of qa · x and −qa · x′, respectively, +due to the Larmor precession frequency being proportional to the magnetic field. Consequently, the +MR signal intensity (divided by the intensity with qa = 0) is given by +E(a) +∆ (qa) = +� +Ω +dx ρ(x) +� +Ω +dx′ p(x′, ∆|x) e−iqa·(x′−x) , +(2) +where ρ(x) is the initial spin density and for simplicity, we assumed short pulses that encode the +instantaneous positions of the particles. Conventional experiments for measuring self-diffusion start at +the steady state, i.e., with ρ(x′) = limt→∞ p(x′, t|x) in the absence of sources and relaxation sinks. +The signal is just the Fourier transform of the ensemble averaged propagator (EAP) defined by +¯P∆(xnet) = +� +Ω +dx ρ(x) p(x + xnet, ∆|x) , +(3) +where xnet = x′ − x is the net displacement vector. Thus, EAP can be computed from the inverse +Fourier transform of the signal +¯P∆(xnet) = +1 +(2π)d +� +Rd dqa eiqa·xnet E(a) +∆ (qa) . +(4) +The EAP is a substantially compromised version of the propagator, indicating the likelihood of net +displacements averaged for all spins irrespective of where they are within the structure. Despite this +limitation, it exhibits very interesting features enabling some understanding of the underlying structure, +thus has been widely utilized in characterizing porous media [9--11] as well as tissues [12, 13]. +Recently, Laun et al. introduced another two-pulse experiment, one pulse being long, the other +narrow [8] as shown in Fig. 1b. Assuming closed pores and uniform structure within, the particles visit +Page 2 + +Diffusion within pores revealed by MR +¨Ozarslan et al. +qa +−qa +qb +q +−qb +−q−q' +q' +Δ +Δ +(a) +(b) +(c) +Figure 1: The diffusion encoding pulse sequences considered. (a) Stejskal-Tanner sequence [6] allows +the measurement of the ensemble average propagator. (b) The gradient waveform introduced by +Laun et al. [8] enables measurement of the long diffusion time limit of the propagator. (c) The pulse +sequence introduced here makes it possible to map the diffusion propagator. +every site within the pore with equal probability during the application of the long pulse. Thus, the +positional average of each and every trajectory is very tightly distributed around the pore’s center-of- +mass. As such, the long pulse has no effect other than diminishing the integral of the waveform, which +is a necessary condition for making the signal independent of the pore’s position within the specimen. +As a result, the signal from all pores add up, generating a detectable signal level even for a specimen +comprising small amount of fluid. If the second pulse is short, the sequence simply introduces a phase +shift proportional to each spin’s location. The total signal for a connected pore is then given by +E(b)(qb) = +� +˜Ω +dx ˜ρ(x) e−iqb·x , +(5) +where x is the position of the spin with respect to the pore’s center-of-mass located at xcm while +˜ρ(x) = ρ(x + xcm) and ˜Ω indicates the domain translated so that the center of mass of the pore is at +the origin. Thus, the sequence is indeed ‘‘an imaging experiment in disguise,’’ [8] making it possible +to obtain the image of the pore indicator function through an inverse Fourier transform of E(b)(qb). +In more general terms, the obtained quantity is the steady-state distribution of the fluid [14], thus not +informative of the diffusion process. +Measuring the diffusion propagator +Here, we consider the sequence in Figure 1c, which combines the key elements of the two sequences +discussed above. The long pulse is there so that the integral of the waveform vanishes, and contributions +from all pores are independent of their position within the sample. The two subsequent pulses q and q′ +Page 3 + +Diffusion within pores revealed by MR +¨Ozarslan et al. +introduce phase shifts that depend on the particles’ positions during their application (in a frame of +reference whose origin is at xcm---the center of mass of the fluid filling up the pore), denoted by x and +x′, respectively. When the second and third pulses are short, the signal is given by +E(c) +∆ (q, q′) = +� +˜Ω +dx ˜ρ(x) +� +˜Ω +dx′ ˜p(x′, ∆|x) e−i(q·x+q′·x′) , +(6) +where ˜p(x′, ∆|x) = p(x′ + xcm, ∆|x + xcm). The propagator, can be obtained via the 2d-dimensional +inverse Fourier transform of the signal +W∆(x, x′) := +1 +(2π)2d +� +Rd dq +� +Rd dq′ E(c) +∆ (q, q′) ei(q·x+q′·x′) +(7) +along with an estimate of ˜ρ(x), which is made available by the d-dimensional inverse Fourier transform +of the subset of the data with q′ = 0. In other words, the diffusion propagator is given by +˜p(x′, ∆|x) = +� +Rd dq eiq·x � +Rd dq′ eiq′·x′ E(c) +∆ (q, q′) +(2π)d � +Rd dq eiq·x E(c) +∆ (q, 0) +. +(8) +Structure within the pore +We demonstrate the estimation of the diffusion propagator of a simulated one-dimensional pore +(interval). The pore is partitioned into two exchanging compartments, with diffusion coefficients DL +and DR, separated by a membrane of permeability w. The walls of the pore are purely reflective. The +simulations summarized in Figure 2 illustrates the agreement of the reconstructed propagator (second +row) with the true propagators (top row) at three different time intervals. The associated EAPs are +depicted in the bottom row. The presence of a membrane within the pore space is conspicuous in the +estimated propagators while the EAPs are not descriptive. +Structural dispersity +The propagator-sensitive sequence of Figure 1c can be used to characterize porous media having +structural dispersity. To this end, we consider such a specimen having N isolated pores where the nth +pore has the non-attenuated signal fraction fn. The Fourier transforms of the signals E(c) +∆ (q, q′) and +E(c) +∆ (q, 0) yield +W∆(x, x′) = +N +� +n=1,2,3,... +fn ˜ρn(x) ˜pn(x′, ∆|x) +(9a) +˜ρ(x) = +N +� +n=1,2,3,... +fn ˜ρn(x) , +(9b) +which are just weighted averages of the respective quantities for all pores translated so that the pores’ +centers of mass coincide. Numerous quantities can be introduced for characterizing the underlying +dispersity in the specimen. For example, a dimensionless ‘variance map’ can be obtained through the +expression +σg(x) := (W∞(x, x) − ˜ρ(x)2)1/2 +ρmax +, +(10) +Page 4 + +Diffusion within pores revealed by MR +¨Ozarslan et al. +where ρmax := ˜ρ(xm) is the maximum value of ρ(x). A dispersity index can be introduced through +DI := σ2 +g(xm), which is equal to ⟨V −1⟩⟨V ⟩ − 1 in the absence of external forces when all pores +contribute at xm and the signal fraction fn is proportional to the pore volume Vn. Here, ⟨·⟩ denotes +averaging over all pores. We shall now consider general x = r ˆu and x′ = r′ ˆu′ where r = |x|, +r′ = |x′|, and ˆu and ˆu′ indicate the directions of x and x′, respectively. One can define the quantity +Φ(ˆu, ˆu′) := +� ∞ +0 +W∞(r ˆu, r ˆu′) r2d−1 dr , +(11) +which is constant for a medium composed of isotropic pores. It has peaks at ˆu′ = ˆu when the pores are +anisotropic. If the pore shape has antipodal symmetry, ˆu′ = −ˆu will exhibit another peak. Similarly, +for shapes that exhibit other symmetries (e.g., cross or star-shaped pores) there will be other peaks. +In Figure 3, we illustrate the maps of the quantities in (9b)-(11) for ten different media with different +compositions of two-dimensional pores. +Figure 2: Representative snapshots of the simulated propagator estimation experiment for exchanging +intervals of length LL and LR having diffusivity DL and DR, for the left and right comnpartments, +respectively. Shown from top to bottom are: true propagator, estimated propagator, and EAP (true +and estimated). The density plots are of p′(x′, ∆|x) = tanh +p(x′,∆|x) +2/(LL+LR) for better depiction. The true +propagator is computed by its (truncated) spectral decomposition. Membrane position is emphasized +by dashed lines. The relaxation time scales τL = L2 +L/π2DL and τR = L2 +R/π2DR correspond (roughly) +to the process of diffusion within the compartments, and τex = √DLDR/w2 to the exchange between +them. +Page 5 + +Diffusion within pores revealed by MR +¨Ozarslan et al. +Figure 3: Maps derived from the long time diffusion measurements for different specimens. The axes +of the Φ maps vary between −π and π. The DI values were estimated to be 0, 0.28, 0.28, 3.8 × 10−5, +5.0 × 10−5, 0.41, 3.9 × 10−5, 4.3 × 10−5, 1.2 × 10−5, 1.5 × 10−5 (top to bottom). +‘Hearing the drum’ +In 1966, Kac posed the now famous question ‘‘Can one hear the shape of a drum?’’ [15], which +pertains to recovering the geometry of an enclosing boundary from the eigenspectrum of the Laplacian +Page 6 + +0000 +00000000000 +00000X ++ +X +X+xxx++++XXXX+ ++++X +XXX +X +X +X +X +?+ADiffusion within pores revealed by MR +¨Ozarslan et al. +operator. The pulse sequence of Laun et al. (depicted in Figure 1b) demonstrated that the shape can be +recovered from the MR signal. By enabling the measurement of the diffusion propagator, our gradient +waveform illustrated in Figure 1c provides access to the diffusion dynamics within the pore and indeed +to the spectrum of the Laplacian. To see this, one can exploit the eigenfunction expansion of the +propagator, which leads to the expression +� +Ω +p(x, t|x) dx = +� ∞ +0 +g(λ) e−λt dλ . +(12) +Clearly, the density of states, g(λ), is accessible from the propagator through an inverse Laplace +transform while the propagator is obtained from the signal of the waveform in Figure 1c through +Eq. (8). In Figure 4, we illustrate the recovery of the density of states from simulated signals for the +one-dimensional scenario involving diffusion in the direction perpendicular to two parallel plates. +Figure 4: How magnetic resonance ‘‘hears the drum.’’ (a) First row: MR signal profiles obtained +using the diffusion encoding in Figure 1c. Second row: Propagators obtained via Eq. (8). (b) +Left-hand-side of Eq. (12) plotted against time and (c) its inverse Laplace transform revealing the +density of states function. The numerical Laplace inversion [16] was performed using the package at +https://github.com/caizkun/pyilt. +Page 7 + +t= 0.001ms +t = 0.031ms +t = 3.000ms +a +4 +4 +4 +2 +2 +2 +[t-ur],b +[t-wr],b +[t-wn],b +0 +0 +0 +-2 +-2 +-2 +-4 - +-4 - +-4 +-4　-2 +0 +2 +4 +-4 +¥2 +4 +-2 +0 +2 +4 +q [μm-1] +q [μm-1] +q [μm-1] +2 +2 +2 +[un],x +[wr],x +[wn] +0 +0 +0 +X +-2 +-2 +-2 +-2 +0 +2 +-2 +0 +2 +-2 +0 +2 +x [μm] +x [μm] +x [μm] +109 +6000 +c +Reconstructed Spectrum +5000 +106 +Expected peaks +X +4000 +103 +X +X +X +XXXXX +3000 +100 +2000 +10-3 +1000 +10-6 +0 +10-9 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +103 +104 +105 +106 +t[ms] +入[ms-1]Diffusion within pores revealed by MR +¨Ozarslan et al. +Concluding remarks +In conclusion, we introduced a new technique, which facilitates the characterization of the diffusion +process within pores in full. This is accomplished by mapping the diffusion propagator through Fourier +transforms and can be related to the density of states function making the shape of the drum ‘‘heard’’ +by magnetic resonance. The technique allows for mapping the structure within closed pores as well as +characterizing disperse specimens with unprecedented detail. +Acknowledgments +E ¨O thanks Carl-Fredrik Westin for a stimulating conversation, and Nicolas Moutal and Denis +Grebenkov for sharing their code on diffusion separated by semi-permeable membranes [17]. +References +[1] E. ¨Ozarslan, ‘‘Recovering almost everything diffusion could reveal,’’ (2021), arXiv:2105.00145 . +[2] A. M. Ordinola Santisteban and E. ¨Ozarslan, ‘‘Magnetic resonance measurement and reconstruction of the +diffusion propagator,’’ (2021), arXiv:2106.16181 . +[3] E. ¨Ozarslan and M. Herberthson, ‘‘Demystifying magnetic resonance measurements of the true diffusion +propagator,’’ (2021), arXiv:2112.15584 . +[4] E. L. Hahn, Phys Rev 80, 580 (1950). +[5] H. Y. Carr and E. M. Purcell, Phys Rev 94, 630 (1954). +[6] E. O. Stejskal and J. E. Tanner, J Chem Phys 42, 288 (1965). +[7] P. C. Lauterbur, Nature 242, 190 (1973). +[8] F. B. Laun, T. A. Kuder, W. Semmler, and B. Stieltjes, Phys Rev Lett 107, 048102 (2011). +[9] J. K¨arger and W. Heink, J Magn Reson 51, 1 (1983). +[10] P. T. Callaghan, A. Coy, D. MacGowan, K. J. Packer, and F. O. Zelaya, Nature 351, 467 (1991). +[11] P. P. Mitra, P. N. Sen, L. M. Schwartz, and P. Le Doussal, Phys Rev Lett 68, 3555 (1992). +[12] D. G. Cory and A. N. Garroway, Magn Reson Med 14, 435 (1990). +[13] V. J. Wedeen, P. Hagmann, W.-Y. I. Tseng, T. G. Reese, and R. M. Weisskoff, Magn Reson Med 54, +1377 (2005). +[14] E. ¨Ozarslan, K. S¸ims¸ek, C. Yolcu, and C. F. Westin, in Proc Intl Soc Mag Reson Med, Vol. 25 (2017) p. +1830. +[15] M. Kac, Am Math Mon 73, 1 (1966). +[16] S. W. Provencher, Comput Phys Commun 27, 213 (1982). +[17] N. Moutal and D. Grebenkov, J Sci Comput 81, 1630 (2019) +Page 8 + diff --git a/L9AyT4oBgHgl3EQfT_dJ/content/tmp_files/load_file.txt b/L9AyT4oBgHgl3EQfT_dJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e757745c254358db854cc9fe5c1ab5d3f0e60353 --- /dev/null +++ b/L9AyT4oBgHgl3EQfT_dJ/content/tmp_files/load_file.txt @@ -0,0 +1,231 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf,len=230 +page_content='Diffusion within pores fully revealed by magnetic resonance* Evren ¨Ozarslan,1∗ Cem Yolcu,1 Alfredo Ordinola,1 Deneb Boito,1 Tom Dela Haije,2 Mathias Højgaard Jensen,2 Magnus Herberthson3 1Department of Biomedical Engineering, Link¨oping University, Link¨oping, Sweden 2Department of Computer Science, University of Copenhagen, Copenhagen, Denmark 3Department of Mathematics, Link¨oping University, Link¨oping, Sweden ∗To whom correspondence should be addressed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' E-mail: evren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='ozarslan@liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Abstract Probing the transport of fluids within confined domains is important in many areas including material science, catalysis, food science, and cell biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The diffusion propagator fully charac- terizes the diffusion process, which is highly sensitive to the confining boundaries as well as the structure within enclosed pores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' While magnetic resonance has been used extensively to observe various features of the diffusion process, its full characterization has been elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Here, we address this challenge by employing a special sequence of magnetic field gradient pulses for measuring the diffusion propagator, which allows for ‘listening to the drum’ and determining not only the pore’s shape but also diffusive dynamics within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The diffusion propagator indicates the probability that a particle located at position x moves to x′ between two specified times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The diffusion propagator fully describes the diffusive motion in environments having restricting or semi-permeable walls, spatially varying diffusivity, external forces, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Let us consider a time-invariant diffusion scenario in d dimensions within a closed and connected domain Ω under the dimensionless potential energy field U(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The diffusion propagator for a time interval of duration t, p(x′, t|x), is then the solution to the system of equations ∇ · � D(x′)e−U(x′)∇eU(x′)p(x′, t|x) � = ∂p(x′, t|x) ∂t (1a) lim t→0 p(x′, t|x) = δ(x′ − x) (1b) ˆn · D(x′) e−U(x′) ∇eU(x′)p(x′, t|x) = 0, x′ ∈ ∂Ω , (1c) where ∇ is a vector of partial derivatives with respect to the components of x′, and ˆn is the surface normal at x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The first of these is the diffusion equation with diffusion tensor D(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The initial condition is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' (1b), while the last equation is the reflective boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' In this For earlier preprints on this technique, the reader is referred to [1--3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='00116v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='mtrl-sci] 31 Dec 2022 Diffusion within pores revealed by MR ¨Ozarslan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' example, U(x), D(x) and Ω are quantities describing the fluid properties or a static picture of the environment all of which give rise to the particular diffusive dynamics, which is captured by the propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' If the diffusion propagator is available, the diffusion tensor and the potential landscape can be determined, respectively, from its short-time and long-time behaviors, while Ω is given by its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Clearly, the diffusive process is an indirect yet powerful means of recovering the structure of the medium, making it relevant to many disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Magnetic resonance has been the method of choice for many characterization studies due to its noninvasive nature and exquisite sensitivity to diffusion, which has been realized since its early days [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' In a typical MR experiment, the specimen is subjected to a magnetic field Bz whose direction defines the z-axis by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The magnetic moments of the spin-bearing particles exhibit coherence, synergistically yielding a magnetization vector that develops along the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' By applying electromagnetic radiation at a specific frequency, magnetization due to the nuclei of the atoms of interest can be tilted towards the xy-plane, upon which it undergoes Larmor precession at an angular frequency given by ω = γBz, where γ is the gyromagnetic ratio, which is specific to the particular atomic nuclei being examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Such precession leads to changing magnetic flux around it, inducing a potential difference in a nearby antenna, which is referred to as the MR signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' During the course of the MR experiment, different particles acquire different phase shifts � − � ω(x, t) dt � due to the differences in the local magnetic field and experimental manipulations of Bz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' One such manipulation introduced by Stejskal and Tanner in 1965 involves incorporating pulsed magnetic field gradients (∇Bz) into MR acquisitions for performing diffusion measurements in a controllable way [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' gradient pulses have also been the building blocks of MR imaging [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Stejskal and Tanner’s experiment (see Figure 1a) featuring two gradient pulses of equal duration is still the most widely employed diffusion encoding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Here, qa denotes the integral of the gradient vector over its duration, multiplied by γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' A spin bearing particle, whose average positions during the application of the first and second pulses denoted by x and x′, suffers phase shifts of qa · x and −qa · x′, respectively, due to the Larmor precession frequency being proportional to the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Consequently, the MR signal intensity (divided by the intensity with qa = 0) is given by E(a) ∆ (qa) = � Ω dx ρ(x) � Ω dx′ p(x′, ∆|x) e−iqa·(x′−x) , (2) where ρ(x) is the initial spin density and for simplicity, we assumed short pulses that encode the instantaneous positions of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Conventional experiments for measuring self-diffusion start at the steady state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=', with ρ(x′) = limt→∞ p(x′, t|x) in the absence of sources and relaxation sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The signal is just the Fourier transform of the ensemble averaged propagator (EAP) defined by ¯P∆(xnet) = � Ω dx ρ(x) p(x + xnet, ∆|x) , (3) where xnet = x′ − x is the net displacement vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Thus, EAP can be computed from the inverse Fourier transform of the signal ¯P∆(xnet) = 1 (2π)d � Rd dqa eiqa·xnet E(a) ∆ (qa) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' (4) The EAP is a substantially compromised version of the propagator, indicating the likelihood of net displacements averaged for all spins irrespective of where they are within the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Despite this limitation, it exhibits very interesting features enabling some understanding of the underlying structure, thus has been widely utilized in characterizing porous media [9--11] as well as tissues [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Recently, Laun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' introduced another two-pulse experiment, one pulse being long, the other narrow [8] as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Assuming closed pores and uniform structure within, the particles visit Page 2 Diffusion within pores revealed by MR ¨Ozarslan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=" qa −qa qb q −qb −q−q' q' Δ Δ (a) (b) (c) Figure 1: The diffusion encoding pulse sequences considered." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' (a) Stejskal-Tanner sequence [6] allows the measurement of the ensemble average propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' (b) The gradient waveform introduced by Laun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' [8] enables measurement of the long diffusion time limit of the propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' (c) The pulse sequence introduced here makes it possible to map the diffusion propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' every site within the pore with equal probability during the application of the long pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Thus, the positional average of each and every trajectory is very tightly distributed around the pore’s center-of- mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' As such, the long pulse has no effect other than diminishing the integral of the waveform, which is a necessary condition for making the signal independent of the pore’s position within the specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' As a result, the signal from all pores add up, generating a detectable signal level even for a specimen comprising small amount of fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' If the second pulse is short, the sequence simply introduces a phase shift proportional to each spin’s location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The total signal for a connected pore is then given by E(b)(qb) = � ˜Ω dx ˜ρ(x) e−iqb·x , (5) where x is the position of the spin with respect to the pore’s center-of-mass located at xcm while ˜ρ(x) = ρ(x + xcm) and ˜Ω indicates the domain translated so that the center of mass of the pore is at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Thus, the sequence is indeed ‘‘an imaging experiment in disguise,’’ [8] making it possible to obtain the image of the pore indicator function through an inverse Fourier transform of E(b)(qb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' In more general terms, the obtained quantity is the steady-state distribution of the fluid [14], thus not informative of the diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Measuring the diffusion propagator Here, we consider the sequence in Figure 1c, which combines the key elements of the two sequences discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The long pulse is there so that the integral of the waveform vanishes, and contributions from all pores are independent of their position within the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The two subsequent pulses q and q′ Page 3 Diffusion within pores revealed by MR ¨Ozarslan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' introduce phase shifts that depend on the particles’ positions during their application (in a frame of reference whose origin is at xcm---the center of mass of the fluid filling up the pore), denoted by x and x′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' When the second and third pulses are short, the signal is given by E(c) ∆ (q, q′) = � ˜Ω dx ˜ρ(x) � ˜Ω dx′ ˜p(x′, ∆|x) e−i(q·x+q′·x′) , (6) where ˜p(x′, ∆|x) = p(x′ + xcm, ∆|x + xcm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The propagator, can be obtained via the 2d-dimensional inverse Fourier transform of the signal W∆(x, x′) := 1 (2π)2d � Rd dq � Rd dq′ E(c) ∆ (q, q′) ei(q·x+q′·x′) (7) along with an estimate of ˜ρ(x), which is made available by the d-dimensional inverse Fourier transform of the subset of the data with q′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' In other words, the diffusion propagator is given by ˜p(x′, ∆|x) = � Rd dq eiq·x � Rd dq′ eiq′·x′ E(c) ∆ (q, q′) (2π)d � Rd dq eiq·x E(c) ∆ (q, 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' (8) Structure within the pore We demonstrate the estimation of the diffusion propagator of a simulated one-dimensional pore (interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The pore is partitioned into two exchanging compartments, with diffusion coefficients DL and DR, separated by a membrane of permeability w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The walls of the pore are purely reflective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The simulations summarized in Figure 2 illustrates the agreement of the reconstructed propagator (second row) with the true propagators (top row) at three different time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The associated EAPs are depicted in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The presence of a membrane within the pore space is conspicuous in the estimated propagators while the EAPs are not descriptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Structural dispersity The propagator-sensitive sequence of Figure 1c can be used to characterize porous media having structural dispersity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' To this end, we consider such a specimen having N isolated pores where the nth pore has the non-attenuated signal fraction fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The Fourier transforms of the signals E(c) ∆ (q, q′) and E(c) ∆ (q, 0) yield W∆(x, x′) = N � n=1,2,3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' fn ˜ρn(x) ˜pn(x′, ∆|x) (9a) ˜ρ(x) = N � n=1,2,3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' fn ˜ρn(x) , (9b) which are just weighted averages of the respective quantities for all pores translated so that the pores’ centers of mass coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Numerous quantities can be introduced for characterizing the underlying dispersity in the specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' For example, a dimensionless ‘variance map’ can be obtained through the expression σg(x) := (W∞(x, x) − ˜ρ(x)2)1/2 ρmax , (10) Page 4 Diffusion within pores revealed by MR ¨Ozarslan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' where ρmax := ˜ρ(xm) is the maximum value of ρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' A dispersity index can be introduced through DI := σ2 g(xm), which is equal to ⟨V −1⟩⟨V ⟩ − 1 in the absence of external forces when all pores contribute at xm and the signal fraction fn is proportional to the pore volume Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Here, ⟨·⟩ denotes averaging over all pores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' We shall now consider general x = r ˆu and x′ = r′ ˆu′ where r = |x|, r′ = |x′|, and ˆu and ˆu′ indicate the directions of x and x′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' One can define the quantity Φ(ˆu, ˆu′) := � ∞ 0 W∞(r ˆu, r ˆu′) r2d−1 dr , (11) which is constant for a medium composed of isotropic pores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' It has peaks at ˆu′ = ˆu when the pores are anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' If the pore shape has antipodal symmetry, ˆu′ = −ˆu will exhibit another peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Similarly, for shapes that exhibit other symmetries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=', cross or star-shaped pores) there will be other peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' In Figure 3, we illustrate the maps of the quantities in (9b)-(11) for ten different media with different compositions of two-dimensional pores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Figure 2: Representative snapshots of the simulated propagator estimation experiment for exchanging intervals of length LL and LR having diffusivity DL and DR, for the left and right comnpartments, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Shown from top to bottom are: true propagator, estimated propagator, and EAP (true and estimated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The density plots are of p′(x′, ∆|x) = tanh p(x′,∆|x) 2/(LL+LR) for better depiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The true propagator is computed by its (truncated) spectral decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Membrane position is emphasized by dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The relaxation time scales τL = L2 L/π2DL and τR = L2 R/π2DR correspond (roughly) to the process of diffusion within the compartments, and τex = √DLDR/w2 to the exchange between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Page 5 Diffusion within pores revealed by MR ¨Ozarslan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Figure 3: Maps derived from the long time diffusion measurements for different specimens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The axes of the Φ maps vary between −π and π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The DI values were estimated to be 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='28, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='28, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='8 × 10−5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='0 × 10−5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='41, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='9 × 10−5, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='3 × 10−5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='2 × 10−5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='5 × 10−5 (top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' ‘Hearing the drum’ In 1966, Kac posed the now famous question ‘‘Can one hear the shape of a drum?’’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' [15], which pertains to recovering the geometry of an enclosing boundary from the eigenspectrum of the Laplacian Page 6 0000 00000000000 00000X + X X+xxx++++XXXX+ +++X XXX X X X X ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='+ADiffusion within pores revealed by MR ¨Ozarslan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The pulse sequence of Laun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' (depicted in Figure 1b) demonstrated that the shape can be recovered from the MR signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' By enabling the measurement of the diffusion propagator, our gradient waveform illustrated in Figure 1c provides access to the diffusion dynamics within the pore and indeed to the spectrum of the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' To see this, one can exploit the eigenfunction expansion of the propagator, which leads to the expression � Ω p(x, t|x) dx = � ∞ 0 g(λ) e−λt dλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' (12) Clearly, the density of states, g(λ), is accessible from the propagator through an inverse Laplace transform while the propagator is obtained from the signal of the waveform in Figure 1c through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' In Figure 4, we illustrate the recovery of the density of states from simulated signals for the one-dimensional scenario involving diffusion in the direction perpendicular to two parallel plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Figure 4: How magnetic resonance ‘‘hears the drum.’’ (a) First row: MR signal profiles obtained using the diffusion encoding in Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Second row: Propagators obtained via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' (b) Left-hand-side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' (12) plotted against time and (c) its inverse Laplace transform revealing the density of states function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The numerical Laplace inversion [16] was performed using the package at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='com/caizkun/pyilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Page 7 t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='001ms t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='031ms t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='000ms a 4 4 4 2 2 2 [t-ur],b [t-wr],b [t-wn],b 0 0 0 2 2 2 4 - 4 - 4 4 -2 0 2 4 4 ¥2 4 2 0 2 4 q [μm-1] q [μm-1] q [μm-1] 2 2 2 [un],x [wr],x [wn] 0 0 0 X 2 2 2 2 0 2 2 0 2 2 0 2 x [μm] x [μm] x [μm] 109 6000 c Reconstructed Spectrum 5000 106 Expected peaks X 4000 103 X X X XXXXX 3000 100 2000 10-3 1000 10-6 0 10-9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content='0 103 104 105 106 t[ms] 入[ms-1]Diffusion within pores revealed by MR ¨Ozarslan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' Concluding remarks In conclusion, we introduced a new technique, which facilitates the characterization of the diffusion process within pores in full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' This is accomplished by mapping the diffusion propagator through Fourier transforms and can be related to the density of states function making the shape of the drum ‘‘heard’’ by magnetic resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} +page_content=' The technique allows for mapping the structure within closed pores as well as characterizing disperse specimens with unprecedented detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AyT4oBgHgl3EQfT_dJ/content/2301.00116v1.pdf'} 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Let G be a compact Lie group of dimension n. In this work we char- +acterise the membership of classical pseudo-differential operators on G in the trace +class ideal S1(L2(G)), as well as in the setting of the Schatten ideals Sr(L2(G)), +for all r > 0. In particular, we deduce Schatten characterisations of elliptic pseudo- +differential operators of (ρ, δ)-type for the large range 0 ≤ δ < ρ ≤ 1. Additional +necessary and sufficient conditions are given in terms of the matrix-valued symbols +of the operators, which are global functions on the phase space G × �G, with the +momentum variables belonging to the unitary dual �G of G. In terms of the param- +eters (ρ, δ), on the torus Tn, we demonstrate the sharpness of our results showing +the existence of atypical operators in the exotic class Ψ−κ +0,0 (Tn), κ > 0, belonging to +all the Schatten ideals. Additional order criteria are given in the setting of classical +pseudo-differential operators. We present also some open problems in this setting. +Contents +1. +Introduction +2 +1.1. +Outline +2 +1.2. +Historical aspects +2 +1.3. +Exotic examples and the main result +3 +1.4. +Open problems +6 +1.5. +Organisation of the manuscript +6 +2. +Preliminaries +7 +2.1. +The Fourier analysis of a compact Lie group +7 +2.2. +The quantisation formula +8 +2.3. +Global H¨ormander classes on compact Lie groups +10 +3. +Schatten Properties +12 +3.1. +Schatten properties for operators. Limited regularity symbols +12 +3.2. +Schatten properties of elliptic operators. The (ρ, δ)-case +16 +3.3. +Schatten properties of non-elliptic operators. Classical symbols +19 +3.4. +Classical operators in Schatten classes and their principal symbols +28 +2020 Mathematics Subject Classification. 35S30, 42B20; Secondary 42B37, 42B35. +Key words and phrases. Schatten von-Neumann classes, H¨ormander classes, Compact Lie groups, +global symbols. +The authors were supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial +Differential Equations, by the Methusalem programme of the Ghent University Special Research +Fund (BOF) (Grant number 01M01021) and by the dyCon Project 2015 H2020-694126. Marianna +Chatzakou is also supported by the FWO Fellowship grant No 12B1223N. Michael Ruzhansky is +also supported by EPSRC grant EP/R003025/2. +1 +arXiv:2301.04044v1 [math.FA] 10 Jan 2023 + +2 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +3.5. +Order of classical operators vs order of global symbols +28 +References +31 +1. Introduction +1.1. Outline. Let M be an orientable compact manifold without boundary with +volume element dx, and let us consider the Hilbert space L2(M) = L2(M, dx). For +any m ∈ R and for 0 ≤ δ < ρ ≤ 1, let Ψm +ρ,δ(M) be the H¨ormander class of continuous +linear operators on C∞(M), in local coordinates having the form +Af(x) = ∫ +Rn ∫ +Rn e2πi(x−y,θ)a(x, θ)f(y)dydθ, +(1.1) +and defined by those symbols a := a(x, θ) ∈ C∞(Rn × Rn) satisfying the estimates +|∂β +x∂α +ξ a(x, θ)| ≲α,β,K (1 + |θ|)m−ρ|α|+δ|β| +(1.2) +uniformly in x over compact subsets K ⊆ Rn. It is well-known that the class Ψm +ρ,δ(M) +is invariant under changes of coordinates if ρ > 1−δ. On the other hand, when M has +a group structure compatible with its differential structure, namely, when M = G is a +compact Lie group, in [39,41] one introduced the notion of a global symbol allowing +the construction of new classes of pseudo-differential operators Ψm +ρ,δ(G) also when +0 ≤ ρ ≤ 1 − δ, and providing a new description of the H¨ormander classes in the case +where ρ > 1 − δ. In this work we investigate necessary and sufficient conditions in +order to guarantee the inclusion of the H¨ormander classes on compact Lie groups +in the Schatten von-Neumann classes Sr(L2(G)), namely, we will investigate sharp +conditions allowing for the inclusion +Ψm +ρ,δ(G) ⊆ Sr(L2(G)). +(1.3) +We recall that for any r > 0, a compact operator T : L2(G) → L2(G) belongs to +the Schatten von-Neumann ideal Sr(L2(G)), if the sequence of its singular values +{sn(T)}n∈N (formed by the eigenvalues of the operator +√ +T ∗T) belongs to ℓr(N), that +is, if �∞ +n=1 sn(T)r < ∞. +1.2. Historical aspects. It is well known that an elliptic pseudo-differential opera- +tor A ∈ Ψm +1,0(M) of order m ∈ R, belongs to the ideal Sr(L2(M)), r > 0, if and only +if m < −n/r. Once removed the ellipticity condition the problem of finding order +criteria for classifying pseudo-differential operators on the ideal Sr(L2(M)) is still an +open problem. However, in the literature, if one considers the problem of classifying +pseudo-differential in the Schatten von-Neumann classes Sr(L2(Rn)), whose symbols +belong to the H¨ormander classes Sm +ρ,δ(Rn), the Beals-Fefferman classes SM1,M2 +Φ,φ +(Rn), +or the H¨ormander classes S(m, g) the subject becomes more classical. Indeed, in [34], +H¨ormander observed that the distribution of the eigenvalues (and then the Schatten +properties) of an elliptic pseudo-differential operator A = Opw(a) is encoded in terms +of the level sets of the symbol a. Indeed, he showed that the spectral formula +N(λ) ∼ +∫ +a(x,ξ)<λ +dxdξ, + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +3 +holds for any λ > 0. Here, N(λ) = #{j : |λj| ≤ λ} denotes the spectral function +of the operator A. The first results of this type can be traced back to H. Weyl for +second order differential operators, and to R. Courant (see [34, Page 297]). Then, in +Theorem 3.9 of [34], H¨ormander proved the following sufficient condition +m ∈ L1(R2n), a ∈ S(m, g) =⇒ Opw(a) ∈ S1(L2(Rn)), +(1.4) +with the metric g and the weight function m satisfying suitable conditions. In [35], +H¨ormander also characterised the L2 continuity of Weyl operators with the symbols +in S(m, g) as +{Opw(a) : a ∈ S(m, g) } ⊆ S∞(L2(Rn)) ⇐⇒ m ∈ L∞. +(1.5) +By adding some additional conditions on m and g, Buzano and Nicola in [1], +extended (1.5) into +{Opw(a) : a ∈ S(m, g) } ⊆ Sp(L2(Rn)) ⇐⇒ m ∈ Lp, +(1.6) +for every p ∈ [1, ∞]. +In [47], it is shown that (1.5) still holds true without the +additional assumptions on m and g in [1]. +In [3] the Schatten characterization +Opw(a) ∈ Sp(L2(Rn)) ⇐⇒ a ∈ Lp, +(1.7) +provided a ∈ S(m, g) and hN +g m ∈ Lp for some N ≥ 0. Here hg ≤ 1 is the Planck’s +function. For further Schatten properties of pseudo-differential operators on Rn, see +e.g. [18,33,37,45,46,48,51] and for Schatten properties on compact manifolds we refer +the reader to the works [5–11,13–15,19–31]. +1.3. Exotic examples and the main result. On the other hand, necessary and +sufficient conditions of the type (1.6) for non-elliptic operators on compact Lie groups +are still an open problem, as in the case of classical pseudo-differential operators +(operators with polyhomogeneous symbols) as well as in the modern setting of the +(ρ, δ)-classes on G, (see [39]) allowing the complete range 0 ≤ δ < ρ ≤ 1. Although +when 0 ≤ ρ < δ ≤ 1, as we will show, the order condition m < −n/r assuring the +membership of an elliptic operator in the class Sr(L2(G)) is a sharp criterion, the +situation changes dramatically if one considers the borderline δ = ρ = 0. Indeed, +in the case of the torus G = Tn with arbitrary dimension n we have discovered the +following strongly atypical situation: +• For any κ > 0, there exists a non-elliptic pseudo-differential operator A in the +exotic class Ψ−κ +0,0 (Tn)\Ψ−κ−ε +0,0 +(Tn), for all ε > 0, that belongs to all the Schatten ideals +Sr(L2(Tn)), with 0 < r < ∞. +We present later this statement in the form of Theorem 3.12. Here we observe that +the classes Ψm +0,0(G) on a compact Lie group G are of interest in PDE when computing +inverses of real vector fields X+c, where the constant term c belongs to an exceptional +set C ⊆ iR, see [43, Page 627] for details. +Now, we are going to discuss our main results and we also will propose some +conjectures related to the inclusion in (1.3) which is the central question of this +manuscript. To continue let us fix the notation and let us introduce the notion of +a (full/global) matrix-valued symbol as developed by the third author and Turunen + +4 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +in [39]. One reason for this is that our criteria will be addressed in terms of such +matrix-valued symbols. +Let us consider the unitary dual �G of the compact Lie group G, which is formed +by all the equivalent classes [ξ] of continuous, unitary, and irreducible representations +ξ : G �→ U(Cℓ), and let ℓ = dξ be the dimension of the representation space. To any +continuous linear operator on C∞(G) and then to any pseudo-differential operator A +in the class Ψm +ρ,δ(G) := Ψm +ρ,δ(G× �G), 0 ≤ δ ≤ ρ ≤ 1, one can associate a matrix-valued +global symbol +a : G × �G → +� +[ξ]∈ �G +Cdξ×dξ, (x, [ξ]) �→ a(x, [ξ]) ∈ Cdξ×dξ, +allowing the global quantisation formula +Af(x) = +� +[ξ]∈ �G +∫ +G +dξTr[ξ(y−1x)a(x, ξ)]f(y)dy, ∀f ∈ C∞(G), ∀x ∈ G. +(1.8) +The problem of finding criteria to assure the membership of a pseudo-differential +operator A in the Schatten classes Sr(L2(G)), in terms of its matrix-valued symbol +a := a(x, [ξ]) has been a source of intensive mathematical activity for around 10 +years, see e.g. [5–11,13–15,19–31]. Contributing to the previous references, the main +results of this work can be summarised in Theorems 1.1, 1.2 and 1.3 below where we +will use the following notations. +• We denote by g the Lie algebra of a compact Lie group G. The mapping +B(X, Y ) = −Tr[ad(X)ad(Y )], X, Y ∈ g, is the Killing form on g × g and we +denote by ||X||g := +� +−B(X, X) the corresponding norm on g associated to +−B.1 +• We denote by LG the positive Laplace Beltrami operator on G, and under the +identification g∗ ∼= g, η �→ ||η||2 +g, η ∈ g∗ \ {0}, denotes its principal symbol. +• The family Sr(L2(G)), 0 < r < ∞, is formed by the Schatten von Neumann +ideals on a compact Lie group G, if 0 < r < ∞, and for r = ∞, Sr(L2(G)) = +B(L2(G)) denotes the algebra of all bounded linear operators on L2(G). +• For every r ∈ (0, ∞), the Schatten norm of a symbol a(x, [ξ]) is given by +∥a(x, [ξ])∥Sr = Tr[|a(x, [ξ])|r] +1 +r , where |a(x, [ξ])| := +� +a(x, [ξ])∗a(x, [ξ]) is de- +fined in terms of the functional calculus of matrices. Note that ∥a(x, [ξ])∥S2 = +∥a(x, [ξ])∥HS is the standard Hilbert-Schmidt norm of matrices. +• For any 1 ≤ p1, p2 < ∞, the space Lp1(G, Sp2( �G)) is defined by those symbols +a := a(x, [ξ]) such that +∥a(·, ·)∥Lp1(G,Sp2( �G)) = +� +∫ +G +∥a(x, ·)∥p1 +Sp2( �G)dx +� 1 +p1 < ∞, +where +∥a(x, ·)∥Sp2( �G) = +� +� � +[ξ]∈ �G +dξ∥a(x, [ξ])∥p2 +Sp2 +� +� +1 +p2 +. +1Since G is a compact Lie group the positive Killing form −B : g×g → C, is positive semi-definite, +i.e. −B(X, X) > 0, ∀X ∈ g \ {0}. + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +5 +• In terms of the ℓp( �G) norm +∥a(x, [ξ])∥ℓp( �G) = +� +� � +[ξ]∈ �G +d +p( 2 +p − 1 +2) +ξ +∥a(x, [ξ])∥p +HS +� +� +1 +p +, 1 ≤ p < ∞, +(1.9) +for any 1 ≤ p1, p2 < ∞, the space Lp1(G, ℓp2( �G)) is defined by those symbols +a := a(x, [ξ]) such that +∥a(·, ·)∥Lp1(G,ℓp2( �G)) = +� +∫ +G +∥a(x, ·)∥p1 +ℓp2( �G)dx +� 1 +p1 < ∞, +we refer the reader to [32] for the embeddings between these two classes of spaces. +The following three theorems summarise our main results. We start with our char- +acterisation of elliptic operators in Schatten classes. +Theorem 1.1 (General symbols). Let G be a compact Lie group of dimension n, +let m ∈ R, r > 0, and let 0 ≤ δ < ρ ≤ 1. Consider an elliptic pseudo-differential +operator A ∈ Ψm +ρ,δ(G × �G). The following conditions are equivalent: +(1) m < 0 and A belongs to the Schatten class of order r > 0, that is A ∈ +Sr(L2(G)); +(2) The Bessel potential of order m belongs to the Schatten class of order r > 0 : +Bm := (1 + LG) +m +2 ∈ Sr(L2(G)). +(3) The matrix-valued symbol of |A| +r +2 satisfies the following summability condition +� +[ξ]∈ �G +dξ ∫ +G +∥σ|A| +r +2 (x, [ξ])∥2 +HSdx < ∞; +(1.10) +(4) The matrix-valued symbol of A satisfies the following summability condition +∫ +G +� +[ξ]∈ �G +dξ∥σA(x, [ξ])∥r +Srdx < ∞; +(1.11) +(5) m < −n/r. +Moreover, if A ∈ Ψm +ρ,δ(G × �G) is not elliptic and m < −n/r, then we have that +A ∈ Sr(L2(G)). +As for classical operators on compact Lie groups we have the following result. +Theorem 1.2 (Classical symbols). Let G be a compact Lie group of dimension n, +let m ∈ R, r > 0, and let 0 ≤ δ < ρ ≤ 1. Let A ∈ Ψm +1,0(G) be a classical pseudo- +differential operator of order m. +(6) If r ∈ [1, ∞) ∩ Z, then A ∈ Sr(L2(G)) if and only if m < −n/r. For r = ∞, +A ∈ B(L2(G)) if and only if m ≤ 0. +(7) If r ∈ (1, ∞) \ Z, and A ∈ Sr(L2(G)), then m ≤ −n/r. Moreover, if A is +elliptic then one has the strict inequality m < −n/r. +Additionally, consider the subclass Ψm +0 (G) in Ψm +cl (G) of operators with homogeneous +symbols of order m. Then the following conditions are equivalent: +(8) Ψm +cl (G) ⊆ Sr(L2(G)), r ∈ (0, ∞]; +(9) Ψm +0 (G) ⊆ Sr(L2(G)), r ∈ (0, ∞]. + +6 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +In the next theorem we consider conditions of limited regularity. +Theorem 1.3 (Symbols of low regularity). Let G be a compact Lie group of di- +mension n. Let us assume that for any [ξ], the symbol a(·, [ξ]) is Haar measurable. +Then: +(10) Assume that a symbol a ∈ Lp(G, ℓp( �G)) for some 1 < p < 2. Then, the +corresponding pseudo-differential operator satisfies A ∈ Sp′(L2(G)), where +p′ = p/(p − 1). +(11) Assume that the matrix-valued symbol a = a(x, [ξ]) satisfies the regularity +condition +∥(1 + LG) +N +2 σA(x, ·)∥L1(G,Sp( �G)) = ∫ +G +∥(1 + LG) +N +2 σA(x, ·)∥Sp( �G)dx < ∞ +(1.12) +where N > n. Then A ∈ Sp(L2(G)) provided that 1 ≤ p < ∞. +1.4. Open problems. In view of the open questions that have arisen in our ap- +proach, and based on the active research on this field in the last 10 years, we propose +the following open problems. +Open problem 1.4. Let 0 ≤ δ < ρ ≤ 1. Prove (or disprove) that if A ∈ Ψm +ρ,δ(G × +�G) is a non-elliptic pseudo-differential operator that belongs to the Schatten class +Sr(L2(G)) then m < −n/r. +Open problem 1.5. Prove (or disprove) that for r ∈ (1, ∞) \ Z, and with A ∈ +Sr(L2(G)) being a non-elliptic operator, then the inequality m ≤ −n/r in (7) of +Theorem 1.2 can be improved to the strict order estimate m < −n/r. +Remark 1.6. We observe that the equivalence (1) ⇐⇒ (3) in Theorem 1.1 was proved +by the third author and J. Delgado in [28]. In particular, in [22] the relation between +the spectral trace and the nuclear trace of operators has been investigated for the +more general notion of nuclear operators and Grothendieck-Lidskii type formulas. +Further analysis involving criteria in terms of matrix-valued symbols was also carried +out on compact Lie groups and on arbitrary compact manifolds in [12,22–31]. +1.5. Organisation of the manuscript. This paper is organised as follows. +• In Section 2 we present the basics of the Fourier analysis on compact Lie +groups used here as well as the preliminaries about the pseudo-differential +calculus on compact Lie groups in terms of the matrix-valued symbols as +developed in [39]. +• Section 3 will be dedicated to the proof of our main Theorems. More precisely: +1. Theorem 1.1 is presented later as Theorem 3.5 of Subsection 3.2. +2. (6) and (7) of Theorem 1.2 are proved in Theorem 3.11 of Subsection 3.3. +3. The equivalence (8) ⇐⇒ (9) of Theorem 1.2 is proved in Proposition 3.13 +of Subsection 3.4. +4. Theorem 1.3 is proved in Subsection 3.1 (see Theorem 3.1 and Theorem +3.3, respectively). +• Finally, in Subsection 3.5 we prove that the order of a matrix-valued symbol +associated to a classical pseudo-differential operator classifies its operator or- +der. For the proof we use the approach developed in this work for the analysis + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +7 +of Schatten operators that involves the average of their principal symbols on +the co-sphere (see Figure 1). +2. Preliminaries +2.1. The Fourier analysis of a compact Lie group. Let dx be the Haar measure +on a compact Lie group G. The Hilbert space L2(G) := L2(G, dx) will be endowed +with the inner product +(f, g) = ∫ +G +f(x)g(x)dx. +The Peter-Weyl theorem gives a spectral decomposition of L2(G) in terms of the +entries of unitary representations of G. In order to present such a result we will give +some preliminaries. +Definition 2.1 (Unitary representation of G). A continuous and unitary represen- +tation of G on Cℓ is any continuous mapping ξ ∈ Hom(G, U(ℓ)), where U(ℓ) is the +Lie group of unitary matrices of order ℓ × ℓ. The integer number ℓ = dξ is called the +dimension of the representation ξ. +Remark 2.2 (Irreducible representations). We recall that: +• a subspace L ⊆ Cdξ is called ξ-invariant if for any x ∈ G, ξ(x)(L) ⊆ L, where +ξ(x)(L) := {ξ(x)v : v ∈ L}. +• The representation ξ is irreducible if its only invariant subspaces are L = ∅ +and L = Cdξ, the trivial ones. +• Any unitary representation ξ is a direct sum of unitary irreducible represen- +tations. We denote it by ξ = ξ1 ⊗· · ·⊗ξk, with ξi, 1 ≤ i ≤ k, being irreducible +representations on factors Cdξi that decompose the representation space +Cdξ = Cdξ1 ⊗ · · · ⊗ Cdξk. +The notion of equivalent representations allows us to define an equivalence relation +in the family of unitary representations. We recall it in the following definition. +Definition 2.3 (Equivalent representations). Two unitary representations +ξ ∈ Hom(G, U(dξ)) and η ∈ Hom(G, U(dη)) +are equivalent if there exists a linear mapping S : Cdξ → Cdη such that for any x ∈ G, +Sξ(x) = η(x)S. The mapping S is called an intertwining operator between ξ and η. +The set of all the intertwining operators between ξ and η is denoted by Hom(ξ, η). +Remark 2.4 (Schur Lemma, 1905). If ξ ∈ Hom(G, U(dξ)) is irreducible, then Hom(ξ, ξ) = +CIdξ is formed by scalar multiples of the identity matrix Idξ of order dξ. +Definition 2.5 (The unitary dual). The relation ∼ on the set of unitary representa- +tions, which we denote by Rep(G), and defined by: ξ ∼ η if and only if ξ and η are +equivalent representations, is an equivalence relation. The quotient set +�G := Rep(G)/∼ +is called the unitary dual of G. Since G is a compact Lie group, �G is a discrete set. +The unitary dual encodes all the Fourier analysis on the group. The Fourier trans- +form is defined as follows. + +8 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +Definition 2.6 (Group Fourier transform). If ξ ∈ Rep(G), the Fourier transform +FG associates to any f ∈ C∞(G) a matrix-valued function FGf defined on Rep(G) +as follows +(FGf)(ξ) ≡ �f(ξ) = +� +G +f(x)ξ(x)∗dx, ξ ∈ Rep(G). +Remark 2.7 (The Fourier inversion formula on a compact Lie group). The discrete +Schwartz space S ( �G) := FG(C∞(G)) is the image of the Fourier transform on the +class of smooth functions. This operator admits a unitary extension from L2(G) into +ℓ2( �G), with +ℓ2( �G) = +� +φ : ∀[ξ] ∈ �G, φ(ξ) ∈ Cdξ×dξ and ∥φ∥ℓ2( �G) < ∞ +� +, +(2.1) +where +∥φ∥ℓ2( �G) := +� +� � +[ξ]∈ �G +dξ∥φ(ξ)∥2 +HS +� +� +1 +2 +. +The norm ∥φ(ξ)∥HS = (Tr(φ(ξ)∗φ(ξ))) +1 +2 is the standard Hilbert-Schmidt norm of +matrices. The Fourier inversion formula takes the form +f(x) = +� +[ξ]∈ �G +dξTr[ξ(x) �f(ξ)], ∀f ∈ L1(G), ∀x ∈ G, +(2.2) +where the summation is understood in the sense that from any equivalence class [ξ] +we choose one (any) unitary representation. +2.2. The quantisation formula. Let A : C∞(G) → C∞(G) be a continuous linear +operator with respect to the standard Fr´echet structure on C∞(G). There is a way +of associating to the operator A a matrix-valued function σA defined on the non- +commutative phase space G × �G to rewrite the operator A in terms of the Fourier +inversion formula and in terms of the Fourier transform. Such a expression is called +the quantisation formula. To introduce it we require the following definition. +Definition 2.8 (Right convolution kernel of an operator). The Schwartz kernel the- +orem associates to A a kernel/distribution KA ∈ D′(G × G) such that +Af(x) = +� +G +KA(x, y)f(y)dy, f ∈ C∞(G). +The distribution defined via RA(x, xy−1) := KA(x, y) that provides the convolution +identity +Af(x) = +� +G +RA(x, xy−1)f(y)dy, f ∈ C∞(G), +is called the right-convolution kernel of A. +Remark 2.9 (The quantisation formula). Now, we will associate a global symbol +σA : G× �G → ∪ℓ∈NCℓ×ℓ to A. Indeed, in view of the identity Af(x) = (f ∗RA(x, ·))(x), +we get +� +Af(ξ) = �RA(x, ξ) �f(ξ). + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +9 +Then we have that +Af(x) = +� +[ξ]∈ �G +dξTr[ξ(x) �RA(x, ξ) �f(ξ)], f ∈ C∞(G). +(2.3) +In view of the identity (2.3), from any equivalence class [ξ] ∈ �G, we can choose one +and only one irreducible unitary representation ξ0 ∈ [ξ], such that the matrix-valued +function +σA(x, [ξ]) ≡ σA(x, ξ0) := �RA(x, ξ0), (x, [ξ]) ∈ G × �G, +(2.4) +satisfies that +Af(x) = +� +[ξ]∈ �G +dξTr[ξ0(x)σA(x, [ξ]) �f(ξ0)], f ∈ C∞(G). +(2.5) +The representation in (2.5) is independent of the choice of the representation ξ0 ∈ +Rep(G) from any equivalent class [ξ] ∈ �G. This is a consequence of the Fourier +inversion formula. So, we can simply write +Af(x) = +� +[ξ]∈ �G +dξTr[ξ(x)σA(x, [ξ]) �f(ξ)], ∀f ∈ C∞(G). +(2.6) +In the following quantisation theorem we observe that the distribution σA in (2.6) +defined on G × �G is unique and can be written in terms of the operator A, see +Theorems 10.4.4 and 10.4.6 of [39]. +Theorem 2.10. Let A : C∞(G) → C∞(G) be a continuous linear operator. The +following statements are equivalent. +• The matrix-valued distribution σA(x, [ξ]) : G × �G → ∪ℓ∈NCℓ×ℓ satisfies that +∀f ∈ C∞(G), ∀x ∈ G, Af(x) = +� +[ξ]∈ �G +dξTr[ξ(x)σA(x, [ξ]) �f(ξ)]. +(2.7) +• We have that ∀(x, [ξ]) ∈ G × �G, σA(x, ξ) = �RA(x, ξ). +• The following identity holds: ∀(x, [ξ]) ∈ G × �G, σA(x, ξ) = ξ(x)∗Aξ(x), where +Aξ(x) := [Aξij(x)] +dξ +i,j=1. +Remark 2.11. In view of the quantisations formulae (2.6) and (2.7), a symbol σA can +be considered as a mapping defined on G× �G or as a mapping defined on G×Rep(G) +by identifying all the values σA(x, ξ) = σA(x, ξ′) = σ(x, [ξ]) when ξ′, ξ ∈ [ξ]. +Example 2.12 (The symbol of a measurable function of the Laplacian). Let X = +{X1, · · · , Xn} be an orthonormal basis of the Lie algebra g. The positive Laplacian +on G is the second order differential operator +LG = − +n +� +j=1 +X2 +j . +(2.8) +The operator LG is independent of the choice of the orthonormal basis X of g, see +e.g. [39]. The L2-spectrum of LG is a discrete set that can be enumerated in terms + +10 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +of the unitary dual �G as +Spectrum(LG) = {λ[ξ] : [ξ] ∈ �G}. +(2.9) +For a Borel function f : R+ +0 → C, the right-convolution kernel Rf(LG) of the operator +f(LG) (defined by the spectral calculus) is determined by the identity +f(LG)φ(x) = φ ∗ Rf(LG)(x), x ∈ G, +(2.10) +where +∀[ξ] ∈ �G, �Rf(LG)([ξ]) = f(λ[ξ])Idξ. +(2.11) +Then the matrix-valued symbol of f(LG) can be determined e.g. using Theorem 2.10 +as follows +σf(LG)(x, ξ) = �Rf(LG)([ξ]). +(2.12) +Since the operator f(LG) is left-invariant the symbol σf(LG)(ξ) = σf(LG)(x, ξ) does +not depend of x ∈ G. Of particular interest for the definition of the global H¨ormander +classes on G, will be the Japanese bracket function +⟨t⟩ := (1 + t) +1 +2, t ≥ −1. +(2.13) +In particular the symbol of the operator ⟨LG⟩ = (1 + LG) +1 +2 is given by +σ⟨LG⟩([ξ]) := ⟨ξ⟩Idξ, ⟨ξ⟩ := ⟨λ[ξ]⟩. +(2.14) +2.3. Global H¨ormander classes on compact Lie groups. In this section we +denote for any linear mapping U on Cn by ∥U∥op the standard operator norm +∥U∥op = ∥U∥End(Cn) := sup +l̸=0 +∥Ul∥e/∥l∥e, +where ∥l∥e = (l2 +1 + · · · + l2 +n) +1 +2 is the Euclidean norm. +For introducing the H¨ormander classes on compact Lie groups we have to measure +the growth of derivatives of symbols in the group variable, for this we use vector +fields X ∈ T(G). To derivate symbols with respect to the discrete variable [ξ] ∈ �G +we use difference operators. Before introducing the H¨ormander classes on compact +Lie groups we have to define these differential/difference operators. +Definition 2.13 (Left-invariant canonical differential operators). If {X1, · · · , Xn} is +an arbitrary family of left-invariant vector fields, we will denote by +Xα +x := Xα1 +1,x · · · Xαn +n,x +an arbitrary canonical differential operator of order m = |α|. +Also, we have to take derivatives with respect to the “discrete” frequency variable +ξ ∈ Rep(G). To do this, we will use the notion of difference operators introduced +in [43]. Indeed, the frequency variable in the symbol σA(x, [ξ]) of a continuous and +linear operator A on C∞(G) is discrete. This is since �G is a discrete space. +Definition 2.14 (Canonical difference operators Dα on the dual �G). If ξ1, ξ2, · · · , ξk, +are fixed irreducible and unitary representations of G, which not necessarily belong +to the same equivalence class, then each coefficient of the matrix +ξℓ(g) − Idξℓ = [ξℓ(g)ij − δij] +dξℓ +i,j=1, +g ∈ G, 1 ≤ ℓ ≤ k, +(2.15) + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +11 +that is each function qℓ +ij(g) := ξℓ(g)ij − δij, g ∈ G, defines a difference operator +Dξℓ,i,j := FG(ξℓ(g)ij − δij)F −1 +G . +(2.16) +We can fix k ≥ dim(G) of these representations in such a way that the corresponding +family of difference operators is admissible, that is, +rank{∇qℓ +i,j(e) : 1 ⩽ ℓ ⩽ k} = dim(G). +To define higher order difference operators of this kind, let us fix a unitary irre- +ducible representation ξℓ. Since the representation is fixed we omit the index ℓ of the +representations ξℓ in the notation that will follow. Then, for any given multi-index +α ∈ N +d2 +ξℓ +0 , with |α| = �dξℓ +i,j=1 αi,j, we write +Dα := Dα11 +1,1 · · · D +αdξℓ ,dξℓ +dξℓdξℓ +for a difference operator of order m = |α|. +Now, we are ready for introducing the global H¨ormander classes on compact Lie +groups. +Definition 2.15 (Global (ρ, δ)-H¨ormander classes in the whole range 0 ≤ δ, ρ ≤ 1). +We say that σ ∈ Sm +ρ,δ(G × �G) if the following symbol inequalities +∥Xβ +x Dασ(x, ξ)∥op ⩽ Cα,β⟨ξ⟩m−ρ|γ|+δ|β|, +(2.17) +are satisfied for all multi-indices β and γ, and for all (x, [ξ]) ∈ G × �G, where ⟨ξ⟩ +denotes the Japanese bracket function at λ[ξ] ∈ Spectrum[LG] defined in 2.14. +The class Ψm +ρ,δ(G× �G) ≡ Op(Sm +ρ,δ(G× �G)) is defined by those continuous and linear +operators on C∞(G) such that σA ∈ Sm +ρ,δ(G × �G). +In the next theorem we describe some fundamental properties of the global H¨ormander +classes of pseudo-differential operators [39]. +Theorem 2.16. Let ρ, δ ∈ [0, 1] be such that 0 ⩽ δ ⩽ ρ ⩽ 1, ρ ̸= 1. Then +Ψ∞ +ρ,δ(G) := ∪m∈RΨm +ρ,δ(G) +is an algebra of operators stable under compositions and adjoints, that is: +1. the mapping +A �→ A∗ : Ψm +ρ,δ(G × �G) → Ψm +ρ,δ(G × �G) +is a continuous linear mapping between Fr´echet spaces. +2. The mapping +(A1, A2) �→ A1 ◦ A2 : Ψm1 +ρ,δ(G × �G) × Ψm2 +ρ,δ(G × �G) → Ψm1+m2 +ρ,δ +(G × �G) +is a continuous bilinear mapping between Fr´echet spaces. +Moreover, any operator in the class Ψ0 +ρ,δ(G × �G) admits a bounded extension from +L2(G) to L2(G). + +12 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +Remark 2.17. With 0 ⩽ δ < ρ ⩽ 1 such that ρ ≥ 1−δ, the condition A ∈ Ψm +ρ,δ(G× �G) +where m ∈ R, is equivalent to the fact that, when microlocalising the operator A into +a local coordinate system U, the operator A takes the form +Af(x) = ∫ +Rn ∫ +Rn e2πi(x−y)·ξa(x, ξ)f(y)dydξ, ∀f ∈ C∞ +0 (U), ∀x ∈ Rn, +where the function a = aU, is such that for every compact subset K ⊆ U and for all +α, β ∈ Nn +0, the inequalities +|∂β +x∂α +ξ a(x, ξ)| ⩽ Cα,β,K(1 + |ξ|)m−ρ|α|+δ|β|, +(2.18) +hold uniformly in (x, ξ) ∈ K ×Rn. This characterisation of the H¨ormander classes on +G was proved in [41]. So, for any compact Lie group G, the classes Ψm +ρ,δ(G× �G) agree +with the ones introduced by H¨ormander [36] when 0 ⩽ δ < ρ ⩽ 1 and ρ ≥ 1 − δ. +3. Schatten Properties +In this section we analyse the membership of pseudo-differential operators in the +Schatten classes on L2(G). +3.1. Schatten properties for operators. Limited regularity symbols. In this +section we study the Schatten properties of operators with symbols of limited regu- +larity. Without assumptions of regularity we start with the following criterion. +Theorem 3.1. Assume that for a symbol σA we have σA ∈ Lp(G, ℓp( �G)) for some +1 < p < 2. Then, the corresponding pseudo-differential operator A satisfies that +A ∈ Sp′(L2(G)), where p′ = p/(p − 1). +Proof. Let us consider the following criterion due to Russo (see [38]): +∥A∥Sp′ ≤ (∥K∥p,p′ × ∥K∗∥p,p′)1/2, +(3.1) +where K is the kernel of A and K∗ is the kernel of the adjoint operator A∗, 1 < p < 2, +and +∥K∥p,p′ = +� +�∫ +G +� +∫ +G +|K(x, y)|pdx +� p′ +p +dy +� +� +1 +p′ +. +(3.2) +For a moment, let us assume that A is self-adjoint. Then, K = K∗ and then +∥A∥Sp′ ≤ ∥K∥p,p′. +(3.3) +Observe that the Hausdorff-Young inequality (see e.g. [32, Page 69]) gives +∫ +G +|K(x, y)|p′dy = ∥y �→ F −1 +G (σA(x, ·))(xy−1)∥p′ +Lp′(G;dy) += ∥z �→ F −1 +G (σA(x, ·))(z)∥p′ +Lp′(G;dz) ≤ ∥σA(x, ·)∥p′ +ℓp( �G). +Since p′ > 2 > p the Minkowski integral inequality implies that +∥K∥p,p′ ≤ ∥K∥p′,p = +� +∫ +G +� +∫ +G +|K(x, y)|p′dy +� p +p′ +dx +� 1 +p +≤ +� +∫ +G +∥σA(x, ·)∥p +ℓp( �G)dx +� 1 +p += ∥σA∥Lp(G,ℓp( �G)). + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +13 +So, we have proved the statement in Theorem 3.1 if A is self-adjoint. Now, in the +general case consider the decomposition of A into its real and imaginary part +A = Re(A) + iIm(A). +(3.4) +Note that +∀(x, [ξ]) ∈ G × �G, σA(x, [ξ]) = σRe(A)(x, [ξ]) + iσIm(A)(x, [ξ]), +(3.5) +that +∀x ∈ G, ∥σA(x, [ξ])∥ℓp( �G) ≍ ∥σRe(A)(x, [ξ])∥ℓp( �G) + ∥σIm(A)(x, [ξ])∥ℓp( �G), +(3.6) +and using that +∥A∥Sp′ ≍ ∥Re(A)∥Sp′ + ∥Im(A)∥Sp′, +(3.7) +and the following inequalities (in view of the self-adjointness of Re(A) and Im(A)) +we have that +∥A∥Sp′ ≍ ∥Re(A)∥Sp′ + ∥Im(A)∥Sp′ +≲ ∥σRe(A)(x, [ξ])∥Lp(G,ℓp( �G)) + ∥σIm(A)(x, [ξ])∥Lp(G,ℓp( �G)) +≍ ∥σA∥Lp(G,ℓp( �G)). +The proof of Theorem 3.1 is complete. +□ +Remark 3.2. In terms of the ℓp-Schatten norm on �G one has the Hausdorff-Young +inequality (see e.g. [32, Page 67]) +∥ �f∥Sp′( �G) ≤ ∥f∥Lp(G), 1 ≤ p ≤ 2, +(3.8) +where p′ is the conjugate exponent of p. However, since the group G is compact one +has the following refined versions for this inequality +∥F −1 +G σ∥Lp′(G) ≤ ∥σ∥ℓp( �G), ∥ �f∥ℓp′( �G) ≤ ∥f∥Lp(G), 1 ≤ p ≤ 2. +(3.9) +In particular, the second inequality in (3.9) is sharper than (3.8) because of the +embedding Sp′( �G) ⊂ ℓp′( �G) for 2 ≤ p′ ≤ ∞, see [32, Page 70]. We have used the first +inequality in (3.9), which is the Hausdorff-Young inequality for the inverse Fourier +transform to estimate the inequality +∥K∥p,p′ ≤ ∥σA∥Lp(G,ℓp( �G)) +in the first part of the proof of Theorem 3.1. +In the next result, we assume more regularity in the spatial variable to deduce a +criterion for pseudo-differential operators to belong to the Schatten classes. +Theorem 3.3. Let G be a compact Lie group of dimension n. Let A : C∞(G) → +C∞(G) be a continuous linear operator and assume that its matrix-valued symbol +satisfies the regularity condition +∥(1 + LG) +N +2 σA(x, ·)∥L1(G,Sp( �G)) = ∫ +G +∥(1 + LG) +N +2 σA(x, ·)∥Sp( �G)dx < ∞ +(3.10) +where N > n. Then A ∈ Sp(L2(G)) for all 1 ≤ p < ∞. + +14 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +Proof. Let us consider the matrix-valued symbol σA(x, [ξ]) = (σA,ij(x, [ξ])) +dξ +i,j=1. The +Fourier inversion formula allows one to write +σA,ij = +� +[η]∈ �G +dηη(x)rs�σA, ij,sr(η, ξ), +(3.11) +where �σA, ij,sr([η], [ξ]) denotes the (s, r)-Fourier coefficient of the function σA,ij(·, [ξ]) +at η ∈ [η] ∈ �G. For any f ∈ C∞(G), the quantisation formula gives the identity +Af(x) = +� +[ξ]∈ �G +dξTr[ξ(x)σA(x, [ξ]) �f(ξ)] = +� +[ξ]∈ �G +dξ +� +i,j,ℓ=1 +dξξij(x)σA, ji(x, [ξ]) �f(ξ)iℓ += +� +[η]∈ �G +dη +� +r,s=1 +� +[ξ]∈ �G +dξ +� +i,j,ℓ=1 +dξξij(x)ηrs(x)�σA, ji, sr([η], [ξ]) �f(ξ)iℓ += +� +[η]∈ �G +dη +� +r,s=1 +� +[ξ]∈ �G +ηrs(x)dξTr[ξ(x)�σA, sr([η], [ξ]) �f(ξ)], +where +�σA, sr([η], [ξ]) := (�σA, ji, sr([η], [ξ])) +dξ +j,i=1. +Let us define the operator Op(�σA, sr([η], ·)) corresponding to the matrix �σA, sr([η], ·) +. Let Mηrs be the multiplication operator associated to the function ηrs. Note that +Mηrs is a bounded operator on L2(G) with operator norm +∥Mηrs∥B(L2(G)) = ∥ηrs∥L∞(G). +(3.12) +Then, we have that +∀f ∈ C∞(G), ∀x ∈ G, Af(x) = +� +[η]∈ �G +dη +� +r,s=1 +Mηrs[Op(�σA, sr([η], ·))f](x). +(3.13) +Note that for all N ∈ N, the symbol +(x, [ξ]) �→ σN,rs(x, [ξ]) := (1 + LG) +N +2 σA, rs(x, [ξ]) +satisfies the Fourier transform identity +�σA, sr([η], [ξ]) = ⟨η⟩−N�σN,rs([η], [ξ]). +(3.14) +This identity allows us to write +∥A∥Sp(L2(G)) ≤ +� +[η]∈ �G +dη +� +r,s=1 +∥MηrsOp(�σA, sr([η], ·))∥Sp(L2(G)) += +� +[η]∈ �G +dη +� +r,s=1 +⟨η⟩−N∥MηrsOp(�σN, sr([η], ·))∥Sp(L2(G)) += +� +[η]∈ �G +dη +� +r,s=1 +⟨η⟩−N∥Mηrs∥B(L2)∥Op(�σN, sr([η], ·))∥Sp(L2(G)) + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +15 += +� +[η]∈ �G +dη +� +r,s=1 +⟨η⟩−N∥ηrs∥L∞(G)∥Op(�σN, sr([η], ·))∥Sp(L2(G)). +Since +∀x ∈ G, |ηrs(x)| ≤ +� +dη +� +r′,s′=1 +|ηr′s′(x)|2 +� 1 +2 += +� +Tr[η(x)η(x)∗] = +√ +dη, +(3.15) +we also have +∀x ∈ G, +dη +� +r,s=1 +|ηrs(x)| ≤ +� +dη +� +r′,s′=1 +|ηr′s′(x)|2 +� 1 +2 +dη = +� +Tr[η(x)η(x)∗]dη = dη +√ +dη, +(3.16) +where the last implies +dη +� +r,s=1 +∥ηrs∥L∞ ≤ dη +√ +dη. +(3.17) +On the other hand, using the triangle inequality for the Sp-norm we have that +∥Op(�σN, sr([η], ·))∥p +Sp(L2(G)) ≲ +� +[ξ]∈ �G +dξ∥�σN, sr([η], [ξ])∥p +Sp( �G) += +� +[ξ]∈ �G +dξ∥ ∫ +G +σN(x, [ξ])ηrs(x)∗dx∥p +Sp( �G) +≤ +� +[ξ]∈ �G +dξ +� +∫ +G +∥η∗ +rs∥L∞(G)∥σN(x, [ξ])∥Sp( �G)dx +�p +≤ +� +[ξ]∈ �G +dξ +√ +d +p +η +� +∫ +G +∥σN(x, [ξ])∥Sp( �G)dx +�p +, +where, in the last line, we have used (3.15) to estimate +∥η∗ +rs∥p +L∞(G) = ∥ηrs∥p +L∞(G) = ∥ηrs∥p +L∞(G) ≤ +√ +d +p +η. +The Minkowski integral inequality for p ≥ 1 implies that +∥Op(�σN, sr([η], ·))∥Sp ≲ +� +� � +[ξ]∈ �G +dξ +√ +d +p +η +� +∫ +G +∥σN(x, [ξ])∥Spdx +�p +� +� +1 +p +≤ +√ +dη ∫ +G +� +� � +[ξ]∈ �G +dξ∥σN(x, [ξ])∥p +Sp +� +� +1 +p +dx += +√ +dη ∫ +G +∥σN(x, ·)∥Sp( �G)dx += +√ +dη∥σN(x, ·)∥L1(G,Sp( �G)). + +16 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +All the analysis above allows the following estimate for the p-Schatten norm of A as +follows: +∥Af∥Sp ≲ +� +[η]∈ �G +dη +� +r,s=1 +⟨η⟩−N∥ηrs∥L∞(G)∥Op(�σN, sr([η], ·))∥Sp +≲ +� +[η]∈ �G +dη +� +r,s=1 +⟨η⟩−N∥ηrs∥L∞(G) +√ +dη∥σN(x, ·)∥L1(G,Sp( �G)) += ∥σN(x, ·)∥L1(G,Sp( �G)) +� +[η]∈ �G +⟨η⟩−N√ +dη +dη +� +r,s=1 +∥ηrs∥L∞(G) +≤ ∥σN(x, ·)∥L1(G,Sp( �G)) +� +[η]∈ �G +⟨η⟩−N√ +dηdη +√ +dη. +Thus, for N > n where n = dim(G) we have that +∥Af∥Sp ≲ ∥σN(x, ·)∥L1(G,Sp( �G)) +� +[η]∈ �G +d2 +η⟨η⟩−N < ∞, +(3.18) +where we have use that (see Lemma 3.8 of [29]) +� +[η]∈ �G +d2 +η⟨η⟩−s < ∞ ⇐⇒ s > n. +The proof of Theorem 3.3 is complete. +□ +The following is a consequence of Theorem 3.3. +Corollary 3.4. Let G be a compact Lie group of dimension n. Let A : C∞(G) → +C∞(G) be a continuous linear operator and assume that its matrix-valued symbol +satisfies the regularity condition +∥(1 + LG) +N +2 σA(x, ·)∥Lp(G,Sp( �G)) = +� +∫ +G +∥(1 + LG) +N +2 σA(x, ·)∥p +Sp( �G)dx +� 1 +p +< ∞ +(3.19) +where N > n. Then A ∈ Sp(L2(G)) provided that 1 ≤ p < ∞. +Proof. Using the H¨older inequality and that the Haar measure of G is normalised we +have that +∫ +G +∥(1 + LG) +N +2 σA(x, ·)∥Sp( �G)dx ≤ +� +∫ +G +∥(1 + LG) +N +2 σA(x, ·)∥p +Sp( �G)dx +� 1 +p +< ∞, +(3.20) +for all N > n. Then, in view of Theorem 3.3 follows the membership of A in the +Schatten class Sp(L2(G)). +□ +3.2. Schatten properties of elliptic operators. The (ρ, δ)-case. Now we charac- +terise the membership of elliptic pseudo-differential operators to the Schatten classes +on L2(G). +Theorem 3.5. Let m ∈ R, r > 0, and let 0 ≤ δ < ρ ≤ 1. Consider an elliptic pseudo- +differential operator A ∈ Ψm +ρ,δ(G × �G). The following conditions are equivalent: + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +17 +(1) m ∈ (−∞, 0) and A belongs to the Schatten class of order r > 0 : A ∈ +Sr(L2(G)); +(2) The Bessel potential of order m belongs to the Schatten class of order r > 0 : +Bm := (1 + LG) +m +2 ∈ Sr(L2(G)). +(3) The following summability condition holds +� +[ξ]∈ �G +dξ ∫ +G +∥σ|A| +r +2 (x, [ξ])∥2 +HSdx < ∞; +(3.21) +(4) The following summability condition holds +∫ +G +� +[ξ]∈ �G +dξ∥σA(x, [ξ])∥r +Srdx < ∞; +(3.22) +(5) m < −n/r. +Proof. We start with the first equivalence (1) ⇐⇒ (2) : +• (1) =⇒ (2). Assume that A ∈ Sr(L2(G)). Let Bs := (1 + LG) +s +2 ∈ Sr(L2(G)) +be the Bessel potential of order s ∈ R. The global functional calculus for +elliptic operators on compact Lie groups (see [42]) implies that +(1 + |A| +1 +|m|)±m ∈ Ψ±m +ρ,δ (G × �G). +(3.23) +In view of the composition properties of the pseudo-differential calculus we +have that +Bm(1 + |A| +1 +|m|)−m ∈ Ψ0 +ρ,δ(G × �G). +The Calder´on-Vaillancourt theorem (see Theorem 5.2 in [42]) implies that +Bm(1 + |A| +1 +|m|)−m is a bounded operator on L2(G). Since Sr(L2(G)) is an +ideal of operators on the algebra B(L2(G)), we have that +Bm = Bm(1 + |A| +1 +|m|)−m(1 + |A| +1 +|m|)m, +(3.24) +belongs to the Schatten class Sr(L2(G)) provided that +(1 + |A| +1 +|m|)m ∈ Sr(L2(G)). +Now, we are going to prove this fact. Since m ≤ 0, if sj(A) is the j-singular +value of A, then the j-singular value sj((1+|A| +1 +|m|)m) of (1+|A| +1 +|m|)m satisfies +the inequality +sj((1 + |A| +1 +|m|)m) ≤ sj(A), +(3.25) +and then the membership of (1 + |A| +1 +|m|)m in Sr(L2(G)) follows from (1). +• (2) ⇐⇒ (5). Observe that +Bm ∈ Sr(L2(G)) ⇐⇒ Bmr/2 ∈ S2(L2(G)). +On the other hand, +∥Bmr/2∥2 +S2(L2(G)) = +� +[ξ]∈ �G +d2 +ξ⟨ξ⟩mr < ∞, +if and only if mr < −n, (see Lemma 3.8 of [29]) or equivalently, if m < −n/r +proving the statement. + +18 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +• (2) =⇒ (1). Let us give a similar proof to the one given for the reverse +statement. Because of (2) we have that m < −n/r < 0. On the other hand, +by the pseudo-differential calculus, we have that +AB−m ∈ Ψ0 +ρ,δ(G × �G). +(3.26) +Moreover, the Calder´on-Vaillancourt theorem (see Theorem 5.2 in [42]) gives +the boundedness of AB−m on L2(G). If Bm ∈ Sr(L2(G)), then using that +Sr(L2(G)) is an ideal in the algebra of the bounded operators on L2(G), we +have that +A = AB−mBm ∈ Sr(L2(G)). +• (1) ⇐⇒ (3). Note that +A ∈ Sr(L2(G)) ⇐⇒ |A| +r +2 ∈ S2(L2(G)). +(3.27) +Let R|A| +r +2 (x, y) and K|A| +r +2 (x, y) be the right-convolution kernel and the Schwartz +kernel of the operator |A| +r +2, respectively. We have the identity +∀x, y ∈ G, RA(x, xy−1) = KA(x, y). +By the Plancherel theorem we have the equivalences +|A| +r +2 ∈ S2(L2(G)) ⇐⇒ K|A| +r +2 (x, y) ∈ L2(G × G) +⇐⇒ ∫ +G +∫ +G +|RA(x, y)|2dydx < ∞ +⇐⇒ ∫ +G +� +[ξ]∈ �G +dξ∥σ|A| +r +2 (x, [ξ])∥2 +HSdx < ∞. +• (3) ⇐⇒ (4). In order to prove the equivalence of these summability conditions +we are going to apply the global functional calculus (see [42]). We start with +the identity +σ|A| +r +2 (x, [ξ]) = |σA(x, [ξ])| +r +2 + rA(x, [ξ]), +(3.28) +where the lower term rA ∈ S +mr +2 −(ρ−δ)(G × �G) because of the asymptotic ex- +pansions. First, note that +∥σ|A| +r +2 (x, [ξ])∥HS ≤ ∥|σA(x, [ξ])| +r +2∥HS + ∥rA(x, [ξ])∥HS += ∥σA(x, [ξ])∥ +r +2 +Sr + ∥rA(x, [ξ])∥HS. +Similarly, since |σA(x, [ξ])| +r +2 = σ|A| +r +2 (x, [ξ]) − rA(x, [ξ]), we have that +∥σA(x, [ξ])∥ +r +2 +Sr = ∥|σA(x, [ξ])| +r +2∥HS ≤ ∥σ|A| +r +2 (x, [ξ])∥HS + ∥rA(x, [ξ])∥HS. +To estimate the remainder ∥rA(x, [ξ])∥HS let us use that rA ∈ S +mr +2 −(ρ−δ)(G × +�G). Then, +∥rA(x, [ξ])∥HS ≤ ∥rA(x, [ξ])⟨ξ⟩−mr/2+(ρ−δ)∥op∥⟨ξ⟩mr/2−(ρ−δ)Idξ∥HS ≲ d +1 +2 +ξ ⟨ξ⟩ +mr +2 −(ρ−δ). +(3.29) +To prove that (3) =⇒ (4), observe that the condition (3) is equivalent to the +fact that A ∈ Sr(L2(G)), from which we deduce that Bm ∈ Sr(L2(G)). In + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +19 +terms of the order m we have that m < −n/r. To prove that (4) holds, note +that +∫ +G +� +[ξ]∈ �G +dξ∥σA(x, [ξ])∥r +Srdx ≤ ∫ +G +� +[ξ]∈ �G +dξ max{2∥σ|A| +r +2 (x, [ξ])∥HS, ∥rA(x, [ξ])∥HS}2 += 4 max +� +� +�∫ +G +� +[ξ]∈ �G +dξ∥σ|A| +r +2 (x, [ξ])∥2 +HSdx, ∫ +G +� +[ξ]∈ �G +dξ∥rA(x, [ξ])∥2 +HS +� +� +� +≲ 4 max +� +� +�∫ +G +� +[ξ]∈ �G +dξ∥σ|A| +r +2 (x, [ξ])∥2 +HSdx, +� +[ξ]∈ �G +d2 +ξ⟨ξ⟩mr−2(ρ−δ) +� +� +� . +Since ρ > δ, we can compare ⟨ξ⟩mr−2(ρ−δ) ≤ ⟨ξ⟩mr, and then +� +[ξ]∈ �G +d2 +ξ⟨ξ⟩mr−2(ρ−δ) ≤ +� +[ξ]∈ �G +d2 +ξ⟨ξ⟩mr < ∞ +(3.30) +because mr < −n. Then, (3) together with the convergence of the series in +(3.30) implies (4). Now, to finish the proof, let us assume (4) and from it let +us deduce (3). To do so we are going to use the inequality +∥σ|A| +r +2 (x, [ξ])∥HS ≤ ∥σA(x, [ξ])∥ +r +2 +Sr + ∥rA(x, [ξ])∥HS. +(3.31) +Then, we argue as follows: +∫ +G +� +[ξ]∈ �G +dξ∥σ|A| +r +2 (x, [ξ])∥2 +HSdx ≤ ∫ +G +� +[ξ]∈ �G +dξ(2 max{∥σA(x, [ξ])∥ +r +2 +Sr, ∥rA(x, [ξ])∥HS})2 += 4 max +� +� +�∫ +G +� +[ξ]∈ �G +dξ∥σA(x, [ξ])∥r +Srdx, ∫ +G +� +[ξ]∈ �G +dξ∥rA(x, [ξ])∥2 +HS +� +� +� +≲ 4 max +� +� +�∫ +G +� +[ξ]∈ �G +dξ∥σ|A| +r +2 (x, [ξ])∥2 +HSdx, +� +[ξ]∈ �G +d2 +ξ⟨ξ⟩mr−2(ρ−δ) +� +� +� +from which we deduce the convergence of the series in (3). +In view of the analysis above the proof of Theorem 3.5 is complete. +□ +3.3. Schatten properties of non-elliptic operators. Classical symbols. We +start this subsection, by showing that the order of an operator can be used as a +sufficient condition for deducing its Schatten-von-Neumann properties, even if the +operator is non-elliptic. +The next corollary is a consequence of Theorem 3.5. +Corollary 3.6. Let m < −n/r, r > 0, and let 0 ≤ δ < ρ ≤ 1. Consider a pseudo- +differential operator A ∈ Ψm +ρ,δ(G × �G). Then, A ∈ Sr(L2(G)). +Proof. Let us give an algebraic argument. Since m < −n/r, then the Bessel potential +satisfies Bm = (1 + LG) +m +2 ∈ Sr(L2(G)). Using again that the class Sr(L2(G)) is an +ideal on the algebra of all bounded operators on L2(G), and that AB−m is bounded on + +20 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +L2(G) (in view of the Calder´on-Vaillancourt theorem), we have that A = AB−mBm +belongs to the Schatten class Sr(L2(G)). +□ +Now, using the local Weyl formula (see [8, Theorem 1.1]) for elliptic operators we +have the following improvement of the equivalence (1) ⇐⇒ (5) in Theorem 3.5 for +trace class operators in the Kohn-Nirenberg algebra. Here, let us consider the norm +|| · ||g on the Lie algebra g induced by the Killing form B(X, Y ) = Tr[ad(X)ad(Y )], +X, Y ∈ g, defined by ||X||g := +� +−B(X, X). +Proposition 3.7. Let A ∈ Ψm +1,0(G) be a classical pseudo-differential operator of order +m ≤ 0. Assume that the average of the principal symbol σloc,A ∈ C∞(T ∗G) of A on +the co-sphere is non-zero, that is +Av[σloc,A] := +∫ +T ∗S(G) +σloc,A(x, η)dµL(x, η) ̸= 0, +(3.32) +with µL denoting the Liouville measure on the spherical vector bundle T ∗S(G) = +{(x, η) ∈ T ∗G : ||η||g = 1}. Then, A ∈ S1(L2(G)) if and only if m < −n. +Proof. From Corollary 3.6 it follows that for m < −n, we have A ∈ S1(L2(G)). Now, +let us prove the converse statement, that is if A ∈ S1(L2(G)) then m < −n. Let us +prove this by assuming that m ≥ −n, and let us get a contradiction. +Note that if A ∈ S1(L2(G)) then for any orthonormal basis (φk)k of L2(G), the +series � +k(Aφk, φk)L2(G) is absolutely convergent and this sum is independent of the +choice of the orthonormal basis (φk). Then, the trace of A is given by +Tr(A) := +� +k +(Aφk, φk)L2(G). +(3.33) +For our purposes, we will consider the orthonormal basis +{d +1 +2 +ξ ξi,j : [ξ] ∈ �G : 1 ≤ i, j ≤ dξ}, +(3.34) +of L2(G) provided by the Peter-Weyl theorem. Consider the spectrum of the positive +Laplacian +Spect(LG) = {|ξ| := λ[ξ] : [ξ] ∈ �G}. +For any λ > 0, we will consider the partial sum +Sλ := +� +|ξ|≤λ +dξ +� +i,j=1 +( ˜A(d +1 +2 +ξ ξij), d +1 +2 +ξ ξij), +(3.35) +and the discussion above allows us to use the identity +Tr(A) = lim +λ→∞ Sλ = lim +λ→∞ +� +|ξ|≤λ +dξ +� +i,j=1 +( ˜A(d +1 +2 +ξ ξij), d +1 +2 +ξ ξij). +(3.36) +Next, we will prove that if m ≥ −n, then the series in (3.36) is not absolutely con- +vergent and then, that A is not of trace class which would contradict our hypothesis. +To simplify the proof, let us consider the real part Re(A) and the imaginary part +Im(A) of A, respectively. We have the identities +Re(A) = (A + A∗)/2, Re(A) = (A − A∗)/2i. +(3.37) + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +21 +To simplify the notation we will write A0 = Re(A), and A1 = Im(A). The following +facts hold: +• Ai ∈ Ψm(G), i = 0, 1, in view of the pseudo-differential calculus properties for +sums operators. +• Ai, i = 0, 1, are self-adjoint operators. +Note that any entry (Ai(ξij), ξij) ∈ R is a real number because of the self-adjointness +of A. Now, let us use a specific property of the Peter-Weyl basis (3.34). +In [8] +the following Local Weyl-formula was obtained for any pseudo-differential operator +˜A ∈ Ψ0(G) : +� +|ξ|≤λ +dξ +� +i,j=1 +( ˜A(d +1 +2 +ξ ξij), d +1 +2 +ξ ξij) = (2π)−nCn, ˜ +Aλn + O(λn−1), +(3.38) +for any λ > 0. Moreover, in [8] the left-hand side of this identity was simplified as +follows +� +|ξ|≤λ +dξ +� +i,j=1 +dξ( ˜Aξij, ξij) = +� +|ξ|≤λ +dξ ∫ +G +Tr[σ ˜ +A(x, ξ)]dx. +(3.39) +Let +˜A0 := Re( ˜A) = ( ˜A + ˜A∗)/2, ˜A1 := Im( ˜A) = ( ˜A − ˜A∗)/2i. +(3.40) +Because these identities are valid for general operators of order m > 0, for k = 0, 1 +we also have that +� +|ξ|≤λ +dξ +� +i,j=1 +( ˜Ak(d +1 +2 +ξ ξij), d +1 +2 +ξ ξij) = (2π)−nCn, ˜ +Akλn + O(λn−1), +(3.41) +for any λ > 0, where the constant Cn, ˜ +Ak is given by +Cn, ˜ +Ak = Av[σloc, ˜ +Ak] := +∫ +T ∗S(G) +σloc, ˜ +Ak(x, η)dµL(x, η). +(3.42) +As before we can write +� +|ξ|≤λ +dξ +� +i,j=1 +dξ( ˜Akξij, ξij) = +� +|ξ|≤λ +dξ ∫ +G +Tr[σ ˜ +Ak(x, ξ)]dx, +(3.43) +where that the left hand side of (3.43) is real valued. Note that in view of (3.41) and +(3.43) we have the identity +� +|ξ|≤λ +dξ ∫ +G +Tr[σ ˜ +Ak(x, ξ)]dx = (2π)−nCn, ˜ +Akλn + O(λn−1). +(3.44) +Note that +� +λ +2 <|ξ|≤λ +dξ ∫ +G +Tr[σ ˜ +Ak(x, ξ)]dx = +� +|ξ|≤λ +dξ ∫ +G +Tr[σ ˜ +Ak(x, ξ)]dx − +� +|ξ|≤ λ +2 +dξ ∫ +G +Tr[σ ˜ +Ak(x, ξ)]dx += (2π)−nCn,Ak(1 − 1 +2n)λn + O(λn−1) += (2π)−n ˜Cn, ˜ +Akλn + O(λn−1). + +22 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +Let us apply the local Weyl formula above to the operator ˜A = AL +− m +2 +G +∈ Ψ0(G). The +matrix-valued symbol of ˜A and ˜Ak, k = 0, 1, are given by +σ ˜ +A(x, [ξ]) = σA(x, [ξ])|ξ|−m, σ ˜ +Ak(x, [ξ]) = σAk(x, [ξ])|ξ|−m, (x, [ξ]) ∈ G × �G, +where +|ξ|−m := +� +λ[ξ] +−m +, if ξ ̸= 1 �G, |1 �G|−m := 0, +where 1 �G is the trivial representation. Because of the self-adjointness of ˜Ak, the left- +hand side (and the right-hand side) of (3.44) is real-valued. Note that the linearity +of the trace gives the identity +Tr(A) = Tr(A0) + iTr(A1). +So, let us estimates the traces Tr(Ak), k = 0, 1. Indeed, +Tr(Ak) = lim +λ→∞ +dξ +� +i,j=1 +dξ(Akξij, ξij) = lim +λ→∞ +� +|ξ|≤λ +dξ ∫ +G +Tr[σAk(x, ξ)]dx += +∞ +� +k=0 +� +[ξ]∈ �G:2k−1<⟨ξ⟩≤2k +dξ ∫ +G +Tr[σAk(x, ξ)]dx += +∞ +� +k=0 +2km +� +[ξ]∈ �G:2k−1<⟨ξ⟩≤2k +dξ2−km ∫ +G +Tr[σAk(x, ξ)]dx +≍ +∞ +� +k=0 +2km +� +[ξ]∈ �G:2k−1<⟨ξ⟩≤2k +dξ⟨ξ⟩−m ∫ +G +Tr[σAk(x, ξ)]dx += +∞ +� +k=0 +2km +� +[ξ]∈ �G:2k−1<⟨ξ⟩≤2k +dξ ∫ +G +Tr[σAk(x, ξ)⟨ξ⟩−m]dx +≍ +∞ +� +k=0 +2km +� +[ξ]∈ �G:2k−1<⟨ξ⟩≤2k +dξ ∫ +G +Tr[σAk(x, ξ)|ξ|−m]dx +≍ +∞ +� +k=0 +2km +� +[ξ]∈ �G:2k−1<|ξ|≤2k +dξ ∫ +G +Tr[σ ˜ +Ak(x, ξ)]dx. +Now, using the local Weyl formula we get +Tr(Ak) ≍ +∞ +� +k=0 +2km +� +[ξ]∈ �G:2k−1<|ξ|≤2k +dξ ∫ +G +Tr[σ ˜ +Ak(x, ξ)]dx +≍ +∞ +� +k=0 +2km � +(2π)−n ˜Cn, ˜ +Ak2kn + O(2k(n−1)� +≍ +∞ +� +k=0 +� +˜Cn, ˜ +Ak2k(n+m) + O(2k(n+m−1)� +. + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +23 +Now, it is clear that if m ≥ −n,then the geometric series +∞ +� +k=0 +� +˜Cn, ˜ +Ak2k(n+m) + O(2k(n+m−1)� +diverges for k = 0 or for k = 1, which also implies that the trace of T diverges. To +see this observe that the constant in (3.42) cannot be equal to zero simultaneously +for k = 0, 1. Indeed, from (3.32) we have that +Av[σloc,A] = ˜Cn, ˜ +A0 + i ˜Cn, ˜ +A1 ̸= 0. +(3.45) +We illustrate this geometric fact in Figure 1 below. Note that we have used that on +Figure 1. The average condition on the co-sphere in the case of a +real-valued principal symbol σA. In this case the positive part of the +symbol dominates the region where the symbol is negative. The green +curve represents the level curve at height zero (that occurs along the +zero section). +the co-sphere T ∗S(G) one has the identity +∀(x, η) ∈ T ∗S(G), σ ˜ +A(x, η) = σA(x, η)∥η∥−m = σA(x, η), +(3.46) +because the fact that (x, η) ∈ T ∗S(G) implies that x ∈ M and that the norm ∥η∥ of +η on the corresponding fiber is equal to one. +Thus we have proved that T is not in the ideal S1(L2(G)) which contradicts our +initial hypothesis. The proof of Proposition 3.7 is complete. +□ +As a consequence of the previous corollary we have the following characterisation +of the trace class pseudo-differential operators. +Theorem 3.8. Let A ∈ Ψm +1,0(G) be a classical pseudo-differential operator of order +m ≤ 0. Then, A ∈ S1(L2(G)) if and only if m < −n. + +T*S(G) +Oloc,A(C, n) > 0 +Oloc,A(C, n) = 0 +loc,A(, n) < 024 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +Proof. Note that if the average of the principal symbol σloc,A ∈ C∞(T ∗G) of A on the +co-sphere is non-zero, that is +Av[σloc,A] := +∫ +T ∗S(G) +σloc,A(x, η)dµL(x, η) ̸= 0, +(3.47) +then the statement follows from Proposition 3.7. On the other hand, assume that +A ∈ S1(L2(G)) and that +Av[σloc,A] := +∫ +T ∗S(G) +σloc,A(x, η)dµL(x, η) = 0. +(3.48) +Define the operator ˜A = A + (LG)−n−ε, ε > 0. Note that ˜A ∈ S1(L2(G)) since +∥ ˜A∥S1 ≤ ∥A∥S1 + ∥˜(LG)−n−ε∥S1 < ∞. Moreover by definition of ˜A we have +Av[σloc, ˜ +A] := +∫ +T ∗S(G) +||η||−n−ε +g +dµL(x, η) = Vol(T ∗S(G)) ̸= 0. +(3.49) +Note that if m ≥ −n, then the order of ˜A would be larger than −n. However, the +condition in (3.49) together with the fact that ˜A ∈ S1(L2(G)) would imply that +m < −n in view of Proposition 3.7, which is a contradiction. Thus, one has that +m < −n as expected. The proof is complete. +□ +Now, we will use the ideal properties of the Schatten classes to derive the extension +of Proposition 3.7 to the case r > 1. +Proposition 3.9. Let A ∈ Ψm +1,0(G) be a classical pseudo-differential operator of order +m ∈ R. Assume that the average of the principal symbol σloc,A ∈ C∞(T ∗G) of A on +the co-sphere is non-zero, that is +Av[σloc,A] := +∫ +T ∗S(G) +σloc,A(x, η)dµL(x, η) ̸= 0. +(3.50) +For any 1 < r < ∞, if A ∈ Sr(L2(G)) then m ≤ −n/r. Moreover, for r = ∞, +A ∈ S∞(L2(G)) = B(L2(G)) if and only if m ≤ 0. +Proof. Let us consider first, the case where 1 < r < ∞. Let ε > 0. Consider the +operator Ls/2 +G +where s satisfies +s := −n +q − ε < −n +q , +and where q := r/(r − 1) is the conjugate exponent of r. Then, q is given by the +identity 1 = 1 +q + 1 +r. Note that since r > 1, one has that q > 0. Proposition 3.7 implies +that Ls/2 +G +∈ Sq(L2(G)). Since q and r and conjugate exponents, we have that +˜A := ALs/2 +G +∈ S1(G). +Note that the principal symbol σloc, ˜ +A satisfies the identity +∀(x, η) ∈ T ∗G \ {0}, σloc, ˜ +A(x, η) = σloc,A(x, η)∥η∥s. +(3.51) +In particular, on the co-sphere we have +∀(x, η) ∈ T ∗S, +σloc, ˜ +A(x, η) = σloc,A(x, η). +(3.52) + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +25 +Then, (3.50) allows to deduce the non-vanishing property +Av[σloc,A] := +∫ +T ∗S(G) +σloc, ˜ +A(x, η)dµL(x, η) ̸= 0, +allowing the use of the Proposition 3.7 in order to conclude that the order m + s of +˜A ∈ S1 satisfies the inequality m + s < −n. Then, we have that +m < −s − n = n +q + ε − n = n +�1 +q − 1 +� ++ ε = −n +r + ε , +where the last hold tru for any ϵ > 0. Taking ε → 0+ we have that m ≤ −n/r. +Now, let us assume that r = ∞. For m ≤ 0, it follows from the Calder´on- +Vaillancourt theorem that A ∈ S∞(L2(G)) = B(L2(G)). Now, let us assume that +A ∈ S∞(L2(G)) = B(L2(G)) and let us give an argument proving that its order +satisfies m ≤ 0. Proceeding by contradiction suppose that m > 0. Note that the +conjugate exponent to r = ∞ is q = 1. Let ε ∈ (0, m) and let s = −n − ε. Consider +the operator Ls/2 +G +∈ S1(L2(G)) where s < −n. Then the operator +˜A := ALs/2 +G +∈ S∞(L2(G)) ◦ S1(L2(G)) ⊆ S1(L2(G)), +is of trace class. Since the principal symbol is given by (3.51) the co-sphere condition +in (3.50) is satisfied and Proposition 3.7 implies that the order of ˜A satisfies the +inequality m + s < −n. We have that +m + s = m − n − ε = (m − ε) − n ≥ −n, +which contradicts the analysis above. So, we must have that m ≤ 0. The proof of +Proposition 3.9 is complete. +□ +Corollary 3.10. Assume that r ∈ (1, ∞) ∩ Z. Let A ∈ Ψm +1,0(G) be a classical pseudo- +differential operator of order m. Assume that the average of the principal symbol +σloc,A ∈ C∞(T ∗G) of A on the co-sphere is non-zero, that is +Av[σloc,A] := +∫ +T ∗S(G) +σloc,A(x, η)dµL(x, η) ̸= 0. +(3.53) +Then, A ∈ Sr(L2(G)) if and only if m < −n/r. +Proof. From Corollary 3.6 it follows that for m < −n/r, 1 < r < ∞, we have +A ∈ Sr(L2(G)). Now, let us prove the converse statement, that is if A ∈ Sr(L2(G)), +then m < −n/r. To do this, let us show that the borderline m = −n/r in Proposition +3.9 is not possible. To do this, observe that if A ∈ Sr, then +Ar = A ◦ A ◦ · · · A ∈ Sr(L2(G)) ◦ Sr(L2(G)) ◦ · · · Sr(L2(G)) ⊆ S1(L2(G)), (3.54) +where the composition is taken r-times. However, the order of the trace class operator +Ar is −n and this contradicts the conclusion in Proposition 3.7. +So, necessarily +m < −n/r. +□ +Theorem 3.11. Let A ∈ Ψm +1,0(G) be a classical pseudo-differential operator of order +m. +(A) If r ∈ [1, ∞) ∩ Z, then A ∈ Sr(L2(G)) if and only if m < −n/r. For r = ∞, +A ∈ B(L2(G)) if and only if m ≤ 0. + +26 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +(B) If r ∈ (1, ∞) \ Z, and A ∈ Sr(L2(G)), then m ≤ −n/r. Moreover, if A is +elliptic then one has the strict inequality m < −n/r. +Proof. Let us prove the statement in (A). For r = 1 this statement has been proved +in Theorem 3.8. +At this stage of the manuscript we only have to prove that if +A ∈ Sr(L2(G)) then m ≤ −n/r, where n/r := 0 for r = ∞. Now, let us consider the +case where 1 < r ≤ ∞, with r ∈ Z. +Note that if the average of the principal symbol σloc,A ∈ C∞(T ∗G) of A on the +co-sphere is non-zero, that is +Av[σloc,A] := +∫ +T ∗S(G) +σloc,A(x, η)dµL(x, η) ̸= 0, +(3.55) +then the statement follows from Corollary 3.10. On the other hand, assume that +A ∈ S1(L2(G)) and that +Av[σloc,A] := +∫ +T ∗S(G) +σloc,A(x, η)dµL(x, η) = 0. +(3.56) +Define the operator ˜A = A + (LG)−n/r−ε, ε > 0, where n/r := 0 whenever r = ∞. +Note that ˜A ∈ Sr(L2(G)) since ∥ ˜A∥Sr ≤ ∥ ˜A∥Sr +∥˜(LG)−n/r−ε∥Sr < ∞. On the other +hand note that +Av[σloc, ˜ +A] := +∫ +T ∗S(G) +||η||−n/r−ε +g +dµL(x, η) = Vol(T ∗S(G)) ̸= 0. +(3.57) +Now that if m ≥ −n/r, then the order of ˜A must be larger than −n/r (or strictly +larger than zero for r = ∞) but the condition in (3.57) together with the fact that +˜A ∈ Sr(L2(G)) implies that m < −n/r in view of Proposition 3.7, which is a con- +tradiction. Thus, one has that m < −n/r for 1 < r < ∞ and m ≥ 0 for r = ∞, as +expected. +Now, let us prove the statement in (B). It can be done following the proof of +Proposition 3.9. +However, let us give another argument. +For this, consider r ∈ +(1, ∞) \ Z. Then, the integer part [r] of r satisfies the strict inequalities +[r] < r < [r] + 1. +(3.58) +Let ε > 0 and define the parameter +s = n +� 1 +[r] − 1 +r +� ++ ε. +(3.59) +Since A ∈ Sr and for any ε′ > 0, (1+LG)− s +2 ∈ S n +s +ε′, we will prove that there exists +ε′ +0 > 0 such that As := A(1 + LG)− s +2 ∈ S[r]. To do this, we have to guarantee, that +there exists ε′ +0 > 0 such that, the H¨older property +1 +[r] = 1 +r + +1 +n +s + ε′ +0 +(3.60) +holds because its validity would imply that As ∈ Sr ◦ S n +s +ε′ +0 ⊆ S[r]. To prove that +the equation in (3.60) admits a solution, define the continuous function +g(ε′) := +1 +n +s + ε′. +(3.61) + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +27 +Note that +g(0) = s +n > 1 +[r] − 1 +r > 0 = g(∞) := lim +ε′→∞ g(ε′). +(3.62) +By Bolzano’s theorem, there exists ε′ +0 ∈ (0, ∞) such that g(ε′ +0) = +1 +[r]− 1 +r as desired. We +have now proved the existence of ε′ +0 in in (3.60) and its implication As ∈ S[r]. Hence +by the result in Part (A), the order m−s of As, satisfies the inequality m−s < −n/[r], +or equivalently m < s − (n/[r]). In view of (3.59), we have proved that +∀ε > 0, m < s = n +� 1 +[r] − 1 +r +� ++ ε − n +[r] = −n +r + ε, +(3.63) +which certainly implies that m ≤ −n/r. Note that if A is elliptic, this inequality can +be improved to an strict inequality, namely we would have m < −n/r, as proved in +Theorem 3.5. The proof is complete. +□ +We end this subsection with the following atypical construction in the case of the +classes Ψm +0,0(Tn) on the n-dimensional torus. +Theorem 3.12. For any κ > 0, there exists a non-elliptic pseudo-differential oper- +ator A in the exotic class Ψ−κ +0,0 (Tn) \ Ψ−κ−ε +0,0 +(Tn), for all ε > 0, that belongs to all the +Schatten ideals Sr(L2(Tn)), where r > 0. +Proof. Let κ > 0, and let (ej)n +j=1 be the canonical orthonormal basis of Rn, and +consider the set +D = {2ke1 = (2k, 0, · · · , 0) : k ∈ N}. +(3.64) +Define the symbol +a(ξ) = ⟨ξ⟩−κ, ξ ∈ D; a(ξ) = 0, ξ ∈ Zn \ D. +(3.65) +Observe that for any α ∈ Nn +0, we have +|∆αa(ξ)| ≤ +� +β≤α +�α +β +� +|a(ξ + β)| ≤ +� +β≤α +�α +β +� +⟨ξ + β⟩−κ ≲α (1 + |ξ|)−κ, ξ ∈ Zn. +To see that the right-hand side of this inequality is the best possible, we consider the +case where α = e1 and when ξk = 2ke1 ∈ D. Indeed, note that +|∆e1a(ξk)| = |a(ξk + e1) − a(ξk)| = |⟨2ke1⟩−κ| = ⟨ξk⟩−κ. +This argument proves that the estimate |∆e1a(ξ)| ≤ ⟨ξ⟩−κ, is sharp and even that it +is an equality when ξk ∈ D. +To prove that the operator A = Op(a) associated to the symbol a belongs to the +Schatten class Sr(L2(Tn)) note that the sequence of singular values of A is determined +by the values of its symbol, that is +sξ(A) = |a(ξ)|, ξ ∈ Zn, +(3.66) +are all the singular values of A. Consequently +� +ξ∈Zn +sξ(A)r = +� +ξk∈D +sξ(A)r ≍ +∞ +� +k=1 +2−kκr < ∞. + +28 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +Now, to prove that A is not elliptic and that it is of order m = −κ, observe that the +best estimate from below that a satisfies is the following +|a(ξ)| ≥ 0, +(3.67) +with equality when ξ ∈ Zn \ D. The proof of this atypical case is complete. +□ +3.4. Classical operators in Schatten classes and their principal symbols. +We consider the subclass Ψm +0 (G) in Ψm/2 +cl +(G) of operators with homogeneous symbols +of order m. First we have the following result where we show that only the principal +homogeneous component of the principal symbols contains the information about the +membership of a classical operator to any Schatten class S (L2(G)). +Proposition 3.13. Let r ∈ (0, ∞] and m ∈ R. Then the following conditions are +equivalent: +(1) Ψm +cl (G) ⊆ Sr(L2(G)); +(2) Ψm +0 (G) ⊆ Sr(L2(G)). +Proof. Evidently, (1) implies (2). Suppose that (2) holds. We shall prove that (1) is +true. If N is large enough, then Ψm−N ⊆ Sr(L2(G)). Observing that the operator +B = +√ +L +−1 +G is continuous on L2(G), we get that Ψm +0 Bk ∈ Sr(L2(G)) for any k ∈ N. +Since, the principal symbol of Bk, which is given by +σBk(x, η) = ∥η∥−k +g , ∀x ∈ G, ∀η ∈ g \ {0}, +is homogeneous of order −k, we have that +Ψm +0 Bk = Ψm−k +0 +mod Ψ−∞(G × �G). +Note that Op(Ψ−∞(G × �G)) ⊆ Sr(L2(G)). Then, we get +Ψm +cl = +N−1 +� +k=0 +Ψm +0 Bk + Ψm−N +cl +mod Ψ−∞(G × �G) ⊆ Sr(L2(G)), +in view of our hypothesis Ψm +0 ⊆ Sr(L2(G)), which shows that (1) holds when (2) +holds. This gives the result. +□ +In view of Theorem 3.11 we have the following characterisation. +Corollary 3.14. Let A ∈ Ψm +0 (G) be of order m ∈ R. +(A). If r ∈ [1, ∞) ∩ Z, then A ∈ Sr(L2(G)) if and only if m < −n/r. For r = ∞, +A ∈ B(L2(G)) if and only if m ≤ 0. +(B). If r ∈ (1, ∞) \ Z, and A ∈ Sr(L2(G)), then m ≤ −n/r. Moreover, if A is +elliptic then one has the strict inequality m < −n/r. +3.5. Order of classical operators vs order of global symbols. As an application +of the methods developed in this manuscript we prove that the order of a symbol +characterises the order of the operator. +Proposition 3.15. Let µ, t ∈ R and let A ∈ Ψµ +1,0(G) be a classical pseudo-differential +operator such that its global symbols satisfies +∀(x, [ξ]) ∈ G × �G, ∥σA(x, [ξ])∥op ≤ C⟨ξ⟩t. +(3.68) + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +29 +Then, A ∈ Ψt +1,0(G), provided that the average of its principal symbol σloc,A ∈ C∞(T ∗G) +on the co-sphere is non-zero: +Av[σloc,A] := +∫ +T ∗S(G) +σloc,A(x, η)dµL(x, η) ̸= 0. +(3.69) +Proof. Let us consider two cases: t = 0 and t ̸= 0. +• Case 1: t = 0. Assume that A ∈ Ψµ +1,0(G). It is clear that if µ ≤ 0 we have +nothing to prove. So, the relevant case is when µ > 0. First, let us prove the +statement in the case where µ ∈ N. To do it, we will proceed using induction. +So, let us start by proving that the result is true if µ = 1. Then, assuming +that A ∈ Ψ1 +1,0(G) and with the additional condition +∀(x, [ξ]) ∈ G × �G, ∥σA(x, [ξ])∥op ≤ C, +(3.70) +we have to deduce that A ∈ Ψ0 +1,0(G). To prove this, we will use Corollary 2.2 +of [40] to deduce that A is bounded on L2(G), and then from Proposition 3.9 +we use the L2(G)-boundedness of A and the average condition in (3.69) to +deduce that its order µ ≤ 0. This proves the case µ = 1. Now, let us state our +induction hypothesis: +– Let us assume that µ0 ∈ N is such that any classical pseudo-differential +operator ˜A ∈ Ψµ0 +1,0(G) with a bounded operator norm of its matrix-valued +symbol +∀(x, [ξ]) ∈ G × �G, ∥σ ˜ +A(x, [ξ])∥op ≤ C, +(3.71) +and with its principal symbol satisfying the average condition (3.69) be- +longs to the class Ψ0 +1,0(G). +Now, let A ∈ Ψµ0+1 +1,0 +(G) be a classical pseudo-differential operator whose sym- +bol satisfies (3.69) and (3.70). Define ˜A = A(1+LG)− 1 +2. Note that ˜A ∈ Ψr0 +1,0(G) +and that its symbol satisfies (3.71) since +∥σ ˜ +A(x, [ξ])∥op = ∥σA(x, [ξ])⟨ξ⟩−1∥op ≤ ∥σA(x, [ξ])∥op ≤ C. +Our induction hypothesis implies that ˜A belongs to the class Ψ0 +1,0(G). But +then, the Calder´on-Vaillancourt theorem implies that ˜A is bounded on L2(G). +Since on the co-sphere T ∗S(G) the principal symbols of ˜A and of A agree, ˜A +satisfies the average condition (3.69). +Then, Proposition 3.9 implies that +µ0 ≤ 0. Since +A(1 + LG)− 1 +2 ∈ Ψµ0 +1,0(G) ⊆ Ψ0 +1,0(G), +then +A ∈ Ψµ0+1 +1,0 +(G) ⊆ Ψ1 +1,0(G), +and together with the hypothesis (3.70) we can use the base case µ = 1 in +the inductive process to establish that A ∈ Ψ0 +1,0(G). So, by the mathematical +induction we have proved the statement in Theorem 3.15 in the case where +µ ∈ N. Now, in the general case if [µ] is the integer part of every µ > 0, and +if A ∈ Ψµ +1,0(G) is a classical pseudo-differential operator satisfying (3.70), we +have that A ∈ Ψ[µ]+1 +1,0 +(G) and then the better conclusion A ∈ Ψ0(G) follows + +30 +D. CARDONA, M. CHATZAKOU, M. RUZHANSKY, AND J. TOFT +from the statement of Theorem 3.15 for integer orders. Thus, we have proved +Theorem 3.15 in the case t = 0. +• Case 2. +t ̸= 0. Assume that A ∈ Ψµ +1,0(G) has a matrix-valued symbol +satisfying (3.72) and the average condition in (3.69). +Then, ˜A = A(1 + +LG)− t +2 ∈ Ψr−t +1,0 (G) satisfies (3.69) and its matrix-valued symbol satisfies the +inequality (3.70). From the first part of the poof, we deduce that ˜A = A(1 + +LG)− t +2 ∈ Ψ0 +1,0(G). Then, the pseudo-differential calculus implies that A ∈ +Ψt +1,0(G). +The proof of Proposition 3.15 is complete. +□ +Now, we will remove the geometric average condition in Proposition 3.15 to prove +a general statement on compact Lie groups. +Theorem 3.16. Let µ, t ∈ R and let A ∈ Ψµ +1,0(G) be a classical pseudo-differential +operator such that its global symbol satisfies +∀(x, [ξ]) ∈ G × �G, ∥σA(x, [ξ])∥op ≤ C⟨ξ⟩t. +(3.72) +Then A ∈ Ψt +1,0(G). +Proof. Note that if the average of the principal symbol σloc,A ∈ C∞(T ∗G) of A on the +co-sphere is non-zero: +Av[σloc,A] := +∫ +T ∗S(G) +σloc,A(x, η)dµL(x, η) ̸= 0, +(3.73) +the statement follows from Proposition 3.15. On the other hand, if the principal +symbol of A has average zero on T ∗S(G), that is, +Av[σloc,A] := +∫ +T ∗S(G) +σloc,A(x, η)dµL(x, η) = 0, +(3.74) +we define the operator ˜A := A + (LG) +t +2. Is it clear that the principal symbol of ˜A is +given by +σloc, ˜ +A := σloc,A + ∥ξ∥t +g. +Note that the matrix-valued symbol of ˜A satisfies also +σ ˜ +A(x, [ξ]) = σA(x, [ξ]) + |ξ|tIdξ, (x, [ξ]) ∈ G × �G. +(3.75) +From our hypothesis we have the estimate +∥σ ˜ +A(x, [ξ])∥op ≤ ˜C⟨ξ⟩t, (x, [ξ]) ∈ G × �G, +and the average condition +Av[σloc, ˜ +A] := +∫ +T ∗S(G) +σloc,A(x, η)dµL(x, η) + +∫ +T ∗S(G) +||η||t +gdµL(x, η) = Vol(T ∗G) ̸= 0. +(3.76) +From Proposition 3.15 follows that ˜A ∈ Ψt +1,0(G) and using the property (LG) +t +2 ∈ +Ψt +1,0(G), one obtains that A = ˜A − (LG) +t +2 ∈ Ψt +1,0(G). The proof is complete. +□ + +SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS +31 +References +1. Buzano, E., Nicola, N.: Pseudo-differential operators and schatten-von Neumann classes. In: +Boggiatto, P., Ashino, R., Wong, M.W. (eds.) 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I. +Duv´an Cardona: +Department of Mathematics: Analysis, Logic and Discrete Mathematics +Ghent University, Belgium +E-mail address duvan.cardonasanchez@ugent.be +Marianna Chatzakou: +Department of Mathematics: Analysis, Logic and Discrete Mathematics +Ghent University, Belgium +E-mail address Marianna.Chatzakou@UGent.be +Michael Ruzhansky: +Department of Mathematics: Analysis, Logic and Discrete Mathematics +Ghent University, Belgium +and +School of Mathematical Sciences +Queen Mary University of London +United Kingdom +E-mail address michael.ruzhansky@ugent.be +Joachim Toft: +Department of Mathematics +Linnæus University +V¨axj¨o-Sweden +E-mail address joachim.toft@lnu.se + diff --git a/N9E2T4oBgHgl3EQfrAg0/content/tmp_files/load_file.txt b/N9E2T4oBgHgl3EQfrAg0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..424a74c7223bc604975e7ff8bf9be96197ef218d --- /dev/null +++ b/N9E2T4oBgHgl3EQfrAg0/content/tmp_files/load_file.txt @@ -0,0 +1,1411 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf,len=1410 +page_content='SCHATTEN-VON NEUMANN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS DUV´AN CARDONA, MARIANNA CHATZAKOU, MICHAEL RUZHANSKY, AND JOACHIM TOFT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let G be a compact Lie group of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In this work we char- acterise the membership of classical pseudo-differential operators on G in the trace class ideal S1(L2(G)), as well as in the setting of the Schatten ideals Sr(L2(G)), for all r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In particular, we deduce Schatten characterisations of elliptic pseudo- differential operators of (ρ, δ)-type for the large range 0 ≤ δ < ρ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Additional necessary and sufficient conditions are given in terms of the matrix-valued symbols of the operators, which are global functions on the phase space G × �G, with the momentum variables belonging to the unitary dual �G of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In terms of the param- eters (ρ, δ), on the torus Tn, we demonstrate the sharpness of our results showing the existence of atypical operators in the exotic class Ψ−κ 0,0 (Tn), κ > 0, belonging to all the Schatten ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Additional order criteria are given in the setting of classical pseudo-differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We present also some open problems in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Outline 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Historical aspects 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Exotic examples and the main result 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Open problems 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Organisation of the manuscript 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Preliminaries 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The Fourier analysis of a compact Lie group 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The quantisation formula 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Global H¨ormander classes on compact Lie groups 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Schatten Properties 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Schatten properties for operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Limited regularity symbols 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Schatten properties of elliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The (ρ, δ)-case 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Schatten properties of non-elliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Classical symbols 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Classical operators in Schatten classes and their principal symbols 28 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 35S30, 42B20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Secondary 42B37, 42B35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Schatten von-Neumann classes, H¨ormander classes, Compact Lie groups, global symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The authors were supported by the FWO Odysseus 1 grant G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='0H94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='18N: Analysis and Partial Differential Equations, by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021) and by the dyCon Project 2015 H2020-694126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Marianna Chatzakou is also supported by the FWO Fellowship grant No 12B1223N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Michael Ruzhansky is also supported by EPSRC grant EP/R003025/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='04044v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='FA] 10 Jan 2023 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Order of classical operators vs order of global symbols 28 References 31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let M be an orientable compact manifold without boundary with volume element dx, and let us consider the Hilbert space L2(M) = L2(M, dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For any m ∈ R and for 0 ≤ δ < ρ ≤ 1, let Ψm ρ,δ(M) be the H¨ormander class of continuous linear operators on C∞(M), in local coordinates having the form Af(x) = ∫ Rn ∫ Rn e2πi(x−y,θ)a(x, θ)f(y)dydθ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1) and defined by those symbols a := a(x, θ) ∈ C∞(Rn × Rn) satisfying the estimates |∂β x∂α ξ a(x, θ)| ≲α,β,K (1 + |θ|)m−ρ|α|+δ|β| (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2) uniformly in x over compact subsets K ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' It is well-known that the class Ψm ρ,δ(M) is invariant under changes of coordinates if ρ > 1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' On the other hand, when M has a group structure compatible with its differential structure, namely, when M = G is a compact Lie group, in [39,41] one introduced the notion of a global symbol allowing the construction of new classes of pseudo-differential operators Ψm ρ,δ(G) also when 0 ≤ ρ ≤ 1 − δ, and providing a new description of the H¨ormander classes in the case where ρ > 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In this work we investigate necessary and sufficient conditions in order to guarantee the inclusion of the H¨ormander classes on compact Lie groups in the Schatten von-Neumann classes Sr(L2(G)), namely, we will investigate sharp conditions allowing for the inclusion Ψm ρ,δ(G) ⊆ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3) We recall that for any r > 0, a compact operator T : L2(G) → L2(G) belongs to the Schatten von-Neumann ideal Sr(L2(G)), if the sequence of its singular values {sn(T)}n∈N (formed by the eigenvalues of the operator √ T ∗T) belongs to ℓr(N), that is, if �∞ n=1 sn(T)r < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Historical aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' It is well known that an elliptic pseudo-differential opera- tor A ∈ Ψm 1,0(M) of order m ∈ R, belongs to the ideal Sr(L2(M)), r > 0, if and only if m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Once removed the ellipticity condition the problem of finding order criteria for classifying pseudo-differential operators on the ideal Sr(L2(M)) is still an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' However, in the literature, if one considers the problem of classifying pseudo-differential in the Schatten von-Neumann classes Sr(L2(Rn)), whose symbols belong to the H¨ormander classes Sm ρ,δ(Rn), the Beals-Fefferman classes SM1,M2 Φ,φ (Rn), or the H¨ormander classes S(m, g) the subject becomes more classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Indeed, in [34], H¨ormander observed that the distribution of the eigenvalues (and then the Schatten properties) of an elliptic pseudo-differential operator A = Opw(a) is encoded in terms of the level sets of the symbol a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Indeed, he showed that the spectral formula N(λ) ∼ ∫ a(x,ξ)<λ dxdξ, SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 3 holds for any λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Here, N(λ) = #{j : |λj| ≤ λ} denotes the spectral function of the operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The first results of this type can be traced back to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Weyl for second order differential operators, and to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Courant (see [34, Page 297]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='9 of [34], H¨ormander proved the following sufficient condition m ∈ L1(R2n), a ∈ S(m, g) =⇒ Opw(a) ∈ S1(L2(Rn)), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4) with the metric g and the weight function m satisfying suitable conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In [35], H¨ormander also characterised the L2 continuity of Weyl operators with the symbols in S(m, g) as {Opw(a) : a ∈ S(m, g) } ⊆ S∞(L2(Rn)) ⇐⇒ m ∈ L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5) By adding some additional conditions on m and g, Buzano and Nicola in [1], extended (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5) into {Opw(a) : a ∈ S(m, g) } ⊆ Sp(L2(Rn)) ⇐⇒ m ∈ Lp, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='6) for every p ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In [47], it is shown that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5) still holds true without the additional assumptions on m and g in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In [3] the Schatten characterization Opw(a) ∈ Sp(L2(Rn)) ⇐⇒ a ∈ Lp, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7) provided a ∈ S(m, g) and hN g m ∈ Lp for some N ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Here hg ≤ 1 is the Planck’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For further Schatten properties of pseudo-differential operators on Rn, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' [18,33,37,45,46,48,51] and for Schatten properties on compact manifolds we refer the reader to the works [5–11,13–15,19–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Exotic examples and the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' On the other hand, necessary and sufficient conditions of the type (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='6) for non-elliptic operators on compact Lie groups are still an open problem, as in the case of classical pseudo-differential operators (operators with polyhomogeneous symbols) as well as in the modern setting of the (ρ, δ)-classes on G, (see [39]) allowing the complete range 0 ≤ δ < ρ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Although when 0 ≤ ρ < δ ≤ 1, as we will show, the order condition m < −n/r assuring the membership of an elliptic operator in the class Sr(L2(G)) is a sharp criterion, the situation changes dramatically if one considers the borderline δ = ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Indeed, in the case of the torus G = Tn with arbitrary dimension n we have discovered the following strongly atypical situation: For any κ > 0, there exists a non-elliptic pseudo-differential operator A in the exotic class Ψ−κ 0,0 (Tn)\\Ψ−κ−ε 0,0 (Tn), for all ε > 0, that belongs to all the Schatten ideals Sr(L2(Tn)), with 0 < r < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We present later this statement in the form of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Here we observe that the classes Ψm 0,0(G) on a compact Lie group G are of interest in PDE when computing inverses of real vector fields X+c, where the constant term c belongs to an exceptional set C ⊆ iR, see [43, Page 627] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, we are going to discuss our main results and we also will propose some conjectures related to the inclusion in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3) which is the central question of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To continue let us fix the notation and let us introduce the notion of a (full/global) matrix-valued symbol as developed by the third author and Turunen 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' One reason for this is that our criteria will be addressed in terms of such matrix-valued symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let us consider the unitary dual �G of the compact Lie group G, which is formed by all the equivalent classes [ξ] of continuous, unitary, and irreducible representations ξ : G �→ U(Cℓ), and let ℓ = dξ be the dimension of the representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To any continuous linear operator on C∞(G) and then to any pseudo-differential operator A in the class Ψm ρ,δ(G) := Ψm ρ,δ(G× �G), 0 ≤ δ ≤ ρ ≤ 1, one can associate a matrix-valued global symbol a : G × �G → � [ξ]∈ �G Cdξ×dξ, (x, [ξ]) �→ a(x, [ξ]) ∈ Cdξ×dξ, allowing the global quantisation formula Af(x) = � [ξ]∈ �G ∫ G dξTr[ξ(y−1x)a(x, ξ)]f(y)dy, ∀f ∈ C∞(G), ∀x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='8) The problem of finding criteria to assure the membership of a pseudo-differential operator A in the Schatten classes Sr(L2(G)), in terms of its matrix-valued symbol a := a(x, [ξ]) has been a source of intensive mathematical activity for around 10 years, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' [5–11,13–15,19–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Contributing to the previous references, the main results of this work can be summarised in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3 below where we will use the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We denote by g the Lie algebra of a compact Lie group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The mapping B(X, Y ) = −Tr[ad(X)ad(Y )], X, Y ∈ g, is the Killing form on g × g and we denote by ||X||g := � −B(X, X) the corresponding norm on g associated to −B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1 We denote by LG the positive Laplace Beltrami operator on G, and under the identification g∗ ∼= g, η �→ ||η||2 g, η ∈ g∗ \\ {0}, denotes its principal symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The family Sr(L2(G)), 0 < r < ∞, is formed by the Schatten von Neumann ideals on a compact Lie group G, if 0 < r < ∞, and for r = ∞, Sr(L2(G)) = B(L2(G)) denotes the algebra of all bounded linear operators on L2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For every r ∈ (0, ∞), the Schatten norm of a symbol a(x, [ξ]) is given by ∥a(x, [ξ])∥Sr = Tr[|a(x, [ξ])|r] 1 r , where |a(x, [ξ])| := � a(x, [ξ])∗a(x, [ξ]) is de- fined in terms of the functional calculus of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that ∥a(x, [ξ])∥S2 = ∥a(x, [ξ])∥HS is the standard Hilbert-Schmidt norm of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For any 1 ≤ p1, p2 < ∞, the space Lp1(G, Sp2( �G)) is defined by those symbols a := a(x, [ξ]) such that ∥a(·, ·)∥Lp1(G,Sp2( �G)) = � ∫ G ∥a(x, ·)∥p1 Sp2( �G)dx � 1 p1 < ∞, where ∥a(x, ·)∥Sp2( �G) = � � � [ξ]∈ �G dξ∥a(x, [ξ])∥p2 Sp2 � � 1 p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 1Since G is a compact Lie group the positive Killing form −B : g×g → C, is positive semi-definite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' −B(X, X) > 0, ∀X ∈ g \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 5 In terms of the ℓp( �G) norm ∥a(x, [ξ])∥ℓp( �G) = � � � [ξ]∈ �G d p( 2 p − 1 2) ξ ∥a(x, [ξ])∥p HS � � 1 p , 1 ≤ p < ∞, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='9) for any 1 ≤ p1, p2 < ∞, the space Lp1(G, ℓp2( �G)) is defined by those symbols a := a(x, [ξ]) such that ∥a(·, ·)∥Lp1(G,ℓp2( �G)) = � ∫ G ∥a(x, ·)∥p1 ℓp2( �G)dx � 1 p1 < ∞, we refer the reader to [32] for the embeddings between these two classes of spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The following three theorems summarise our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We start with our char- acterisation of elliptic operators in Schatten classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1 (General symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let G be a compact Lie group of dimension n, let m ∈ R, r > 0, and let 0 ≤ δ < ρ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Consider an elliptic pseudo-differential operator A ∈ Ψm ρ,δ(G × �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The following conditions are equivalent: (1) m < 0 and A belongs to the Schatten class of order r > 0, that is A ∈ Sr(L2(G));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2) The Bessel potential of order m belongs to the Schatten class of order r > 0 : Bm := (1 + LG) m 2 ∈ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3) The matrix-valued symbol of |A| r 2 satisfies the following summability condition � [ξ]∈ �G dξ ∫ G ∥σ|A| r 2 (x, [ξ])∥2 HSdx < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='10) (4) The matrix-valued symbol of A satisfies the following summability condition ∫ G � [ξ]∈ �G dξ∥σA(x, [ξ])∥r Srdx < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='11) (5) m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Moreover, if A ∈ Ψm ρ,δ(G × �G) is not elliptic and m < −n/r, then we have that A ∈ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' As for classical operators on compact Lie groups we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2 (Classical symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let G be a compact Lie group of dimension n, let m ∈ R, r > 0, and let 0 ≤ δ < ρ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let A ∈ Ψm 1,0(G) be a classical pseudo- differential operator of order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (6) If r ∈ [1, ∞) ∩ Z, then A ∈ Sr(L2(G)) if and only if m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For r = ∞, A ∈ B(L2(G)) if and only if m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (7) If r ∈ (1, ∞) \\ Z, and A ∈ Sr(L2(G)), then m ≤ −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Moreover, if A is elliptic then one has the strict inequality m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Additionally, consider the subclass Ψm 0 (G) in Ψm cl (G) of operators with homogeneous symbols of order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then the following conditions are equivalent: (8) Ψm cl (G) ⊆ Sr(L2(G)), r ∈ (0, ∞];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (9) Ψm 0 (G) ⊆ Sr(L2(G)), r ∈ (0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT In the next theorem we consider conditions of limited regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3 (Symbols of low regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let G be a compact Lie group of di- mension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let us assume that for any [ξ], the symbol a(·, [ξ]) is Haar measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then: (10) Assume that a symbol a ∈ Lp(G, ℓp( �G)) for some 1 < p < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, the corresponding pseudo-differential operator satisfies A ∈ Sp′(L2(G)), where p′ = p/(p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (11) Assume that the matrix-valued symbol a = a(x, [ξ]) satisfies the regularity condition ∥(1 + LG) N 2 σA(x, ·)∥L1(G,Sp( �G)) = ∫ G ∥(1 + LG) N 2 σA(x, ·)∥Sp( �G)dx < ∞ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='12) where N > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then A ∈ Sp(L2(G)) provided that 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In view of the open questions that have arisen in our ap- proach, and based on the active research on this field in the last 10 years, we propose the following open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Open problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let 0 ≤ δ < ρ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Prove (or disprove) that if A ∈ Ψm ρ,δ(G × �G) is a non-elliptic pseudo-differential operator that belongs to the Schatten class Sr(L2(G)) then m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Open problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Prove (or disprove) that for r ∈ (1, ∞) \\ Z, and with A ∈ Sr(L2(G)) being a non-elliptic operator, then the inequality m ≤ −n/r in (7) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2 can be improved to the strict order estimate m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We observe that the equivalence (1) ⇐⇒ (3) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1 was proved by the third author and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Delgado in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In particular, in [22] the relation between the spectral trace and the nuclear trace of operators has been investigated for the more general notion of nuclear operators and Grothendieck-Lidskii type formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Further analysis involving criteria in terms of matrix-valued symbols was also carried out on compact Lie groups and on arbitrary compact manifolds in [12,22–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Organisation of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' This paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In Section 2 we present the basics of the Fourier analysis on compact Lie groups used here as well as the preliminaries about the pseudo-differential calculus on compact Lie groups in terms of the matrix-valued symbols as developed in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Section 3 will be dedicated to the proof of our main Theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' More precisely: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1 is presented later as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5 of Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (6) and (7) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2 are proved in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='11 of Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The equivalence (8) ⇐⇒ (9) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2 is proved in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='13 of Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3 is proved in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1 (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Finally, in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5 we prove that the order of a matrix-valued symbol associated to a classical pseudo-differential operator classifies its operator or- der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For the proof we use the approach developed in this work for the analysis SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 7 of Schatten operators that involves the average of their principal symbols on the co-sphere (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The Fourier analysis of a compact Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let dx be the Haar measure on a compact Lie group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The Hilbert space L2(G) := L2(G, dx) will be endowed with the inner product (f, g) = ∫ G f(x)g(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The Peter-Weyl theorem gives a spectral decomposition of L2(G) in terms of the entries of unitary representations of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In order to present such a result we will give some preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1 (Unitary representation of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' A continuous and unitary represen- tation of G on Cℓ is any continuous mapping ξ ∈ Hom(G, U(ℓ)), where U(ℓ) is the Lie group of unitary matrices of order ℓ × ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The integer number ℓ = dξ is called the dimension of the representation ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2 (Irreducible representations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We recall that: a subspace L ⊆ Cdξ is called ξ-invariant if for any x ∈ G, ξ(x)(L) ⊆ L, where ξ(x)(L) := {ξ(x)v : v ∈ L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The representation ξ is irreducible if its only invariant subspaces are L = ∅ and L = Cdξ, the trivial ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Any unitary representation ξ is a direct sum of unitary irreducible represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We denote it by ξ = ξ1 ⊗· · ·⊗ξk, with ξi, 1 ≤ i ≤ k, being irreducible representations on factors Cdξi that decompose the representation space Cdξ = Cdξ1 ⊗ · · · ⊗ Cdξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The notion of equivalent representations allows us to define an equivalence relation in the family of unitary representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We recall it in the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3 (Equivalent representations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Two unitary representations ξ ∈ Hom(G, U(dξ)) and η ∈ Hom(G, U(dη)) are equivalent if there exists a linear mapping S : Cdξ → Cdη such that for any x ∈ G, Sξ(x) = η(x)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The mapping S is called an intertwining operator between ξ and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The set of all the intertwining operators between ξ and η is denoted by Hom(ξ, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4 (Schur Lemma, 1905).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' If ξ ∈ Hom(G, U(dξ)) is irreducible, then Hom(ξ, ξ) = CIdξ is formed by scalar multiples of the identity matrix Idξ of order dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5 (The unitary dual).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The relation ∼ on the set of unitary representa- tions, which we denote by Rep(G), and defined by: ξ ∼ η if and only if ξ and η are equivalent representations, is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The quotient set �G := Rep(G)/∼ is called the unitary dual of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Since G is a compact Lie group, �G is a discrete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The unitary dual encodes all the Fourier analysis on the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The Fourier trans- form is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='6 (Group Fourier transform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' If ξ ∈ Rep(G), the Fourier transform FG associates to any f ∈ C∞(G) a matrix-valued function FGf defined on Rep(G) as follows (FGf)(ξ) ≡ �f(ξ) = � G f(x)ξ(x)∗dx, ξ ∈ Rep(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7 (The Fourier inversion formula on a compact Lie group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The discrete Schwartz space S ( �G) := FG(C∞(G)) is the image of the Fourier transform on the class of smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' This operator admits a unitary extension from L2(G) into ℓ2( �G), with ℓ2( �G) = � φ : ∀[ξ] ∈ �G, φ(ξ) ∈ Cdξ×dξ and ∥φ∥ℓ2( �G) < ∞ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1) where ∥φ∥ℓ2( �G) := � � � [ξ]∈ �G dξ∥φ(ξ)∥2 HS � � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The norm ∥φ(ξ)∥HS = (Tr(φ(ξ)∗φ(ξ))) 1 2 is the standard Hilbert-Schmidt norm of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The Fourier inversion formula takes the form f(x) = � [ξ]∈ �G dξTr[ξ(x) �f(ξ)], ∀f ∈ L1(G), ∀x ∈ G, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2) where the summation is understood in the sense that from any equivalence class [ξ] we choose one (any) unitary representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The quantisation formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let A : C∞(G) → C∞(G) be a continuous linear operator with respect to the standard Fr´echet structure on C∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' There is a way of associating to the operator A a matrix-valued function σA defined on the non- commutative phase space G × �G to rewrite the operator A in terms of the Fourier inversion formula and in terms of the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Such a expression is called the quantisation formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To introduce it we require the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='8 (Right convolution kernel of an operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The Schwartz kernel the- orem associates to A a kernel/distribution KA ∈ D′(G × G) such that Af(x) = � G KA(x, y)f(y)dy, f ∈ C∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The distribution defined via RA(x, xy−1) := KA(x, y) that provides the convolution identity Af(x) = � G RA(x, xy−1)f(y)dy, f ∈ C∞(G), is called the right-convolution kernel of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='9 (The quantisation formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, we will associate a global symbol σA : G× �G → ∪ℓ∈NCℓ×ℓ to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Indeed, in view of the identity Af(x) = (f ∗RA(x, ·))(x), we get � Af(ξ) = �RA(x, ξ) �f(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 9 Then we have that Af(x) = � [ξ]∈ �G dξTr[ξ(x) �RA(x, ξ) �f(ξ)], f ∈ C∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3) In view of the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3), from any equivalence class [ξ] ∈ �G, we can choose one and only one irreducible unitary representation ξ0 ∈ [ξ], such that the matrix-valued function σA(x, [ξ]) ≡ σA(x, ξ0) := �RA(x, ξ0), (x, [ξ]) ∈ G × �G, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4) satisfies that Af(x) = � [ξ]∈ �G dξTr[ξ0(x)σA(x, [ξ]) �f(ξ0)], f ∈ C∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5) The representation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5) is independent of the choice of the representation ξ0 ∈ Rep(G) from any equivalent class [ξ] ∈ �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' This is a consequence of the Fourier inversion formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' So, we can simply write Af(x) = � [ξ]∈ �G dξTr[ξ(x)σA(x, [ξ]) �f(ξ)], ∀f ∈ C∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='6) In the following quantisation theorem we observe that the distribution σA in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='6) defined on G × �G is unique and can be written in terms of the operator A, see Theorems 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='6 of [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let A : C∞(G) → C∞(G) be a continuous linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The matrix-valued distribution σA(x, [ξ]) : G × �G → ∪ℓ∈NCℓ×ℓ satisfies that ∀f ∈ C∞(G), ∀x ∈ G, Af(x) = � [ξ]∈ �G dξTr[ξ(x)σA(x, [ξ]) �f(ξ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7) We have that ∀(x, [ξ]) ∈ G × �G, σA(x, ξ) = �RA(x, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The following identity holds: ∀(x, [ξ]) ∈ G × �G, σA(x, ξ) = ξ(x)∗Aξ(x), where Aξ(x) := [Aξij(x)] dξ i,j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In view of the quantisations formulae (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7), a symbol σA can be considered as a mapping defined on G× �G or as a mapping defined on G×Rep(G) by identifying all the values σA(x, ξ) = σA(x, ξ′) = σ(x, [ξ]) when ξ′, ξ ∈ [ξ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='12 (The symbol of a measurable function of the Laplacian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let X = {X1, · · · , Xn} be an orthonormal basis of the Lie algebra g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The positive Laplacian on G is the second order differential operator LG = − n � j=1 X2 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='8) The operator LG is independent of the choice of the orthonormal basis X of g, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The L2-spectrum of LG is a discrete set that can be enumerated in terms 10 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT of the unitary dual �G as Spectrum(LG) = {λ[ξ] : [ξ] ∈ �G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='9) For a Borel function f : R+ 0 → C, the right-convolution kernel Rf(LG) of the operator f(LG) (defined by the spectral calculus) is determined by the identity f(LG)φ(x) = φ ∗ Rf(LG)(x), x ∈ G, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='10) where ∀[ξ] ∈ �G, �Rf(LG)([ξ]) = f(λ[ξ])Idξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='11) Then the matrix-valued symbol of f(LG) can be determined e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='10 as follows σf(LG)(x, ξ) = �Rf(LG)([ξ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='12) Since the operator f(LG) is left-invariant the symbol σf(LG)(ξ) = σf(LG)(x, ξ) does not depend of x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Of particular interest for the definition of the global H¨ormander classes on G, will be the Japanese bracket function ⟨t⟩ := (1 + t) 1 2, t ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='13) In particular the symbol of the operator ⟨LG⟩ = (1 + LG) 1 2 is given by σ⟨LG⟩([ξ]) := ⟨ξ⟩Idξ, ⟨ξ⟩ := ⟨λ[ξ]⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='14) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Global H¨ormander classes on compact Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In this section we denote for any linear mapping U on Cn by ∥U∥op the standard operator norm ∥U∥op = ∥U∥End(Cn) := sup l̸=0 ∥Ul∥e/∥l∥e, where ∥l∥e = (l2 1 + · · · + l2 n) 1 2 is the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For introducing the H¨ormander classes on compact Lie groups we have to measure the growth of derivatives of symbols in the group variable, for this we use vector fields X ∈ T(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To derivate symbols with respect to the discrete variable [ξ] ∈ �G we use difference operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Before introducing the H¨ormander classes on compact Lie groups we have to define these differential/difference operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='13 (Left-invariant canonical differential operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' If {X1, · · · , Xn} is an arbitrary family of left-invariant vector fields, we will denote by Xα x := Xα1 1,x · · · Xαn n,x an arbitrary canonical differential operator of order m = |α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Also, we have to take derivatives with respect to the “discrete” frequency variable ξ ∈ Rep(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To do this, we will use the notion of difference operators introduced in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Indeed, the frequency variable in the symbol σA(x, [ξ]) of a continuous and linear operator A on C∞(G) is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' This is since �G is a discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='14 (Canonical difference operators Dα on the dual �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' If ξ1, ξ2, · · · , ξk, are fixed irreducible and unitary representations of G, which not necessarily belong to the same equivalence class, then each coefficient of the matrix ξℓ(g) − Idξℓ = [ξℓ(g)ij − δij] dξℓ i,j=1, g ∈ G, 1 ≤ ℓ ≤ k, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='15) SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 11 that is each function qℓ ij(g) := ξℓ(g)ij − δij, g ∈ G, defines a difference operator Dξℓ,i,j := FG(ξℓ(g)ij − δij)F −1 G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='16) We can fix k ≥ dim(G) of these representations in such a way that the corresponding family of difference operators is admissible, that is, rank{∇qℓ i,j(e) : 1 ⩽ ℓ ⩽ k} = dim(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To define higher order difference operators of this kind, let us fix a unitary irre- ducible representation ξℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Since the representation is fixed we omit the index ℓ of the representations ξℓ in the notation that will follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, for any given multi-index α ∈ N d2 ξℓ 0 , with |α| = �dξℓ i,j=1 αi,j, we write Dα := Dα11 1,1 · · · D αdξℓ ,dξℓ dξℓdξℓ for a difference operator of order m = |α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, we are ready for introducing the global H¨ormander classes on compact Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='15 (Global (ρ, δ)-H¨ormander classes in the whole range 0 ≤ δ, ρ ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We say that σ ∈ Sm ρ,δ(G × �G) if the following symbol inequalities ∥Xβ x Dασ(x, ξ)∥op ⩽ Cα,β⟨ξ⟩m−ρ|γ|+δ|β|, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='17) are satisfied for all multi-indices β and γ, and for all (x, [ξ]) ∈ G × �G, where ⟨ξ⟩ denotes the Japanese bracket function at λ[ξ] ∈ Spectrum[LG] defined in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The class Ψm ρ,δ(G× �G) ≡ Op(Sm ρ,δ(G× �G)) is defined by those continuous and linear operators on C∞(G) such that σA ∈ Sm ρ,δ(G × �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In the next theorem we describe some fundamental properties of the global H¨ormander classes of pseudo-differential operators [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let ρ, δ ∈ [0, 1] be such that 0 ⩽ δ ⩽ ρ ⩽ 1, ρ ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then Ψ∞ ρ,δ(G) := ∪m∈RΨm ρ,δ(G) is an algebra of operators stable under compositions and adjoints, that is: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' the mapping A �→ A∗ : Ψm ρ,δ(G × �G) → Ψm ρ,δ(G × �G) is a continuous linear mapping between Fr´echet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The mapping (A1, A2) �→ A1 ◦ A2 : Ψm1 ρ,δ(G × �G) × Ψm2 ρ,δ(G × �G) → Ψm1+m2 ρ,δ (G × �G) is a continuous bilinear mapping between Fr´echet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Moreover, any operator in the class Ψ0 ρ,δ(G × �G) admits a bounded extension from L2(G) to L2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 12 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' With 0 ⩽ δ < ρ ⩽ 1 such that ρ ≥ 1−δ, the condition A ∈ Ψm ρ,δ(G× �G) where m ∈ R, is equivalent to the fact that, when microlocalising the operator A into a local coordinate system U, the operator A takes the form Af(x) = ∫ Rn ∫ Rn e2πi(x−y)·ξa(x, ξ)f(y)dydξ, ∀f ∈ C∞ 0 (U), ∀x ∈ Rn, where the function a = aU, is such that for every compact subset K ⊆ U and for all α, β ∈ Nn 0, the inequalities |∂β x∂α ξ a(x, ξ)| ⩽ Cα,β,K(1 + |ξ|)m−ρ|α|+δ|β|, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='18) hold uniformly in (x, ξ) ∈ K ×Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' This characterisation of the H¨ormander classes on G was proved in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' So, for any compact Lie group G, the classes Ψm ρ,δ(G× �G) agree with the ones introduced by H¨ormander [36] when 0 ⩽ δ < ρ ⩽ 1 and ρ ≥ 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Schatten Properties In this section we analyse the membership of pseudo-differential operators in the Schatten classes on L2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Schatten properties for operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Limited regularity symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In this section we study the Schatten properties of operators with symbols of limited regu- larity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Without assumptions of regularity we start with the following criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Assume that for a symbol σA we have σA ∈ Lp(G, ℓp( �G)) for some 1 < p < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, the corresponding pseudo-differential operator A satisfies that A ∈ Sp′(L2(G)), where p′ = p/(p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let us consider the following criterion due to Russo (see [38]): ∥A∥Sp′ ≤ (∥K∥p,p′ × ∥K∗∥p,p′)1/2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1) where K is the kernel of A and K∗ is the kernel of the adjoint operator A∗, 1 < p < 2, and ∥K∥p,p′ = � �∫ G � ∫ G |K(x, y)|pdx � p′ p dy � � 1 p′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2) For a moment, let us assume that A is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, K = K∗ and then ∥A∥Sp′ ≤ ∥K∥p,p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3) Observe that the Hausdorff-Young inequality (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' [32, Page 69]) gives ∫ G |K(x, y)|p′dy = ∥y �→ F −1 G (σA(x, ·))(xy−1)∥p′ Lp′(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='dy) = ∥z �→ F −1 G (σA(x, ·))(z)∥p′ Lp′(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='dz) ≤ ∥σA(x, ·)∥p′ ℓp( �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Since p′ > 2 > p the Minkowski integral inequality implies that ∥K∥p,p′ ≤ ∥K∥p′,p = � ∫ G � ∫ G |K(x, y)|p′dy � p p′ dx � 1 p ≤ � ∫ G ∥σA(x, ·)∥p ℓp( �G)dx � 1 p = ∥σA∥Lp(G,ℓp( �G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 13 So, we have proved the statement in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1 if A is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, in the general case consider the decomposition of A into its real and imaginary part A = Re(A) + iIm(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4) Note that ∀(x, [ξ]) ∈ G × �G, σA(x, [ξ]) = σRe(A)(x, [ξ]) + iσIm(A)(x, [ξ]), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5) that ∀x ∈ G, ∥σA(x, [ξ])∥ℓp( �G) ≍ ∥σRe(A)(x, [ξ])∥ℓp( �G) + ∥σIm(A)(x, [ξ])∥ℓp( �G), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='6) and using that ∥A∥Sp′ ≍ ∥Re(A)∥Sp′ + ∥Im(A)∥Sp′, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7) and the following inequalities (in view of the self-adjointness of Re(A) and Im(A)) we have that ∥A∥Sp′ ≍ ∥Re(A)∥Sp′ + ∥Im(A)∥Sp′ ≲ ∥σRe(A)(x, [ξ])∥Lp(G,ℓp( �G)) + ∥σIm(A)(x, [ξ])∥Lp(G,ℓp( �G)) ≍ ∥σA∥Lp(G,ℓp( �G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In terms of the ℓp-Schatten norm on �G one has the Hausdorff-Young inequality (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' [32, Page 67]) ∥ �f∥Sp′( �G) ≤ ∥f∥Lp(G), 1 ≤ p ≤ 2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='8) where p′ is the conjugate exponent of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' However, since the group G is compact one has the following refined versions for this inequality ∥F −1 G σ∥Lp′(G) ≤ ∥σ∥ℓp( �G), ∥ �f∥ℓp′( �G) ≤ ∥f∥Lp(G), 1 ≤ p ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='9) In particular, the second inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='9) is sharper than (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='8) because of the embedding Sp′( �G) ⊂ ℓp′( �G) for 2 ≤ p′ ≤ ∞, see [32, Page 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We have used the first inequality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='9), which is the Hausdorff-Young inequality for the inverse Fourier transform to estimate the inequality ∥K∥p,p′ ≤ ∥σA∥Lp(G,ℓp( �G)) in the first part of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In the next result, we assume more regularity in the spatial variable to deduce a criterion for pseudo-differential operators to belong to the Schatten classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let G be a compact Lie group of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let A : C∞(G) → C∞(G) be a continuous linear operator and assume that its matrix-valued symbol satisfies the regularity condition ∥(1 + LG) N 2 σA(x, ·)∥L1(G,Sp( �G)) = ∫ G ∥(1 + LG) N 2 σA(x, ·)∥Sp( �G)dx < ∞ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='10) where N > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then A ∈ Sp(L2(G)) for all 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 14 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let us consider the matrix-valued symbol σA(x, [ξ]) = (σA,ij(x, [ξ])) dξ i,j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The Fourier inversion formula allows one to write σA,ij = � [η]∈ �G dηη(x)rs�σA, ij,sr(η, ξ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='11) where �σA, ij,sr([η], [ξ]) denotes the (s, r)-Fourier coefficient of the function σA,ij(·, [ξ]) at η ∈ [η] ∈ �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For any f ∈ C∞(G), the quantisation formula gives the identity Af(x) = � [ξ]∈ �G dξTr[ξ(x)σA(x, [ξ]) �f(ξ)] = � [ξ]∈ �G dξ � i,j,ℓ=1 dξξij(x)σA, ji(x, [ξ]) �f(ξ)iℓ = � [η]∈ �G dη � r,s=1 � [ξ]∈ �G dξ � i,j,ℓ=1 dξξij(x)ηrs(x)�σA, ji, sr([η], [ξ]) �f(ξ)iℓ = � [η]∈ �G dη � r,s=1 � [ξ]∈ �G ηrs(x)dξTr[ξ(x)�σA, sr([η], [ξ]) �f(ξ)], where �σA, sr([η], [ξ]) := (�σA, ji, sr([η], [ξ])) dξ j,i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let us define the operator Op(�σA, sr([η], ·)) corresponding to the matrix �σA, sr([η], ·) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let Mηrs be the multiplication operator associated to the function ηrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that Mηrs is a bounded operator on L2(G) with operator norm ∥Mηrs∥B(L2(G)) = ∥ηrs∥L∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='12) Then, we have that ∀f ∈ C∞(G), ∀x ∈ G, Af(x) = � [η]∈ �G dη � r,s=1 Mηrs[Op(�σA, sr([η], ·))f](x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='13) Note that for all N ∈ N, the symbol (x, [ξ]) �→ σN,rs(x, [ξ]) := (1 + LG) N 2 σA, rs(x, [ξ]) satisfies the Fourier transform identity �σA, sr([η], [ξ]) = ⟨η⟩−N�σN,rs([η], [ξ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='14) This identity allows us to write ∥A∥Sp(L2(G)) ≤ � [η]∈ �G dη � r,s=1 ∥MηrsOp(�σA, sr([η], ·))∥Sp(L2(G)) = � [η]∈ �G dη � r,s=1 ⟨η⟩−N∥MηrsOp(�σN, sr([η], ·))∥Sp(L2(G)) = � [η]∈ �G dη � r,s=1 ⟨η⟩−N∥Mηrs∥B(L2)∥Op(�σN, sr([η], ·))∥Sp(L2(G)) SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 15 = � [η]∈ �G dη � r,s=1 ⟨η⟩−N∥ηrs∥L∞(G)∥Op(�σN, sr([η], ·))∥Sp(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Since ∀x ∈ G, |ηrs(x)| ≤ � dη � r′,s′=1 |ηr′s′(x)|2 � 1 2 = � Tr[η(x)η(x)∗] = √ dη, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='15) we also have ∀x ∈ G, dη � r,s=1 |ηrs(x)| ≤ � dη � r′,s′=1 |ηr′s′(x)|2 � 1 2 dη = � Tr[η(x)η(x)∗]dη = dη √ dη, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='16) where the last implies dη � r,s=1 ∥ηrs∥L∞ ≤ dη √ dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='17) On the other hand, using the triangle inequality for the Sp-norm we have that ∥Op(�σN, sr([η], ·))∥p Sp(L2(G)) ≲ � [ξ]∈ �G dξ∥�σN, sr([η], [ξ])∥p Sp( �G) = � [ξ]∈ �G dξ∥ ∫ G σN(x, [ξ])ηrs(x)∗dx∥p Sp( �G) ≤ � [ξ]∈ �G dξ � ∫ G ∥η∗ rs∥L∞(G)∥σN(x, [ξ])∥Sp( �G)dx �p ≤ � [ξ]∈ �G dξ √ d p η � ∫ G ∥σN(x, [ξ])∥Sp( �G)dx �p , where, in the last line, we have used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='15) to estimate ∥η∗ rs∥p L∞(G) = ∥ηrs∥p L∞(G) = ∥ηrs∥p L∞(G) ≤ √ d p η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The Minkowski integral inequality for p ≥ 1 implies that ∥Op(�σN, sr([η], ·))∥Sp ≲ � � � [ξ]∈ �G dξ √ d p η � ∫ G ∥σN(x, [ξ])∥Spdx �p � � 1 p ≤ √ dη ∫ G � � � [ξ]∈ �G dξ∥σN(x, [ξ])∥p Sp � � 1 p dx = √ dη ∫ G ∥σN(x, ·)∥Sp( �G)dx = √ dη∥σN(x, ·)∥L1(G,Sp( �G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 16 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT All the analysis above allows the following estimate for the p-Schatten norm of A as follows: ∥Af∥Sp ≲ � [η]∈ �G dη � r,s=1 ⟨η⟩−N∥ηrs∥L∞(G)∥Op(�σN, sr([η], ·))∥Sp ≲ � [η]∈ �G dη � r,s=1 ⟨η⟩−N∥ηrs∥L∞(G) √ dη∥σN(x, ·)∥L1(G,Sp( �G)) = ∥σN(x, ·)∥L1(G,Sp( �G)) � [η]∈ �G ⟨η⟩−N√ dη dη � r,s=1 ∥ηrs∥L∞(G) ≤ ∥σN(x, ·)∥L1(G,Sp( �G)) � [η]∈ �G ⟨η⟩−N√ dηdη √ dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Thus, for N > n where n = dim(G) we have that ∥Af∥Sp ≲ ∥σN(x, ·)∥L1(G,Sp( �G)) � [η]∈ �G d2 η⟨η⟩−N < ∞, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='18) where we have use that (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='8 of [29]) � [η]∈ �G d2 η⟨η⟩−s < ∞ ⇐⇒ s > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ The following is a consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let G be a compact Lie group of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let A : C∞(G) → C∞(G) be a continuous linear operator and assume that its matrix-valued symbol satisfies the regularity condition ∥(1 + LG) N 2 σA(x, ·)∥Lp(G,Sp( �G)) = � ∫ G ∥(1 + LG) N 2 σA(x, ·)∥p Sp( �G)dx � 1 p < ∞ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='19) where N > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then A ∈ Sp(L2(G)) provided that 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Using the H¨older inequality and that the Haar measure of G is normalised we have that ∫ G ∥(1 + LG) N 2 σA(x, ·)∥Sp( �G)dx ≤ � ∫ G ∥(1 + LG) N 2 σA(x, ·)∥p Sp( �G)dx � 1 p < ∞, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='20) for all N > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, in view of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3 follows the membership of A in the Schatten class Sp(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Schatten properties of elliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The (ρ, δ)-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now we charac- terise the membership of elliptic pseudo-differential operators to the Schatten classes on L2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let m ∈ R, r > 0, and let 0 ≤ δ < ρ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Consider an elliptic pseudo- differential operator A ∈ Ψm ρ,δ(G × �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The following conditions are equivalent: SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 17 (1) m ∈ (−∞, 0) and A belongs to the Schatten class of order r > 0 : A ∈ Sr(L2(G));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2) The Bessel potential of order m belongs to the Schatten class of order r > 0 : Bm := (1 + LG) m 2 ∈ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3) The following summability condition holds � [ξ]∈ �G dξ ∫ G ∥σ|A| r 2 (x, [ξ])∥2 HSdx < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='21) (4) The following summability condition holds ∫ G � [ξ]∈ �G dξ∥σA(x, [ξ])∥r Srdx < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='22) (5) m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We start with the first equivalence (1) ⇐⇒ (2) : (1) =⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Assume that A ∈ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let Bs := (1 + LG) s 2 ∈ Sr(L2(G)) be the Bessel potential of order s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The global functional calculus for elliptic operators on compact Lie groups (see [42]) implies that (1 + |A| 1 |m|)±m ∈ Ψ±m ρ,δ (G × �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='23) In view of the composition properties of the pseudo-differential calculus we have that Bm(1 + |A| 1 |m|)−m ∈ Ψ0 ρ,δ(G × �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The Calder´on-Vaillancourt theorem (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2 in [42]) implies that Bm(1 + |A| 1 |m|)−m is a bounded operator on L2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Since Sr(L2(G)) is an ideal of operators on the algebra B(L2(G)), we have that Bm = Bm(1 + |A| 1 |m|)−m(1 + |A| 1 |m|)m, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='24) belongs to the Schatten class Sr(L2(G)) provided that (1 + |A| 1 |m|)m ∈ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, we are going to prove this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Since m ≤ 0, if sj(A) is the j-singular value of A, then the j-singular value sj((1+|A| 1 |m|)m) of (1+|A| 1 |m|)m satisfies the inequality sj((1 + |A| 1 |m|)m) ≤ sj(A), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='25) and then the membership of (1 + |A| 1 |m|)m in Sr(L2(G)) follows from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2) ⇐⇒ (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Observe that Bm ∈ Sr(L2(G)) ⇐⇒ Bmr/2 ∈ S2(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' On the other hand, ∥Bmr/2∥2 S2(L2(G)) = � [ξ]∈ �G d2 ξ⟨ξ⟩mr < ∞, if and only if mr < −n, (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='8 of [29]) or equivalently, if m < −n/r proving the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 18 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT (2) =⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let us give a similar proof to the one given for the reverse statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Because of (2) we have that m < −n/r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' On the other hand, by the pseudo-differential calculus, we have that AB−m ∈ Ψ0 ρ,δ(G × �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='26) Moreover, the Calder´on-Vaillancourt theorem (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2 in [42]) gives the boundedness of AB−m on L2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' If Bm ∈ Sr(L2(G)), then using that Sr(L2(G)) is an ideal in the algebra of the bounded operators on L2(G), we have that A = AB−mBm ∈ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (1) ⇐⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that A ∈ Sr(L2(G)) ⇐⇒ |A| r 2 ∈ S2(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='27) Let R|A| r 2 (x, y) and K|A| r 2 (x, y) be the right-convolution kernel and the Schwartz kernel of the operator |A| r 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We have the identity ∀x, y ∈ G, RA(x, xy−1) = KA(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' By the Plancherel theorem we have the equivalences |A| r 2 ∈ S2(L2(G)) ⇐⇒ K|A| r 2 (x, y) ∈ L2(G × G) ⇐⇒ ∫ G ∫ G |RA(x, y)|2dydx < ∞ ⇐⇒ ∫ G � [ξ]∈ �G dξ∥σ|A| r 2 (x, [ξ])∥2 HSdx < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3) ⇐⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In order to prove the equivalence of these summability conditions we are going to apply the global functional calculus (see [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We start with the identity σ|A| r 2 (x, [ξ]) = |σA(x, [ξ])| r 2 + rA(x, [ξ]), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='28) where the lower term rA ∈ S mr 2 −(ρ−δ)(G × �G) because of the asymptotic ex- pansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' First, note that ∥σ|A| r 2 (x, [ξ])∥HS ≤ ∥|σA(x, [ξ])| r 2∥HS + ∥rA(x, [ξ])∥HS = ∥σA(x, [ξ])∥ r 2 Sr + ∥rA(x, [ξ])∥HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Similarly, since |σA(x, [ξ])| r 2 = σ|A| r 2 (x, [ξ]) − rA(x, [ξ]), we have that ∥σA(x, [ξ])∥ r 2 Sr = ∥|σA(x, [ξ])| r 2∥HS ≤ ∥σ|A| r 2 (x, [ξ])∥HS + ∥rA(x, [ξ])∥HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To estimate the remainder ∥rA(x, [ξ])∥HS let us use that rA ∈ S mr 2 −(ρ−δ)(G × �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, ∥rA(x, [ξ])∥HS ≤ ∥rA(x, [ξ])⟨ξ⟩−mr/2+(ρ−δ)∥op∥⟨ξ⟩mr/2−(ρ−δ)Idξ∥HS ≲ d 1 2 ξ ⟨ξ⟩ mr 2 −(ρ−δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='29) To prove that (3) =⇒ (4), observe that the condition (3) is equivalent to the fact that A ∈ Sr(L2(G)), from which we deduce that Bm ∈ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 19 terms of the order m we have that m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To prove that (4) holds, note that ∫ G � [ξ]∈ �G dξ∥σA(x, [ξ])∥r Srdx ≤ ∫ G � [ξ]∈ �G dξ max{2∥σ|A| r 2 (x, [ξ])∥HS, ∥rA(x, [ξ])∥HS}2 = 4 max � � �∫ G � [ξ]∈ �G dξ∥σ|A| r 2 (x, [ξ])∥2 HSdx, ∫ G � [ξ]∈ �G dξ∥rA(x, [ξ])∥2 HS � � � ≲ 4 max � � �∫ G � [ξ]∈ �G dξ∥σ|A| r 2 (x, [ξ])∥2 HSdx, � [ξ]∈ �G d2 ξ⟨ξ⟩mr−2(ρ−δ) � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Since ρ > δ, we can compare ⟨ξ⟩mr−2(ρ−δ) ≤ ⟨ξ⟩mr, and then � [ξ]∈ �G d2 ξ⟨ξ⟩mr−2(ρ−δ) ≤ � [ξ]∈ �G d2 ξ⟨ξ⟩mr < ∞ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='30) because mr < −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, (3) together with the convergence of the series in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='30) implies (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, to finish the proof, let us assume (4) and from it let us deduce (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To do so we are going to use the inequality ∥σ|A| r 2 (x, [ξ])∥HS ≤ ∥σA(x, [ξ])∥ r 2 Sr + ∥rA(x, [ξ])∥HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='31) Then, we argue as follows: ∫ G � [ξ]∈ �G dξ∥σ|A| r 2 (x, [ξ])∥2 HSdx ≤ ∫ G � [ξ]∈ �G dξ(2 max{∥σA(x, [ξ])∥ r 2 Sr, ∥rA(x, [ξ])∥HS})2 = 4 max � � �∫ G � [ξ]∈ �G dξ∥σA(x, [ξ])∥r Srdx, ∫ G � [ξ]∈ �G dξ∥rA(x, [ξ])∥2 HS � � � ≲ 4 max � � �∫ G � [ξ]∈ �G dξ∥σ|A| r 2 (x, [ξ])∥2 HSdx, � [ξ]∈ �G d2 ξ⟨ξ⟩mr−2(ρ−δ) � � � from which we deduce the convergence of the series in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In view of the analysis above the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Schatten properties of non-elliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Classical symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We start this subsection, by showing that the order of an operator can be used as a sufficient condition for deducing its Schatten-von-Neumann properties, even if the operator is non-elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The next corollary is a consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let m < −n/r, r > 0, and let 0 ≤ δ < ρ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Consider a pseudo- differential operator A ∈ Ψm ρ,δ(G × �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, A ∈ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let us give an algebraic argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Since m < −n/r, then the Bessel potential satisfies Bm = (1 + LG) m 2 ∈ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Using again that the class Sr(L2(G)) is an ideal on the algebra of all bounded operators on L2(G), and that AB−m is bounded on 20 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT L2(G) (in view of the Calder´on-Vaillancourt theorem), we have that A = AB−mBm belongs to the Schatten class Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ Now, using the local Weyl formula (see [8, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='1]) for elliptic operators we have the following improvement of the equivalence (1) ⇐⇒ (5) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5 for trace class operators in the Kohn-Nirenberg algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Here, let us consider the norm || · ||g on the Lie algebra g induced by the Killing form B(X, Y ) = Tr[ad(X)ad(Y )], X, Y ∈ g, defined by ||X||g := � −B(X, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let A ∈ Ψm 1,0(G) be a classical pseudo-differential operator of order m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Assume that the average of the principal symbol σloc,A ∈ C∞(T ∗G) of A on the co-sphere is non-zero, that is Av[σloc,A] := ∫ T ∗S(G) σloc,A(x, η)dµL(x, η) ̸= 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='32) with µL denoting the Liouville measure on the spherical vector bundle T ∗S(G) = {(x, η) ∈ T ∗G : ||η||g = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, A ∈ S1(L2(G)) if and only if m < −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' From Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='6 it follows that for m < −n, we have A ∈ S1(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, let us prove the converse statement, that is if A ∈ S1(L2(G)) then m < −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let us prove this by assuming that m ≥ −n, and let us get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that if A ∈ S1(L2(G)) then for any orthonormal basis (φk)k of L2(G), the series � k(Aφk, φk)L2(G) is absolutely convergent and this sum is independent of the choice of the orthonormal basis (φk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, the trace of A is given by Tr(A) := � k (Aφk, φk)L2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='33) For our purposes, we will consider the orthonormal basis {d 1 2 ξ ξi,j : [ξ] ∈ �G : 1 ≤ i, j ≤ dξ}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='34) of L2(G) provided by the Peter-Weyl theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Consider the spectrum of the positive Laplacian Spect(LG) = {|ξ| := λ[ξ] : [ξ] ∈ �G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For any λ > 0, we will consider the partial sum Sλ := � |ξ|≤λ dξ � i,j=1 ( ˜A(d 1 2 ξ ξij), d 1 2 ξ ξij), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='35) and the discussion above allows us to use the identity Tr(A) = lim λ→∞ Sλ = lim λ→∞ � |ξ|≤λ dξ � i,j=1 ( ˜A(d 1 2 ξ ξij), d 1 2 ξ ξij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='36) Next, we will prove that if m ≥ −n, then the series in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='36) is not absolutely con- vergent and then, that A is not of trace class which would contradict our hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To simplify the proof, let us consider the real part Re(A) and the imaginary part Im(A) of A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We have the identities Re(A) = (A + A∗)/2, Re(A) = (A − A∗)/2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='37) SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 21 To simplify the notation we will write A0 = Re(A), and A1 = Im(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The following facts hold: Ai ∈ Ψm(G), i = 0, 1, in view of the pseudo-differential calculus properties for sums operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Ai, i = 0, 1, are self-adjoint operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that any entry (Ai(ξij), ξij) ∈ R is a real number because of the self-adjointness of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, let us use a specific property of the Peter-Weyl basis (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In [8] the following Local Weyl-formula was obtained for any pseudo-differential operator ˜A ∈ Ψ0(G) : � |ξ|≤λ dξ � i,j=1 ( ˜A(d 1 2 ξ ξij), d 1 2 ξ ξij) = (2π)−nCn, ˜ Aλn + O(λn−1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='38) for any λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Moreover, in [8] the left-hand side of this identity was simplified as follows � |ξ|≤λ dξ � i,j=1 dξ( ˜Aξij, ξij) = � |ξ|≤λ dξ ∫ G Tr[σ ˜ A(x, ξ)]dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='39) Let ˜A0 := Re( ˜A) = ( ˜A + ˜A∗)/2, ˜A1 := Im( ˜A) = ( ˜A − ˜A∗)/2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='40) Because these identities are valid for general operators of order m > 0, for k = 0, 1 we also have that � |ξ|≤λ dξ � i,j=1 ( ˜Ak(d 1 2 ξ ξij), d 1 2 ξ ξij) = (2π)−nCn, ˜ Akλn + O(λn−1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='41) for any λ > 0, where the constant Cn, ˜ Ak is given by Cn, ˜ Ak = Av[σloc, ˜ Ak] := ∫ T ∗S(G) σloc, ˜ Ak(x, η)dµL(x, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='42) As before we can write � |ξ|≤λ dξ � i,j=1 dξ( ˜Akξij, ξij) = � |ξ|≤λ dξ ∫ G Tr[σ ˜ Ak(x, ξ)]dx, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='43) where that the left hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='43) is real valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='41) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='43) we have the identity � |ξ|≤λ dξ ∫ G Tr[σ ˜ Ak(x, ξ)]dx = (2π)−nCn, ˜ Akλn + O(λn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='44) Note that � λ 2 <|ξ|≤λ dξ ∫ G Tr[σ ˜ Ak(x, ξ)]dx = � |ξ|≤λ dξ ∫ G Tr[σ ˜ Ak(x, ξ)]dx − � |ξ|≤ λ 2 dξ ∫ G Tr[σ ˜ Ak(x, ξ)]dx = (2π)−nCn,Ak(1 − 1 2n)λn + O(λn−1) = (2π)−n ˜Cn, ˜ Akλn + O(λn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 22 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT Let us apply the local Weyl formula above to the operator ˜A = AL − m 2 G ∈ Ψ0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The matrix-valued symbol of ˜A and ˜Ak, k = 0, 1, are given by σ ˜ A(x, [ξ]) = σA(x, [ξ])|ξ|−m, σ ˜ Ak(x, [ξ]) = σAk(x, [ξ])|ξ|−m, (x, [ξ]) ∈ G × �G, where |ξ|−m := � λ[ξ] −m , if ξ ̸= 1 �G, |1 �G|−m := 0, where 1 �G is the trivial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Because of the self-adjointness of ˜Ak, the left- hand side (and the right-hand side) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='44) is real-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that the linearity of the trace gives the identity Tr(A) = Tr(A0) + iTr(A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' So, let us estimates the traces Tr(Ak), k = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Tr(Ak) = lim λ→∞ dξ � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='j=1 dξ(Akξij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' ξij) = lim λ→∞ � |ξ|≤λ dξ ∫ G Tr[σAk(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' ξ)]dx = ∞ � k=0 � [ξ]∈ �G:2k−1<⟨ξ⟩≤2k dξ ∫ G Tr[σAk(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' ξ)]dx = ∞ � k=0 2km � [ξ]∈ �G:2k−1<⟨ξ⟩≤2k dξ2−km ∫ G Tr[σAk(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' ξ)]dx ≍ ∞ � k=0 2km � [ξ]∈ �G:2k−1<⟨ξ⟩≤2k dξ⟨ξ⟩−m ∫ G Tr[σAk(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' ξ)]dx = ∞ � k=0 2km � [ξ]∈ �G:2k−1<⟨ξ⟩≤2k dξ ∫ G Tr[σAk(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' ξ)⟨ξ⟩−m]dx ≍ ∞ � k=0 2km � [ξ]∈ �G:2k−1<⟨ξ⟩≤2k dξ ∫ G Tr[σAk(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' ξ)|ξ|−m]dx ≍ ∞ � k=0 2km � [ξ]∈ �G:2k−1<|ξ|≤2k dξ ∫ G Tr[σ ˜ Ak(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' ξ)]dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, using the local Weyl formula we get Tr(Ak) ≍ ∞ � k=0 2km � [ξ]∈ �G:2k−1<|ξ|≤2k dξ ∫ G Tr[σ ˜ Ak(x, ξ)]dx ≍ ∞ � k=0 2km � (2π)−n ˜Cn, ˜ Ak2kn + O(2k(n−1)� ≍ ∞ � k=0 � ˜Cn, ˜ Ak2k(n+m) + O(2k(n+m−1)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 23 Now, it is clear that if m ≥ −n,then the geometric series ∞ � k=0 � ˜Cn, ˜ Ak2k(n+m) + O(2k(n+m−1)� diverges for k = 0 or for k = 1, which also implies that the trace of T diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To see this observe that the constant in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='42) cannot be equal to zero simultaneously for k = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Indeed, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='32) we have that Av[σloc,A] = ˜Cn, ˜ A0 + i ˜Cn, ˜ A1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='45) We illustrate this geometric fact in Figure 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that we have used that on Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The average condition on the co-sphere in the case of a real-valued principal symbol σA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In this case the positive part of the symbol dominates the region where the symbol is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The green curve represents the level curve at height zero (that occurs along the zero section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' the co-sphere T ∗S(G) one has the identity ∀(x, η) ∈ T ∗S(G), σ ˜ A(x, η) = σA(x, η)∥η∥−m = σA(x, η), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='46) because the fact that (x, η) ∈ T ∗S(G) implies that x ∈ M and that the norm ∥η∥ of η on the corresponding fiber is equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Thus we have proved that T is not in the ideal S1(L2(G)) which contradicts our initial hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ As a consequence of the previous corollary we have the following characterisation of the trace class pseudo-differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let A ∈ Ψm 1,0(G) be a classical pseudo-differential operator of order m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, A ∈ S1(L2(G)) if and only if m < −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' T*S(G) Oloc,A(C, n) > 0 Oloc,A(C, n) = 0 loc,A(, n) < 024 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that if the average of the principal symbol σloc,A ∈ C∞(T ∗G) of A on the co-sphere is non-zero, that is Av[σloc,A] := ∫ T ∗S(G) σloc,A(x, η)dµL(x, η) ̸= 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='47) then the statement follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' On the other hand, assume that A ∈ S1(L2(G)) and that Av[σloc,A] := ∫ T ∗S(G) σloc,A(x, η)dµL(x, η) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='48) Define the operator ˜A = A + (LG)−n−ε, ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that ˜A ∈ S1(L2(G)) since ∥ ˜A∥S1 ≤ ∥A∥S1 + ∥˜(LG)−n−ε∥S1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Moreover by definition of ˜A we have Av[σloc, ˜ A] := ∫ T ∗S(G) ||η||−n−ε g dµL(x, η) = Vol(T ∗S(G)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='49) Note that if m ≥ −n, then the order of ˜A would be larger than −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' However, the condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='49) together with the fact that ˜A ∈ S1(L2(G)) would imply that m < −n in view of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Thus, one has that m < −n as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ Now, we will use the ideal properties of the Schatten classes to derive the extension of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7 to the case r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let A ∈ Ψm 1,0(G) be a classical pseudo-differential operator of order m ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Assume that the average of the principal symbol σloc,A ∈ C∞(T ∗G) of A on the co-sphere is non-zero, that is Av[σloc,A] := ∫ T ∗S(G) σloc,A(x, η)dµL(x, η) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='50) For any 1 < r < ∞, if A ∈ Sr(L2(G)) then m ≤ −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Moreover, for r = ∞, A ∈ S∞(L2(G)) = B(L2(G)) if and only if m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let us consider first, the case where 1 < r < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Consider the operator Ls/2 G where s satisfies s := −n q − ε < −n q , and where q := r/(r − 1) is the conjugate exponent of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, q is given by the identity 1 = 1 q + 1 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that since r > 1, one has that q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7 implies that Ls/2 G ∈ Sq(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Since q and r and conjugate exponents, we have that ˜A := ALs/2 G ∈ S1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that the principal symbol σloc, ˜ A satisfies the identity ∀(x, η) ∈ T ∗G \\ {0}, σloc, ˜ A(x, η) = σloc,A(x, η)∥η∥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='51) In particular, on the co-sphere we have ∀(x, η) ∈ T ∗S, σloc, ˜ A(x, η) = σloc,A(x, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='52) SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 25 Then, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='50) allows to deduce the non-vanishing property Av[σloc,A] := ∫ T ∗S(G) σloc, ˜ A(x, η)dµL(x, η) ̸= 0, allowing the use of the Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7 in order to conclude that the order m + s of ˜A ∈ S1 satisfies the inequality m + s < −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, we have that m < −s − n = n q + ε − n = n �1 q − 1 � + ε = −n r + ε , where the last hold tru for any ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Taking ε → 0+ we have that m ≤ −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, let us assume that r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For m ≤ 0, it follows from the Calder´on- Vaillancourt theorem that A ∈ S∞(L2(G)) = B(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, let us assume that A ∈ S∞(L2(G)) = B(L2(G)) and let us give an argument proving that its order satisfies m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proceeding by contradiction suppose that m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that the conjugate exponent to r = ∞ is q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let ε ∈ (0, m) and let s = −n − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Consider the operator Ls/2 G ∈ S1(L2(G)) where s < −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then the operator ˜A := ALs/2 G ∈ S∞(L2(G)) ◦ S1(L2(G)) ⊆ S1(L2(G)), is of trace class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Since the principal symbol is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='51) the co-sphere condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='50) is satisfied and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7 implies that the order of ˜A satisfies the inequality m + s < −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We have that m + s = m − n − ε = (m − ε) − n ≥ −n, which contradicts the analysis above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' So, we must have that m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='9 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Assume that r ∈ (1, ∞) ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let A ∈ Ψm 1,0(G) be a classical pseudo- differential operator of order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Assume that the average of the principal symbol σloc,A ∈ C∞(T ∗G) of A on the co-sphere is non-zero, that is Av[σloc,A] := ∫ T ∗S(G) σloc,A(x, η)dµL(x, η) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='53) Then, A ∈ Sr(L2(G)) if and only if m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' From Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='6 it follows that for m < −n/r, 1 < r < ∞, we have A ∈ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, let us prove the converse statement, that is if A ∈ Sr(L2(G)), then m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To do this, let us show that the borderline m = −n/r in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='9 is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To do this, observe that if A ∈ Sr, then Ar = A ◦ A ◦ · · · A ∈ Sr(L2(G)) ◦ Sr(L2(G)) ◦ · · · Sr(L2(G)) ⊆ S1(L2(G)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='54) where the composition is taken r-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' However, the order of the trace class operator Ar is −n and this contradicts the conclusion in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' So, necessarily m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let A ∈ Ψm 1,0(G) be a classical pseudo-differential operator of order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (A) If r ∈ [1, ∞) ∩ Z, then A ∈ Sr(L2(G)) if and only if m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For r = ∞, A ∈ B(L2(G)) if and only if m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT (B) If r ∈ (1, ∞) \\ Z, and A ∈ Sr(L2(G)), then m ≤ −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Moreover, if A is elliptic then one has the strict inequality m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let us prove the statement in (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For r = 1 this statement has been proved in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' At this stage of the manuscript we only have to prove that if A ∈ Sr(L2(G)) then m ≤ −n/r, where n/r := 0 for r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, let us consider the case where 1 < r ≤ ∞, with r ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that if the average of the principal symbol σloc,A ∈ C∞(T ∗G) of A on the co-sphere is non-zero, that is Av[σloc,A] := ∫ T ∗S(G) σloc,A(x, η)dµL(x, η) ̸= 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='55) then the statement follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' On the other hand, assume that A ∈ S1(L2(G)) and that Av[σloc,A] := ∫ T ∗S(G) σloc,A(x, η)dµL(x, η) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='56) Define the operator ˜A = A + (LG)−n/r−ε, ε > 0, where n/r := 0 whenever r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that ˜A ∈ Sr(L2(G)) since ∥ ˜A∥Sr ≤ ∥ ˜A∥Sr +∥˜(LG)−n/r−ε∥Sr < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' On the other hand note that Av[σloc, ˜ A] := ∫ T ∗S(G) ||η||−n/r−ε g dµL(x, η) = Vol(T ∗S(G)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='57) Now that if m ≥ −n/r, then the order of ˜A must be larger than −n/r (or strictly larger than zero for r = ∞) but the condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='57) together with the fact that ˜A ∈ Sr(L2(G)) implies that m < −n/r in view of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='7, which is a con- tradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Thus, one has that m < −n/r for 1 < r < ∞ and m ≥ 0 for r = ∞, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, let us prove the statement in (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' It can be done following the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' However, let us give another argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For this, consider r ∈ (1, ∞) \\ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, the integer part [r] of r satisfies the strict inequalities [r] < r < [r] + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='58) Let ε > 0 and define the parameter s = n � 1 [r] − 1 r � + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='59) Since A ∈ Sr and for any ε′ > 0, (1+LG)− s 2 ∈ S n s +ε′, we will prove that there exists ε′ 0 > 0 such that As := A(1 + LG)− s 2 ∈ S[r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To do this, we have to guarantee, that there exists ε′ 0 > 0 such that, the H¨older property 1 [r] = 1 r + 1 n s + ε′ 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='60) holds because its validity would imply that As ∈ Sr ◦ S n s +ε′ 0 ⊆ S[r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To prove that the equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='60) admits a solution, define the continuous function g(ε′) := 1 n s + ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='61) SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 27 Note that g(0) = s n > 1 [r] − 1 r > 0 = g(∞) := lim ε′→∞ g(ε′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='62) By Bolzano’s theorem, there exists ε′ 0 ∈ (0, ∞) such that g(ε′ 0) = 1 [r]− 1 r as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We have now proved the existence of ε′ 0 in in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='60) and its implication As ∈ S[r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Hence by the result in Part (A), the order m−s of As, satisfies the inequality m−s < −n/[r], or equivalently m < s − (n/[r]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='59), we have proved that ∀ε > 0, m < s = n � 1 [r] − 1 r � + ε − n [r] = −n r + ε, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='63) which certainly implies that m ≤ −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that if A is elliptic, this inequality can be improved to an strict inequality, namely we would have m < −n/r, as proved in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ We end this subsection with the following atypical construction in the case of the classes Ψm 0,0(Tn) on the n-dimensional torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For any κ > 0, there exists a non-elliptic pseudo-differential oper- ator A in the exotic class Ψ−κ 0,0 (Tn) \\ Ψ−κ−ε 0,0 (Tn), for all ε > 0, that belongs to all the Schatten ideals Sr(L2(Tn)), where r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let κ > 0, and let (ej)n j=1 be the canonical orthonormal basis of Rn, and consider the set D = {2ke1 = (2k, 0, · · · , 0) : k ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='64) Define the symbol a(ξ) = ⟨ξ⟩−κ, ξ ∈ D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' a(ξ) = 0, ξ ∈ Zn \\ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='65) Observe that for any α ∈ Nn 0, we have |∆αa(ξ)| ≤ � β≤α �α β � |a(ξ + β)| ≤ � β≤α �α β � ⟨ξ + β⟩−κ ≲α (1 + |ξ|)−κ, ξ ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To see that the right-hand side of this inequality is the best possible, we consider the case where α = e1 and when ξk = 2ke1 ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Indeed, note that |∆e1a(ξk)| = |a(ξk + e1) − a(ξk)| = |⟨2ke1⟩−κ| = ⟨ξk⟩−κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' This argument proves that the estimate |∆e1a(ξ)| ≤ ⟨ξ⟩−κ, is sharp and even that it is an equality when ξk ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To prove that the operator A = Op(a) associated to the symbol a belongs to the Schatten class Sr(L2(Tn)) note that the sequence of singular values of A is determined by the values of its symbol, that is sξ(A) = |a(ξ)|, ξ ∈ Zn, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='66) are all the singular values of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Consequently � ξ∈Zn sξ(A)r = � ξk∈D sξ(A)r ≍ ∞ � k=1 2−kκr < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 28 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT Now, to prove that A is not elliptic and that it is of order m = −κ, observe that the best estimate from below that a satisfies is the following |a(ξ)| ≥ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='67) with equality when ξ ∈ Zn \\ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The proof of this atypical case is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Classical operators in Schatten classes and their principal symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We consider the subclass Ψm 0 (G) in Ψm/2 cl (G) of operators with homogeneous symbols of order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' First we have the following result where we show that only the principal homogeneous component of the principal symbols contains the information about the membership of a classical operator to any Schatten class S (L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let r ∈ (0, ∞] and m ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then the following conditions are equivalent: (1) Ψm cl (G) ⊆ Sr(L2(G));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (2) Ψm 0 (G) ⊆ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Evidently, (1) implies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Suppose that (2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' We shall prove that (1) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' If N is large enough, then Ψm−N ⊆ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Observing that the operator B = √ L −1 G is continuous on L2(G), we get that Ψm 0 Bk ∈ Sr(L2(G)) for any k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Since, the principal symbol of Bk, which is given by σBk(x, η) = ∥η∥−k g , ∀x ∈ G, ∀η ∈ g \\ {0}, is homogeneous of order −k, we have that Ψm 0 Bk = Ψm−k 0 mod Ψ−∞(G × �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that Op(Ψ−∞(G × �G)) ⊆ Sr(L2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, we get Ψm cl = N−1 � k=0 Ψm 0 Bk + Ψm−N cl mod Ψ−∞(G × �G) ⊆ Sr(L2(G)), in view of our hypothesis Ψm 0 ⊆ Sr(L2(G)), which shows that (1) holds when (2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' This gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ In view of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='11 we have the following characterisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let A ∈ Ψm 0 (G) be of order m ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' If r ∈ [1, ∞) ∩ Z, then A ∈ Sr(L2(G)) if and only if m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' For r = ∞, A ∈ B(L2(G)) if and only if m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' If r ∈ (1, ∞) \\ Z, and A ∈ Sr(L2(G)), then m ≤ −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Moreover, if A is elliptic then one has the strict inequality m < −n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Order of classical operators vs order of global symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' As an application of the methods developed in this manuscript we prove that the order of a symbol characterises the order of the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let µ, t ∈ R and let A ∈ Ψµ 1,0(G) be a classical pseudo-differential operator such that its global symbols satisfies ∀(x, [ξ]) ∈ G × �G, ∥σA(x, [ξ])∥op ≤ C⟨ξ⟩t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='68) SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 29 Then, A ∈ Ψt 1,0(G), provided that the average of its principal symbol σloc,A ∈ C∞(T ∗G) on the co-sphere is non-zero: Av[σloc,A] := ∫ T ∗S(G) σloc,A(x, η)dµL(x, η) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='69) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let us consider two cases: t = 0 and t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Case 1: t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Assume that A ∈ Ψµ 1,0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' It is clear that if µ ≤ 0 we have nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' So, the relevant case is when µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' First, let us prove the statement in the case where µ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To do it, we will proceed using induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' So, let us start by proving that the result is true if µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, assuming that A ∈ Ψ1 1,0(G) and with the additional condition ∀(x, [ξ]) ∈ G × �G, ∥σA(x, [ξ])∥op ≤ C, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='70) we have to deduce that A ∈ Ψ0 1,0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' To prove this, we will use Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='2 of [40] to deduce that A is bounded on L2(G), and then from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='9 we use the L2(G)-boundedness of A and the average condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='69) to deduce that its order µ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' This proves the case µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, let us state our induction hypothesis: – Let us assume that µ0 ∈ N is such that any classical pseudo-differential operator ˜A ∈ Ψµ0 1,0(G) with a bounded operator norm of its matrix-valued symbol ∀(x, [ξ]) ∈ G × �G, ∥σ ˜ A(x, [ξ])∥op ≤ C, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='71) and with its principal symbol satisfying the average condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='69) be- longs to the class Ψ0 1,0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, let A ∈ Ψµ0+1 1,0 (G) be a classical pseudo-differential operator whose sym- bol satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='69) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Define ˜A = A(1+LG)− 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that ˜A ∈ Ψr0 1,0(G) and that its symbol satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='71) since ∥σ ˜ A(x, [ξ])∥op = ∥σA(x, [ξ])⟨ξ⟩−1∥op ≤ ∥σA(x, [ξ])∥op ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Our induction hypothesis implies that ˜A belongs to the class Ψ0 1,0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' But then, the Calder´on-Vaillancourt theorem implies that ˜A is bounded on L2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Since on the co-sphere T ∗S(G) the principal symbols of ˜A and of A agree, ˜A satisfies the average condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='9 implies that µ0 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Since A(1 + LG)− 1 2 ∈ Ψµ0 1,0(G) ⊆ Ψ0 1,0(G), then A ∈ Ψµ0+1 1,0 (G) ⊆ Ψ1 1,0(G), and together with the hypothesis (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='70) we can use the base case µ = 1 in the inductive process to establish that A ∈ Ψ0 1,0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' So, by the mathematical induction we have proved the statement in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='15 in the case where µ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Now, in the general case if [µ] is the integer part of every µ > 0, and if A ∈ Ψµ 1,0(G) is a classical pseudo-differential operator satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='70), we have that A ∈ Ψ[µ]+1 1,0 (G) and then the better conclusion A ∈ Ψ0(G) follows 30 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CARDONA, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' CHATZAKOU, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' RUZHANSKY, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' TOFT from the statement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='15 for integer orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Thus, we have proved Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='15 in the case t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Assume that A ∈ Ψµ 1,0(G) has a matrix-valued symbol satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='72) and the average condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, ˜A = A(1 + LG)− t 2 ∈ Ψr−t 1,0 (G) satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='69) and its matrix-valued symbol satisfies the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' From the first part of the poof, we deduce that ˜A = A(1 + LG)− t 2 ∈ Ψ0 1,0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Then, the pseudo-differential calculus implies that A ∈ Ψt 1,0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='15 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ Now, we will remove the geometric average condition in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='15 to prove a general statement on compact Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Let µ, t ∈ R and let A ∈ Ψµ 1,0(G) be a classical pseudo-differential operator such that its global symbol satisfies ∀(x, [ξ]) ∈ G × �G, ∥σA(x, [ξ])∥op ≤ C⟨ξ⟩t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='72) Then A ∈ Ψt 1,0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that if the average of the principal symbol σloc,A ∈ C∞(T ∗G) of A on the co-sphere is non-zero: Av[σloc,A] := ∫ T ∗S(G) σloc,A(x, η)dµL(x, η) ̸= 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='73) the statement follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' On the other hand, if the principal symbol of A has average zero on T ∗S(G), that is, Av[σloc,A] := ∫ T ∗S(G) σloc,A(x, η)dµL(x, η) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='74) we define the operator ˜A := A + (LG) t 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Is it clear that the principal symbol of ˜A is given by σloc, ˜ A := σloc,A + ∥ξ∥t g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Note that the matrix-valued symbol of ˜A satisfies also σ ˜ A(x, [ξ]) = σA(x, [ξ]) + |ξ|tIdξ, (x, [ξ]) ∈ G × �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='75) From our hypothesis we have the estimate ∥σ ˜ A(x, [ξ])∥op ≤ ˜C⟨ξ⟩t, (x, [ξ]) ∈ G × �G, and the average condition Av[σloc, ˜ A] := ∫ T ∗S(G) σloc,A(x, η)dµL(x, η) + ∫ T ∗S(G) ||η||t gdµL(x, η) = Vol(T ∗G) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='76) From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='15 follows that ˜A ∈ Ψt 1,0(G) and using the property (LG) t 2 ∈ Ψt 1,0(G), one obtains that A = ˜A − (LG) t 2 ∈ Ψt 1,0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' □ SCHATTEN PROPERTIES FOR H¨ORMANDER CLASSES ON COMPACT LIE GROUPS 31 References 1.' metadata={'source': 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operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 1967 Singular Integrals (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=', Chicago, Ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=', 1966) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' 288–307, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=', Providence, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content=' Duv´an Cardona: Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University, Belgium E-mail address duvan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='cardonasanchez@ugent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='be Marianna Chatzakou: Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University, Belgium E-mail address Marianna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='Chatzakou@UGent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='be Michael Ruzhansky: Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University, Belgium and School of Mathematical Sciences Queen Mary University of London United Kingdom E-mail address michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='ruzhansky@ugent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='be Joachim Toft: Department of Mathematics Linnæus University V¨axj¨o-Sweden E-mail address joachim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='toft@lnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} +page_content='se' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E2T4oBgHgl3EQfrAg0/content/2301.04044v1.pdf'} diff --git a/NtFIT4oBgHgl3EQfdSsx/content/tmp_files/2301.11269v1.pdf.txt b/NtFIT4oBgHgl3EQfdSsx/content/tmp_files/2301.11269v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f74d9e8f5f624055f8458569eca7eb4447677d3 --- /dev/null +++ b/NtFIT4oBgHgl3EQfdSsx/content/tmp_files/2301.11269v1.pdf.txt @@ -0,0 +1,1086 @@ +arXiv:2301.11269v1 [math.OC] 26 Jan 2023 +On Low-Rank Convex-Convex Quadratic +Fractional Programming +Ilya Krishtal and Brendan Miller +Abstract We present an efficient algorithm for solving fractional programming +problems whose objective functions are the ratio of a low-rank quadratic to a positive +definite quadratic with convex constraints. The proposed algorithm for these convex- +convex problems is based on the Shen-Yu Quadratic Transform [13] which finds +stationary points of concave-convex sum-of-ratios problems. We further use elements +of the algorithm proposed in [13] and the classic Dinkelbach approach to ensure +convergence. We show that our algorithm performs better than previous algorithms +for low-rank problems. +1 Introduction +Methods of fractional programming encompass a large range of techniques to solve +problems of the form +max +퐹(푥) = +푀 +� +푚=1 +푁푚(푥) +퐷푚(푥) +s.t. +푥 ∈ 푋 ⊂ R푛 +(1) +where 푁푚(푥), 퐷푚(푥) : R푛 → R are continuous functions, and 푋 is a closed, +convex set. A convention of the field, which we adopt throughout this paper, is that +푁푚(푥) ≥ 0 and 퐷푚(푥) > 0 for all 푥 ∈ 푋. The problem is called single-ratio if +푀 = 1 and a sum-of-ratios fractional programming problem if 푀 > 1. Fractional +programming problems arise in many different applications, such as finance and +portfolio analysis, government contracting, and engineering (see [1, 2, 10, 11]). We +were led to a single-ratio problem in the process of studying the problem of optimal +sensor placement for dynamical sampling on graphs [8]. In particular, the relative +Ilya Krishtal · Brendan Miller +Northern Illinois University, Dekalb, IL 60435 USA +e-mail: ikrishtal@niu.edu (IK), bmiller14@niu.edu (BM) +1 + +2 +Ilya Krishtal and Brendan Miller +error of reconstructing a signal on a graph from spatio-temporal samples in the +presence of noise can be bounded above by a ratio of two quadratics depending on +the eigenvalues of a certain frame operator. +Fractional programs have a long history. Below we outline a few milestones in +the development of the theory. +In 1962, Charnes and Cooper showed in [3] that linear fractional programming +problems, which are single-ratio fractional programming problems whose numerator, +denominator, and constraints are linear, can be solved by simplex method. +In 1967, Dinkelbach [4] showed that any concave-convex single-ratio fractional +programming problem can be solved efficiently by consecutively solving several +concave maximization problems. The approach used by Dinkelbach was so useful +that it has become standard and even seen many applications to fractional programs +which are not concave-convex. In the latter case, however, the method involves +solving a succession of non-concave maximization problems and becomes rather +expensive. +In 1997, Lo and MacKinlay [11] proposed the problem of maximizing the ratio +of two convex quadratic functions in the context of portfolio analysis. The following +decade saw two papers published [7, 14] which presented algorithms for solving +this problem exactly. Both follow the standard Dinkelbach approach, with different +methods of handling the most expensive aspect of the problem: the need to solve +multiple non-convex quadratic programming problems. To do so, the former paper +by Gotoh and Konnoh implemented a branch-and-bound technique while the latter +by Yamamoto and Konnoh iteratively approximated the quadratic by a piece-wise +linear function, which is maximized by standard mixed integer linear programming +techniques. +In recent years, there has also been much research on the sum-of-ratios problem. +For example, [1,2,5] all produce branch-and-bound type algorithms for solving the +concave-convex sum-of-ratios problem. Similarly, [10, 12] propose efficient algo- +rithms for solving quadratically constrained quadratic sum-of-ratios type problems. +Of particular interest to us is the Shen-Yu Quadratic Transform introduced in [13] +which, like the Dinkelbach method, iteratively solves a concave programming prob- +lem to converge to a stationary point of the concave-convex sum-of-ratios problem. +In this paper, we propose an algorithm for solving the single-ratio convex-convex +quadratic programming problem which can be effectively utilized when the numer- +ator has low rank. We divide the feasible region into several subregions in which +the Shen-Yu Quadratic Transform method can be applied successively until global +convergence. Since the methods of Shen and Yu only guarantee convergence to a +stationary point, we use the mixed integer linear programming techniques employed +in [14] to check if a given stationary point is a global optimum. Although, as we +will see empirically, this rather expensive procedure can often be omitted altogether +(with a minimal chance of error). This yields a very efficient algorithm which, with +high probability, converges to the global solution of a convex-convex quadratic pro- +gramming problem by successively solving concave maximization problems. The +only known algorithms for solving such problems involve successively maximizing +a nonconvex quadratic programming problems. + +On Low-Rank Convex-Convex Quadratic Fractional Programming +3 +The remainder of this paper is organized as follows. In Section 2, we state the +problem we focus on and outline the classic Dinkelbach approach for ratio maxi- +mization. We then convert our problem into a sum-of-ratios problem and recall a +recent Shen-Yu scheme for solving such problems. Section 3 is the centerpiece of the +paper. In it, we present a natural way of subdividing the feasible region of our prob- +lem into a number of subregions thereby replacing a convex-convex sum-of-ratios +problem with a finite number of much simpler concave-convex sum-of-ratios prob- +lems. This results in a region checking algorithm (Algorithm 3) which encompasses +our approach to solving single-ratio quadratic convex-convex problems. Numeri- +cal experiments described in Section 4 illustrate the effectiveness of our approach +in comparison with other algorithms. Finally, concluding remarks are presented in +Section 5. +2 Basic Analysis +We study the problem of the form +max +퐹(푥) = 푥푇 푄푥 +푥푇 푃푥 +s.t. +퐴푥 ≤ 푏 +(2) +where 푄 is an 푛×푛 positive semidefinite matrix, 푃 is an 푛×푛 positive definite matrix, +퐴 ∈ R푇 ×푛, 푏 ∈ R푇 . We will denote the feasible region of this problem by 푋 ⊂ R푛. +Such optimization problems are nonconcave in general, and thus require expensive +algorithms to solve. We recall the standard Dinkelbach method which utilizes the +function +휋(휆) := max +푥∈푋 푥푇 (푄 − 휆푃)푥, 휆 > 0. +It is convenient to introduce the following notation: +푥(휆) = arg max +푥∈푋 +푥푇 (푄 − 휆푃)푥, 휆 > 0. +Theorem 1 The function 휋(휆) is convex and strictly decreasing in 휆. Furthermore, +휋(휆) = 0 if and only if 푥(휆) maximizes (2) in 푋. +The algorithms of [7,14] are root-finding algorithms which use a scheme devel- +oped by Ibaraki [9] to search for the root of 휋. These algorithms become expensive +because computing 휋(휆) is a nonconvex quadratic programming problem, whose +difficulty is larger for smaller values of 휆. We will utilize this Theorem of Dinkel- +bach in our algorithm, but only to check if a local maximum of (2) is indeed a global +maximum. We summarize the Ibaraki scheme here as Algorithm 1 for reference, but +omit the explanation of convergence. + +4 +Ilya Krishtal and Brendan Miller +input : Matrices 푄 and 푃, linear inequality constraints 퐴푥 ≤ 푏, and a tolerance 휀. +1 Find 휆푢 with 휋 (휆푢) < 0 and 휆푙 with 휋 (휆푙) > 0; +2 repeat +3 +Compute 휆 as follows: +휆 = +� +− 휋(휆푢) +Δ휋 ++ 휆푢 +if 푥(휆푢)푇 푃푥(휆푢) + Δ휋 ≠ 0 +휋(휆푢) +푥(휆푢)푇 푃푥(휆푢) + 휆푢 +otherwise, +where Δ휋 = ( 휋 (휆푢) − 휋 (휆푙))/(휆푢 − 휆푙); +4 +Compute 휋 (휆); +5 +Update 휆푙 = 휆 if 휋 (휆) > 0, or 휆푢 = 휆 if 휋 (휆) < 0; +6 until | 휋 (휆) | < 휀; +Algorithm 1: Interpolated Binary Search (Ibaraki scheme) +The algorithms presented in [7,14] both utilize Algorithm 1, but employ different +methods of computing 휋(휆). We will use contemporary software to solve these +non-convex quadratic programming problems to ensure the most accurate solutions. +Suppose now that 푄 can be written as +푄 = +푀 +� +푚=1 +푞푚푞푇 +푚, +which is a sum of rank-onematrices. Note that푄 always admits such a decomposition +as we can take 푀 = 푟푎푛푘(푄) and 푞푚 = √휆푚푣푚 where 푣푚 is the eigenvector of 푄 +associated to the nonzero eigenvalue휆푚. Then we may rewrite the objective function +in the following way: +퐹(푥) = +�푀 +푚=1 푥푇 (푞푚푞푇 +푚)푥 +푥푇 푃푥 += +푀 +� +푚=1 +⟨푞푚, 푥⟩2 +푥푇 푃푥 . +(3) +This reformulation of the objective function converts a single-ratio fractional pro- +gramming problem (2) into a convex-convex sum-of-ratios fractional programming +problem which we will refer to as (3). +Next, we summarize a method for suboptimally solving sum-of-ratios fractional +programming problems (1) that was introduced in [13]. +Definition 1 (See [13]) Given the sum-of-ratios fractional programming problem +(1), the Shen-Yu Quadratic Transform of this problem is defined to be +푔(푥, 푦) = +푀 +� +푚=1 +� +2푦푚 +� +푁푚(푥) − 푦2 +푚퐷푚(푥) +� +. +(4) +Theorem 2 ( [13]) + +On Low-Rank Convex-Convex Quadratic Fractional Programming +5 +The sum-of-ratios problem (1) and its quadratic transform (4) maximization +problem are equivalent. More precisely, the maximum of (4) occurs at (푥∗, 푦∗) +where 푥∗ maximizes (1) and 푦∗ = (푦∗ +푚)푀 +푚=1 satisfies +푦∗ +푚 = +� +푁푚(푥∗) +퐷푚(푥∗) . +The following Lemma is useful for deriving the Shen-Yu scheme. +Lemma 1 Let 푔(푥, 푦) be as in (4). If 푥0 ∈ 푋, then 푦0 := arg max푦∈R푀 푔(푥0, 푦) can +be found analytically and is given by 푦0 = (푦0 +푚)푀 +푚=1, where +푦0 +푚 = +� +푁푚(푥0) +퐷푚(푥0) . +Proof It suffices to maximize each term in the sum in (4) individually. Clearly, the +vertex of the quadratic 푓 (푦푚) = 2푦푚 +� +푁푚(푥0) − 푦2 +푚퐷(푥0) occurs at the point +푦0 +푚 = +� +푁푚(푥0) +퐷푚(푥0) +as desired. +□ +We may now state and prove a theorem from which, when taken together with +Theorem 2, an algorithm for suboptimally solving (1) is naturally derived. +Theorem 3 (Shen, Yu [13]) Consider the sum-of-ratios problem (1) and suppose +that 푥0 ∈ 푋. Let 푦0 = (푦0 +푚)푀 +푚=1 with 푦0 +푚 = +� +푁푚(푥0)/퐷푚(푥0). If +푥∗ = arg max +푥∈푋 +푔(푥, 푦0) = arg max +푥∈푋 +푀 +� +푚=1 +� +2푦0 +푚 +� +푁푚(푥) − (푦0 +푚)2퐷푚(푥) +� +, +then 퐹(푥∗) ≥ 퐹(푥0). +Proof Let 푥0, 푦0, and 푥∗ be as above and set +푦∗ +푚 = +� +푁푚(푥∗) +퐷푚(푥∗) +and 푦∗ = (푦∗ +푚)푀 +푚=1. Then we have the following string of inequalities +퐹(푥0) = 푔(푥0, 푦0) +≤ 푔(푥∗, 푦0) +≤ 푔(푥∗, 푦∗) += 퐹(푥∗) + +6 +Ilya Krishtal and Brendan Miller +where the third line follows from Lemma 1, and the last by a direct computation. +The proof is complete. +□ +The above Theorem guarantees that replacing 푥0 by 푥∗ and 푦0 by 푦∗ improves the +value of the objective function with each iteration, and thus this method (with the +scheme written explicitly in Algorithm 2) converges. It is clear that the algorithm +converges to a local maximum, say (푥∗, 푦∗), of the Quadratic Transform 푔. It follows +easily that the value 푥∗ is indeed a local maximum of the objective 퐹 as well. +input : Functions 푁푚 and 퐷푚, and a compact, convex set 푋. +1 Find an initial 푥0 ∈ 푋; +2 repeat +3 +set 푦0 +푚 = +� +푁푚(푥0)/퐷푚 (푥0); +4 +solve 푥∗ = arg max푥∈푋 +�푀 +푚=1 +� +2푦0 +푚 +� +푁푚(푥) − (푦0 +푚)2퐷푚(푥) +� +; +5 +update 푥0 = 푥∗; +6 until convergence; +Algorithm 2: Shen-Yu Iterative Algorithm +Corollary 1 Algorithm 2 converges to a stationary point of the sum-of-ratios frac- +tional programming problem (1). +Although Algorithm 2 is guaranteed to converge to a stationary point, its use- +fulness is limited to the difficulty of maximizing the Quadratic Transform over the +variable 푥. In the case when each 푁푚 is concave and each 퐷푚 is convex, this can be +done by any method of concave programming. When the objective has the form (3), +the Quadratic Transform becomes +푔(푥, 푦) = +푀 +� +푚=1 +2푦푚|⟨푞푚, 푥⟩| − 푦2 +푚푥푇 푃푥 +(5) +which is not, in general, concave in 푥. However, in the event that the absolute values +around each linear term in the sum can be dropped (i.e. each ⟨푞푚, 푥⟩ is either +nonnegative or nonpositve valued on 푋), then we may apply Algorithm 2 effectively. +3 Region checking algorithm +Henceforth we will assume the objective function 퐹 is as in (2) and, therefore, it +can be rewritten in the sum-of-ratios form (3). The first observation to make is +that combining Algorithm 2 with the Dinkelbach method yields an algorithm which +converges to the global maximum of (3). Indeed, if 푥∗ is the local optimum found +by Algorithm 2, we may set 휆∗ = 퐹(푥∗) and compute both 휋(휆∗) and 푥(휆∗). If + +On Low-Rank Convex-Convex Quadratic Fractional Programming +7 +|휋(휆∗)| > 휀 where 휀 is some tolerance, we again run Algorithm 2 with 푥(휆∗) as +the initial feasible point and repeat. This method is guaranteed to converge to the +globally optimal solution since +푥(휆∗)푇 (푄 − 휆∗푃)푥(휆∗) > 푥∗푇 (푄 − 휆∗푃)푥∗ +implies that +퐹(푥(휆∗)) > 퐹(푥∗) ≥ 퐹(푥0) +where 푥0 is the initial feasible point used in Algorithm 2. +Finding the value of 휋(휆∗) and the vector 푥(휆∗) can be achieved by the methods +introduced in [7] and refined in [14], but maximizing the Quadratic Transform +as in Algorithm 2 cannot. To circumvent this issue, we divide the feasible region +into at most 2푟푎푛푘(푄) subregions and perform this algorithm independently in each +subregion. Suppose, as before, that +푄 = +푀 +� +푚=1 +푞푚푞푇 +푚, +and for each 1 ≤ 푚 ≤ 푀 define +푅0 +푚 = {푥 ∈ 푋 | ⟨푞푚, 푥⟩ ≤ 0}, 푅1 +푚 = {푥 ∈ 푋 | ⟨푞푚, 푥⟩ ≥ 0}. +For each binary sequence (푛푚)푀 +푚=1 = 푛 ∈ {0, 1}푀, we denote by 푅푛 a (possibly +empty) subregion of 푋 given by +푅푛 = +푀 +� +푚=1 +푅푛푚 +푚 . +A straightforward observation then yields the following result. +Lemma 2 The Quadratic Transform (5) of (3) is concave in the 푥 variable over each +nonempty subregion 푅푛 for 푛 ∈ {0, 1}푀. +Proof Let 푔(푥, 푦) be as in (5) and fix a binary sequence 푛 ∈ {0, 1}푀. Then, by +definition of 푅푛, we see that for each 1 ≤ 푚 ≤ 푀 we have either ⟨푞푚, 푥⟩ ≥ 0 or +⟨푞푚, 푥⟩ ≤ 0 for all 푥 ∈ 푅푛. This implies that |⟨푞푚, 푥⟩| is linear over 푅푛, and hence +푔(푥, 푦) is a sum of concave functions, which is itself concave. +□ +We can now formulate our method for solving the convex-convex quadratic frac- +tional programming problem as Algorithm 3. +Remark 1 The utility of Algorithm 3 is, in full generality, limited by the rank of +푄 or, more precisely, by the number of nonempty subregions (which is controlled +by the rank of 푄). It is also worth noting that different decompositions of 푄 may +yield different numbers of nonempty subregions of the feasible region. We leave the +question of how to find better decompositions of 푄 beyond the scope of this paper. + +8 +Ilya Krishtal and Brendan Miller +input : Matrices 푄 = �푀 +푚=1 푞푚푞푇 +푚, 푃, linear constraints 퐴푥 ≤ 푏, and a tolerance 휀. +output +: +Global Solution of (2) +1 for 푛 ∈ {0, 1}푀 do +2 +if 푅푛 ≠ ∅ then +3 +Find an initial 푥푛 ∈ 푅푛; +4 +repeat +5 +Find stationary point 푥∗ of (2) via Algorithm 2 with initial point 푥푛; +6 +Set 휆 = 퐹 (푥∗); +7 +Compute 휋 (휆) and 푥(휆); +8 +Update 푥푛 = 푥(휆); +9 +until | 휋 (휆) | < 휀; +10 +end +11 end +12 Determine max 퐹 (푥푛); +Algorithm 3: Region-Checking Algorithm for Convex-Convex Quadratic +Fractional Programming +We do, however, mention explicitly the case when 푄 is a totally nonnegative matrix +and the optimization problem in question is +max +퐹(푥) = 푥푇 푄푥 +푥푇 푃푥 +s.t. +퐴푥 = 푏 +0 ≤ 푥 ≤ 훼. +In this case, we write 푄 = 퐿퐷퐿푇 for 퐷 a diagonal matrix and 퐿 a lower triangular +matrix. This gives a decomposition of 푄 as +푄 = +푀 +� +푚=1 +푑푚ℓ푚ℓ푇 +푚, +where ℓ푚 and 푑푚 are, respectively, the 푚푡ℎ column of 퐿 and 푚푡ℎ diagonal entry of +퐷. Since 푄 is totally nonnegative, the entries of each vector ℓ푚 are nonnegative [6], +in which case ⟨ℓ푚, 푥⟩ ≥ 0 for all feasible 푥. Thus, there is only one subregion 푅푛 of +the feasible region which is nonempty. +It is quite possible that Algorithm 3 is considerably slower than the algorithms +proposed in [7]and [14]given thepotentiallylargenumberoftimes 휋(휆)iscomputed. +However, as we will show empirically in the next section, often the first local +maximum found by Algorithm 3 in a given region is, in fact, the global maximum +of the region. Thus, we will also compare the efficiency and accuracy of Algorithm +3 without computing 휋(휆) and assuming each 푥∗ found by Algorithm 2 is a global +maximum of the region. This modification of Algorithm 3 is Algorithm 4 below. + +On Low-Rank Convex-Convex Quadratic Fractional Programming +9 +input : Matrices 푄 = �푀 +푚=1 푞푚푞푇 +푚, 푃, linear constraints 퐴푥 ≤ 푏, and a tolerance 휀. +output +: +Local Solution of (2) +1 for 푛 ∈ {0, 1}푀 do +2 +if 푅푛 ≠ ∅ then +3 +Find an initial 푥푛 ∈ 푅푛; +4 +Find stationary point 푥∗ of (2) via Algorithm 2 with initial point 푥푛; +5 +Update 푥푛 = 푥∗; +6 +end +7 end +8 Determine max 퐹 (푥푛); +Algorithm 4: Fast Region-Checking Algorithm +Remark 2 Suppose 푄 = 푞푞푇 is a rank-one matrix. We write the objective function +as +퐹(푥) = ⟨푞, 푥⟩2 +푥푇 푃푥 . +In this case, there are only two subregions of the feasible region: 푅0 and 푅1. Also, +we may equivalently maximize the square-root of the objective, which is given by +� +퐹(푥) = +� +−⟨푞, 푥⟩/ +√ +푥푇 푃푥 +푥 ∈ 푅0 +⟨푞, 푥⟩/ +√ +푥푇 푃푥 +푥 ∈ 푅1. +Thus, +� +퐹(푥) is a concave-convexfractionalprogrammingproblemin each subregion +of the feasible region, and hence can be solved by two applications of Algorithm 1 +where computing 휋(휆) is a concave programming problem. This method is superior +to Algorithm 4 as it was shown in [13] that Algorithm 2 is slower than the standard +Dinkelbach method for standard single-ratio concave-convex fractional programs. +4 Numerical Experiments +In this section we conduct several numerical experiments on the following optimiza- +tion problem: +max +퐹(푥) = 푥푇 푄푥 +푥푇 푃푥 +s.t. +퐴푥 ≤ 푏, 퐴 ∈ R푇 ×푛, 푏 ∈ R푇 +푛 +� +푖=1 +푥푖 = 1 +0 ≤ 푥푖 ≤ 0.1, 푖 = 1, ..., 푛 + +10 +Ilya Krishtal and Brendan Miller +where 퐴 and 푏 have random entries in the interval [0, 1] so that the vector +(1/푛, ..., 1/푛) is feasible, and 푄 = 푋푇 푋 and 푃 = 푌푇푌 where 푋 and 푌 are, re- +spectively, an 푀 × 푛 and an 푛 × 푛 random matrix with entries in [0, 1]. Unless +otherwise specified, we will always decompose 푄 as a sum of rank one matrices +according to its eigendecomposition as noted in Section 2. +We first demonstrate the efficiency and accuracy of Algorithms 3 and 4 against +Algorithm 1 for various combinations of (푛, 푀,푇). Next, we examine the average +number of nonempty subregions in Algorithms 3 and 4. Finally, we demonstrate the +accuracy of Algorithm 4 in a full-rank example (i.e. 푟 = 푛, which has several local +maxima) when there are a small number of subregions (see Remark 1). +All computation was done in MATLAB (on AMD A6-7400K Radeon R5 4.09 +GHz processor), using Gurobi 9.5 interface to solve the nonconvex quadratic pro- +gramming problems involved in Algorithms 1 and 3. The tolerance 휀 is always set +as 휀 = 10−3. All values in the forthcoming tables are averages of five tests. +4.1 Algorithm Comparison +We first give a demonstration of how differing combinations of (푛, 푀,푇) affect the +computation time of Algorithm 3 in a single nonempty region of the feasible set. +CPU Time (sec) +Iterations +푇 = 1 +푇 = 10 +푇 = 30 +푇 = 50 +푇 = 1 +푇 = 10 +푇 = 30 +푇 = 50 +푛 = 25 +푀 = 2 +0.4657 +0.3689 +0.7372 +0.7063 +1.2 +1 +1 +1 +5 +1.074 +0.5517 +1.183 +0.7725 +1 +1.2 +1 +1 +7 +1.037 +2.339 +1.473 +1.337 +1 +1 +1 +1 +10 +0.5639 +1.253 +9.06 +1.849 +1.2 +1 +1.4 +1.2 +50 +2 +2.022 +2.512 +2.403 +5.308 +1.2 +1.2 +1 +1.2 +5 +3.208 +6.842 +15.71 +19.66 +1 +1 +1.2 +1 +7 +60.85 +5.055 +6.284 +31.49 +1.4 +1 +1.2 +1 +10 +8.032 +126.5 +663.4 +49.52 +1.2 +1 +1.4 +1 +75 +2 +4.261 +7.612 +11.04 +7.077 +1 +1 +1.2 +1 +5 +5.299 +32.95 +58.67 +79.35 +1 +1.2 +1.4 +1 +7 +125.9 +62.65 +48.03 +273.3 +1.4 +1.2 +1.4 +1 +10 +23.8 +20.38 +93.34 +904.7 +1 +1 +1 +1.4 +Table 1: Efficiency of Algorithm 3 for one region +Table 1 shows that the computation time for Algorithm 3 increases sharply with +the number of inequality constraints due to the increasing complexity of solving the + +On Low-Rank Convex-Convex Quadratic Fractional Programming +11 +nonconvex quadratic subproblems. Likewise, the computation time increases with +both the number of variables and the rank of 푄, albeit not as sharply. Second, none of +the problems solved in this experiment took more than two iterations of Algorithm +2 to converge. In fact, 209 of the 240 problems solved in this experiment converged +in just one iteration of Algorithm 2. This suggests that the expensive procedure of +computing 휋(휆) can safely be dropped from the algorithm, if a low probability of +missing the exact solution may be tolerated by the application. +In the next two experiments, we fix 푇 = 10. We now compare Algorithm 3 with +the Algorithm 1. We show the results for Algorithm 3 both converging to the global +solution and forcing only one iteration per region without computing 휋(휆). +CPU Time (sec) +Alg 4 Error +Alg 1 +Alg 3 +Alg 4 +Error (%) +푛 = 10 +푀 = 2 +0.01372 +0.07733 +0.09616 +0 +3 +0.006822 +0.01969 +0.1347 +0 +4 +0.005291 +0.02188 +0.2637 +0 +5 +0.01414 +0.02462 +0.5047 +0 +7 +0.01159 +– +2.006 +0 +10 +0.01003 +– +16.67 +0 +25 +2 +0.5673 +0.9934 +0.1193 +0 +3 +0.6599 +1.266 +0.1932 +0.08 +4 +1.068 +3.975 +0.3618 +0 +5 +1.105 +26.85 +0.7185 +0.06 +7 +0.7177 +– +2.976 +0.02 +10 +0.8667 +– +23.35 +0.02 +50 +2 +4.932 +4.649 +0.1331 +0.04 +3 +5.258 +11.6 +0.2104 +0.05 +4 +9.485 +72.51 +0.4216 +0.02 +5 +7.618 +223.9 +0.91 +0.01 +7 +5.573 +– +3.18 +0.03 +10 +8.004 +– +26.5 +0 +75 +2 +12.63 +17.12 +0.1229 +0.01 +3 +14.43 +273.3 +0.346 +0 +4 +30.47 +211.2 +0.531 +0.2 +5 +45.63 +868.8 +1.249 +0.07 +7 +34.33 +– +4.407 +0 +10 +53.35 +– +35.55 +0.05 +Table 2: Comparison of Algorithms in CPU Time (sec) +Error = (퐹(푥표푝푡) − 퐹(푥))/퐹(푥표푝푡) + +12 +Ilya Krishtal and Brendan Miller +Table 2 shows the comparison of Algorithms 1, 3, and 4. There are several things +to note about these results. First, the error incurred fromperforming only one iteration +per region is negligible; it is always under one percent and quite often is under 0.1 +percent. This implies that one need only performAlgorithm 2 once in each subregion +of the feasible region, making irrelevant the need to compute 휋(휆). Second, for a +given value of 푀, the computation time needed to complete Algorithm 2 increases +steadily as 푛 increases, but does so at a much slower rate for Algorithm 4. +In fact, Table 3 shows that Algorithm 4 can be used efficiently for large scale +problems when the number of subregions is less than 130. For comparison, the +authors in [14] state that the case when 푛 = 500 is within reach via their algorithm +only by employing an elaborate local search. +Algorithm 4 +푛 = 250 +푀 = 7 +푇 = 10 +22.32 +500 +7 +10 +117.34 +750 +7 +10 +357.42 +1000 +7 +10 +743.24 +Table 3: CPU Time (sec) for Algorithm 4 +4.2 Accuracy of Algorithm 4 +The numerical experiments above call into question the accuracy of Algorithm 4 in +high-rank problems with a small number of nonempty regions. Table 1 shows that +there is typically only one local maximum per subregion, and this may seem to be +attributableto the large number of subregionsrelative to the numberof local maxima. +Thus, one might conclude that it may be disadvantageous to choose a decomposition +of 푄 with fewer terms as this leads to fewer possible subregions. We aim to show +that this is not the case. +The number of subregions is, of course, determined by the linear constraints. +Table 4 shows that the numberof subregionsis usually 2푟푎푛푘(푄)−1, and this decreases +only when the number of inequality constraints is much larger than the number of +variables. This implies that the number of subregions checked by the algorithm +increases exponentially with the rank of 푄, and thus the accuracy of Algorithm 4 +could be a product of the brute-force nature of checking each region. +For illustration we construct a full-rank example with a small number of subre- +gions to check. For a given value of 푛, we generate 푛 random, linearly independent +vectors (푞푖)푛 +푖=1 with values in the unit interval [0, 1]. We construct the matrix 푄 as +푄 = +푛 +� +푖=1 +푞푖푞푇 +푖 . + +On Low-Rank Convex-Convex Quadratic Fractional Programming +13 +Number of Subregions +푇 = 푛/2 +푇 = 푛 +푇 = 3푛/2 +푇 = 2푛 +푇 = 5푛/2 +푛 = 30 +푀 = 2 +2 +2 +2 +1 +1 +3 +4 +4 +3.8 +2 +1 +5 +16 +16 +12 +3 +1 +7 +64 +64 +49.4 +11 +1 +50 +2 +2 +2 +2 +1.8 +1 +3 +4 +4 +4 +2.8 +1 +5 +16 +16 +14.4 +5 +1 +7 +64 +64 +64 +19.4 +1 +100 +2 +2 +2 +2 +1.4 +1 +3 +4 +4 +4 +2.8 +1 +5 +16 +16 +16 +6.6 +1 +7 +64 +64 +64 +23.4 +1 +150 +2 +2 +2 +2 +1.2 +1 +3 +4 +4 +4 +2.8 +1 +5 +16 +16 +16 +8 +1 +7 +64 +64 +64 +13.6 +1 +Table 4: Average number of subregions checked by Algorithm 4 +Note that since the 푞푖 are constructed to be linearly independent, the matrix 푄 will +be invertible. Using this decomposition of 푄 and the constraints as before, there will +be only one non-empty subregion of the feasible region. This is because the entries +of each 푞푖 are positive, and hence ⟨푞푖, 푥⟩ ≥ 0 for all feasible 푥. All other matrices +are constructed in the same manner as before. +It is shown in Table 5 that the accuracy of Algorithm 4 remains quite high using +this decomposition of 푄. So, the accuracy of Algorithm 4 should not be attributed +primarily to the number of subregions. Therefore, since choosing a decomposition +of 푄 which results in few nonempty subregions yields a faster algorithm, it is +advantageous to choose one which yields the fewest number of nonempty subregions +of the feasible region. +5 Conclusion +We have presented an efficient and accurate algorithm for globally maximizing +low-rank convex-convex quadratic fractional programming problems. We have also +demonstrated that this algorithm can be utilized in high-rank problems if the numer- + +14 +Ilya Krishtal and Brendan Miller +CPU Time (sec) +Alg 4 Error +Alg 1 +Alg 4 +Error (%) +푛 = 20 +푇 = 1 +0.2189 +0.01942 +0 +10 +0.6094 +0.02111 +0 +30 +0.9902 +0.02257 +0.02936 +50 +1.766 +0.04667 +0 +35 +1 +1.423 +0.02659 +0.08328 +10 +6.945 +0.02921 +0.01075 +30 +7.726 +0.03527 +0.05105 +50 +20.61 +0.05936 +0.02039 +50 +1 +6.25 +0.03352 +0 +10 +6.753 +0.03664 +0.02419 +30 +25.31 +0.04325 +0.01203 +50 +189 +0.09199 +0.01428 +Table 5: CPU Time (sec) and Error in Full Rank Example +ator admits a decomposition which divides the feasible region into a small number +of subregions. Although Algorithm 4 is only guaranteed to converge to a local maxi- +mum of (2), we have shown heuristically that it almost always converges to the global +solution. +To guarantee global convergenceof Algorithm 4, one needs only to have a method +to determine if there is a feasible 푥 ∈ 푋 ∩ 푅 such that 퐹(푥) > 퐹(푥∗) where 푥∗ is +the local maximum found in the subregion 푅 by Algorithm 2. One way this may be +done is by using a solver to maximize the nonconvex quadratic +퐺(푥) = 푥푇 (푄 − 퐹(푥∗)푃)푥 +with the added quadratic constraint that 퐺(푥) > 0, and artificially terminating the +solver once a feasible 푥 is found. If no such 푥 can be found, then 푥∗ is the global +solution in the subregion. +Finally, we remark that Algorithm 4 is also applicable to the quadratic sum-of- +ratios problems, i.e. in the case when the matrix 푃 in the denominator of (3) is +allowed to vary with 푚 (in fact, the denominators need not be quadratic, they just +need to be convex). +Acknowledgement. Both authors of the paper were supported in part by the NSF +grant DMS-2208031. The paper is dedicated to the everlasting memory of Guido +L. Weiss whose research, teaching, and friendship has inspired generations. + +On Low-Rank Convex-Convex Quadratic Fractional Programming +15 +References +1. H. P. Benson. Global optimization algorithm for the nonlinear sum of ratios problem. Journal +of Optimization Theory and Applications, 112(1):1–29, 2002. +2. Harold P. Benson. Using concave envelopes to globally solve the nonlinear sum of ratios +problem. Journal of Global Optimization, 22(1/4):343–364, Jan 2002. +3. A. Charnes and W. W. Cooper. Programming with linear fractional functionals. Naval Research +Logistics Quarterly, 10(1):273–274, 1962. +4. Werner Dinkelbach. +On nonlinear fractional programming. +Management Science, +13(7):492–498, 1967. +5. Lianbo Gao, Shashi K. Mishra, and Jianming Shi. An extension of branch-and-bound algorithm +for solving sum-of-nonlinear-ratios problem. Optimization Letters, 6(2):221–230, 2010. +6. K.R. Goodearl and T.H. Lenagan. Lu decomposition of totally nonnegative matrices. Linear +Algebra and its Applications, 436(7):2554–2566, 2012. +7. Jun-Ya Gotoh and Hiroshi Konno. Maximization of the ratio of two convex quadratic functions +over a polytope. Computational Optimization and Applications, 20(1):43–60, 2001. +8. Longxiu Huang, Deanna Needell, and Sui Tang. Robust recovery of bandlimited graph signals +via randomized dynamical sampling. arXiv:2109.14079, 2021. +9. Toshihide Ibaraki. Parametric approaches to fractional programs. Mathematical Programming, +26(3):345–362, 1983. +10. Hongwei Jiao and Sanyang Liu. An efficient algorithm for quadratic sum-of-ratios fractional +programs problem. +Numerical Functional Analysis and Optimization, 38(11):1426–1445, +2017. +11. Andrew Lo and A. Craig MacKinlay. Maximizing predictability in the stock and bond markets. +Macroeconomic Dynamics, 1997. +12. Shao-Jian Qu, Ke-Cun Zhang, and Jia-Kun Zhao. An efficient algorithm for globally mini- +mizing sum of quadratic ratios problem with nonconvex quadratic constraints. Applied Math- +ematics and Computation, 189(2):1624–1636, 2007. +13. Kaiming Shen and Wei Yu. Fractional programming forcommunication systems—part i: Power +control and beamforming. IEEE Transactions on Signal Processing, 66(10):2616–2630, 2018. +14. R. Yamamoto and H. Konno. An efficient algorithm for solving convex–convex quadratic +fractional programs. +Journal of Optimization Theory and Applications, 133(2):241–255, +2007. + diff --git a/NtFIT4oBgHgl3EQfdSsx/content/tmp_files/load_file.txt b/NtFIT4oBgHgl3EQfdSsx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c65526fca0257cb13250e17b805ff956c79c143b --- /dev/null +++ b/NtFIT4oBgHgl3EQfdSsx/content/tmp_files/load_file.txt @@ -0,0 +1,483 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf,len=482 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='11269v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='OC] 26 Jan 2023 On Low-Rank Convex-Convex Quadratic Fractional Programming Ilya Krishtal and Brendan Miller Abstract We present an efficient algorithm for solving fractional programming problems whose objective functions are the ratio of a low-rank quadratic to a positive definite quadratic with convex constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' The proposed algorithm for these convex- convex problems is based on the Shen-Yu Quadratic Transform [13] which finds stationary points of concave-convex sum-of-ratios problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We further use elements of the algorithm proposed in [13] and the classic Dinkelbach approach to ensure convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We show that our algorithm performs better than previous algorithms for low-rank problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 1 Introduction Methods of fractional programming encompass a large range of techniques to solve problems of the form max 퐹(푥) = 푀 � 푚=1 푁푚(푥) 퐷푚(푥) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 푥 ∈ 푋 ⊂ R푛 (1) where 푁푚(푥), 퐷푚(푥) : R푛 → R are continuous functions, and 푋 is a closed, convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' A convention of the field, which we adopt throughout this paper, is that 푁푚(푥) ≥ 0 and 퐷푚(푥) > 0 for all 푥 ∈ 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' The problem is called single-ratio if 푀 = 1 and a sum-of-ratios fractional programming problem if 푀 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Fractional programming problems arise in many different applications, such as finance and portfolio analysis, government contracting, and engineering (see [1, 2, 10, 11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We were led to a single-ratio problem in the process of studying the problem of optimal sensor placement for dynamical sampling on graphs [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In particular, the relative Ilya Krishtal · Brendan Miller Northern Illinois University, Dekalb, IL 60435 USA e-mail: ikrishtal@niu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='edu (IK), bmiller14@niu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='edu (BM) 1 2 Ilya Krishtal and Brendan Miller error of reconstructing a signal on a graph from spatio-temporal samples in the presence of noise can be bounded above by a ratio of two quadratics depending on the eigenvalues of a certain frame operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Fractional programs have a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Below we outline a few milestones in the development of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In 1962, Charnes and Cooper showed in [3] that linear fractional programming problems, which are single-ratio fractional programming problems whose numerator, denominator, and constraints are linear, can be solved by simplex method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In 1967, Dinkelbach [4] showed that any concave-convex single-ratio fractional programming problem can be solved efficiently by consecutively solving several concave maximization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' The approach used by Dinkelbach was so useful that it has become standard and even seen many applications to fractional programs which are not concave-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In the latter case, however, the method involves solving a succession of non-concave maximization problems and becomes rather expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In 1997, Lo and MacKinlay [11] proposed the problem of maximizing the ratio of two convex quadratic functions in the context of portfolio analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' The following decade saw two papers published [7, 14] which presented algorithms for solving this problem exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Both follow the standard Dinkelbach approach, with different methods of handling the most expensive aspect of the problem: the need to solve multiple non-convex quadratic programming problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' To do so, the former paper by Gotoh and Konnoh implemented a branch-and-bound technique while the latter by Yamamoto and Konnoh iteratively approximated the quadratic by a piece-wise linear function, which is maximized by standard mixed integer linear programming techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In recent years, there has also been much research on the sum-of-ratios problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' For example, [1,2,5] all produce branch-and-bound type algorithms for solving the concave-convex sum-of-ratios problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Similarly, [10, 12] propose efficient algo- rithms for solving quadratically constrained quadratic sum-of-ratios type problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Of particular interest to us is the Shen-Yu Quadratic Transform introduced in [13] which, like the Dinkelbach method, iteratively solves a concave programming prob- lem to converge to a stationary point of the concave-convex sum-of-ratios problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In this paper, we propose an algorithm for solving the single-ratio convex-convex quadratic programming problem which can be effectively utilized when the numer- ator has low rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We divide the feasible region into several subregions in which the Shen-Yu Quadratic Transform method can be applied successively until global convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Since the methods of Shen and Yu only guarantee convergence to a stationary point, we use the mixed integer linear programming techniques employed in [14] to check if a given stationary point is a global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Although, as we will see empirically, this rather expensive procedure can often be omitted altogether (with a minimal chance of error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' This yields a very efficient algorithm which, with high probability, converges to the global solution of a convex-convex quadratic pro- gramming problem by successively solving concave maximization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' The only known algorithms for solving such problems involve successively maximizing a nonconvex quadratic programming problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' On Low-Rank Convex-Convex Quadratic Fractional Programming 3 The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In Section 2, we state the problem we focus on and outline the classic Dinkelbach approach for ratio maxi- mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We then convert our problem into a sum-of-ratios problem and recall a recent Shen-Yu scheme for solving such problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Section 3 is the centerpiece of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In it, we present a natural way of subdividing the feasible region of our prob- lem into a number of subregions thereby replacing a convex-convex sum-of-ratios problem with a finite number of much simpler concave-convex sum-of-ratios prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' This results in a region checking algorithm (Algorithm 3) which encompasses our approach to solving single-ratio quadratic convex-convex problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Numeri- cal experiments described in Section 4 illustrate the effectiveness of our approach in comparison with other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Finally, concluding remarks are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 2 Basic Analysis We study the problem of the form max 퐹(푥) = 푥푇 푄푥 푥푇 푃푥 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 퐴푥 ≤ 푏 (2) where 푄 is an 푛×푛 positive semidefinite matrix, 푃 is an 푛×푛 positive definite matrix, 퐴 ∈ R푇 ×푛, 푏 ∈ R푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We will denote the feasible region of this problem by 푋 ⊂ R푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Such optimization problems are nonconcave in general, and thus require expensive algorithms to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We recall the standard Dinkelbach method which utilizes the function 휋(휆) := max 푥∈푋 푥푇 (푄 − 휆푃)푥, 휆 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' It is convenient to introduce the following notation: 푥(휆) = arg max 푥∈푋 푥푇 (푄 − 휆푃)푥, 휆 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Theorem 1 The function 휋(휆) is convex and strictly decreasing in 휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Furthermore, 휋(휆) = 0 if and only if 푥(휆) maximizes (2) in 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' The algorithms of [7,14] are root-finding algorithms which use a scheme devel- oped by Ibaraki [9] to search for the root of 휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' These algorithms become expensive because computing 휋(휆) is a nonconvex quadratic programming problem, whose difficulty is larger for smaller values of 휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We will utilize this Theorem of Dinkel- bach in our algorithm, but only to check if a local maximum of (2) is indeed a global maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We summarize the Ibaraki scheme here as Algorithm 1 for reference, but omit the explanation of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 4 Ilya Krishtal and Brendan Miller input : Matrices 푄 and 푃, linear inequality constraints 퐴푥 ≤ 푏, and a tolerance 휀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 1 Find 휆푢 with 휋 (휆푢) < 0 and 휆푙 with 휋 (휆푙) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 2 repeat 3 Compute 휆 as follows: 휆 = � − 휋(휆푢) Δ휋 + 휆푢 if 푥(휆푢)푇 푃푥(휆푢) + Δ휋 ≠ 0 휋(휆푢) 푥(휆푢)푇 푃푥(휆푢) + 휆푢 otherwise, where Δ휋 = ( 휋 (휆푢) − 휋 (휆푙))/(휆푢 − 휆푙);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 4 Compute 휋 (휆);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 5 Update 휆푙 = 휆 if 휋 (휆) > 0, or 휆푢 = 휆 if 휋 (휆) < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 6 until | 휋 (휆) | < 휀;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Algorithm 1: Interpolated Binary Search (Ibaraki scheme) The algorithms presented in [7,14] both utilize Algorithm 1, but employ different methods of computing 휋(휆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We will use contemporary software to solve these non-convex quadratic programming problems to ensure the most accurate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Suppose now that 푄 can be written as 푄 = 푀 � 푚=1 푞푚푞푇 푚, which is a sum of rank-onematrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Note that푄 always admits such a decomposition as we can take 푀 = 푟푎푛푘(푄) and 푞푚 = √휆푚푣푚 where 푣푚 is the eigenvector of 푄 associated to the nonzero eigenvalue휆푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Then we may rewrite the objective function in the following way: 퐹(푥) = �푀 푚=1 푥푇 (푞푚푞푇 푚)푥 푥푇 푃푥 = 푀 � 푚=1 ⟨푞푚, 푥⟩2 푥푇 푃푥 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' (3) This reformulation of the objective function converts a single-ratio fractional pro- gramming problem (2) into a convex-convex sum-of-ratios fractional programming problem which we will refer to as (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Next, we summarize a method for suboptimally solving sum-of-ratios fractional programming problems (1) that was introduced in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Definition 1 (See [13]) Given the sum-of-ratios fractional programming problem (1), the Shen-Yu Quadratic Transform of this problem is defined to be 푔(푥, 푦) = 푀 � 푚=1 � 2푦푚 � 푁푚(푥) − 푦2 푚퐷푚(푥) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' (4) Theorem 2 ( [13]) On Low-Rank Convex-Convex Quadratic Fractional Programming 5 The sum-of-ratios problem (1) and its quadratic transform (4) maximization problem are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' More precisely, the maximum of (4) occurs at (푥∗, 푦∗) where 푥∗ maximizes (1) and 푦∗ = (푦∗ 푚)푀 푚=1 satisfies 푦∗ 푚 = � 푁푚(푥∗) 퐷푚(푥∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' The following Lemma is useful for deriving the Shen-Yu scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Lemma 1 Let 푔(푥, 푦) be as in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' If 푥0 ∈ 푋, then 푦0 := arg max푦∈R푀 푔(푥0, 푦) can be found analytically and is given by 푦0 = (푦0 푚)푀 푚=1, where 푦0 푚 = � 푁푚(푥0) 퐷푚(푥0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Proof It suffices to maximize each term in the sum in (4) individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Clearly, the vertex of the quadratic 푓 (푦푚) = 2푦푚 � 푁푚(푥0) − 푦2 푚퐷(푥0) occurs at the point 푦0 푚 = � 푁푚(푥0) 퐷푚(푥0) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' □ We may now state and prove a theorem from which, when taken together with Theorem 2, an algorithm for suboptimally solving (1) is naturally derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Theorem 3 (Shen, Yu [13]) Consider the sum-of-ratios problem (1) and suppose that 푥0 ∈ 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Let 푦0 = (푦0 푚)푀 푚=1 with 푦0 푚 = � 푁푚(푥0)/퐷푚(푥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' If 푥∗ = arg max 푥∈푋 푔(푥, 푦0) = arg max 푥∈푋 푀 � 푚=1 � 2푦0 푚 � 푁푚(푥) − (푦0 푚)2퐷푚(푥) � , then 퐹(푥∗) ≥ 퐹(푥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Proof Let 푥0, 푦0, and 푥∗ be as above and set 푦∗ 푚 = � 푁푚(푥∗) 퐷푚(푥∗) and 푦∗ = (푦∗ 푚)푀 푚=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Then we have the following string of inequalities 퐹(푥0) = 푔(푥0, 푦0) ≤ 푔(푥∗, 푦0) ≤ 푔(푥∗, 푦∗) = 퐹(푥∗) 6 Ilya Krishtal and Brendan Miller where the third line follows from Lemma 1, and the last by a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' □ The above Theorem guarantees that replacing 푥0 by 푥∗ and 푦0 by 푦∗ improves the value of the objective function with each iteration, and thus this method (with the scheme written explicitly in Algorithm 2) converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' It is clear that the algorithm converges to a local maximum, say (푥∗, 푦∗), of the Quadratic Transform 푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' It follows easily that the value 푥∗ is indeed a local maximum of the objective 퐹 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' input : Functions 푁푚 and 퐷푚, and a compact, convex set 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 1 Find an initial 푥0 ∈ 푋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 2 repeat 3 set 푦0 푚 = � 푁푚(푥0)/퐷푚 (푥0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 4 solve 푥∗ = arg max푥∈푋 �푀 푚=1 � 2푦0 푚 � 푁푚(푥) − (푦0 푚)2퐷푚(푥) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 5 update 푥0 = 푥∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 6 until convergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Algorithm 2: Shen-Yu Iterative Algorithm Corollary 1 Algorithm 2 converges to a stationary point of the sum-of-ratios frac- tional programming problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Although Algorithm 2 is guaranteed to converge to a stationary point, its use- fulness is limited to the difficulty of maximizing the Quadratic Transform over the variable 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In the case when each 푁푚 is concave and each 퐷푚 is convex, this can be done by any method of concave programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' When the objective has the form (3), the Quadratic Transform becomes 푔(푥, 푦) = 푀 � 푚=1 2푦푚|⟨푞푚, 푥⟩| − 푦2 푚푥푇 푃푥 (5) which is not, in general, concave in 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' However, in the event that the absolute values around each linear term in the sum can be dropped (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' each ⟨푞푚, 푥⟩ is either nonnegative or nonpositve valued on 푋), then we may apply Algorithm 2 effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 3 Region checking algorithm Henceforth we will assume the objective function 퐹 is as in (2) and, therefore, it can be rewritten in the sum-of-ratios form (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' The first observation to make is that combining Algorithm 2 with the Dinkelbach method yields an algorithm which converges to the global maximum of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Indeed, if 푥∗ is the local optimum found by Algorithm 2, we may set 휆∗ = 퐹(푥∗) and compute both 휋(휆∗) and 푥(휆∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' If On Low-Rank Convex-Convex Quadratic Fractional Programming 7 |휋(휆∗)| > 휀 where 휀 is some tolerance, we again run Algorithm 2 with 푥(휆∗) as the initial feasible point and repeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' This method is guaranteed to converge to the globally optimal solution since 푥(휆∗)푇 (푄 − 휆∗푃)푥(휆∗) > 푥∗푇 (푄 − 휆∗푃)푥∗ implies that 퐹(푥(휆∗)) > 퐹(푥∗) ≥ 퐹(푥0) where 푥0 is the initial feasible point used in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Finding the value of 휋(휆∗) and the vector 푥(휆∗) can be achieved by the methods introduced in [7] and refined in [14], but maximizing the Quadratic Transform as in Algorithm 2 cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' To circumvent this issue, we divide the feasible region into at most 2푟푎푛푘(푄) subregions and perform this algorithm independently in each subregion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Suppose, as before, that 푄 = 푀 � 푚=1 푞푚푞푇 푚, and for each 1 ≤ 푚 ≤ 푀 define 푅0 푚 = {푥 ∈ 푋 | ⟨푞푚, 푥⟩ ≤ 0}, 푅1 푚 = {푥 ∈ 푋 | ⟨푞푚, 푥⟩ ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' For each binary sequence (푛푚)푀 푚=1 = 푛 ∈ {0, 1}푀, we denote by 푅푛 a (possibly empty) subregion of 푋 given by 푅푛 = 푀 � 푚=1 푅푛푚 푚 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' A straightforward observation then yields the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Lemma 2 The Quadratic Transform (5) of (3) is concave in the 푥 variable over each nonempty subregion 푅푛 for 푛 ∈ {0, 1}푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Proof Let 푔(푥, 푦) be as in (5) and fix a binary sequence 푛 ∈ {0, 1}푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Then, by definition of 푅푛, we see that for each 1 ≤ 푚 ≤ 푀 we have either ⟨푞푚, 푥⟩ ≥ 0 or ⟨푞푚, 푥⟩ ≤ 0 for all 푥 ∈ 푅푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' This implies that |⟨푞푚, 푥⟩| is linear over 푅푛, and hence 푔(푥, 푦) is a sum of concave functions, which is itself concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' □ We can now formulate our method for solving the convex-convex quadratic frac- tional programming problem as Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Remark 1 The utility of Algorithm 3 is, in full generality, limited by the rank of 푄 or, more precisely, by the number of nonempty subregions (which is controlled by the rank of 푄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' It is also worth noting that different decompositions of 푄 may yield different numbers of nonempty subregions of the feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We leave the question of how to find better decompositions of 푄 beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 8 Ilya Krishtal and Brendan Miller input : Matrices 푄 = �푀 푚=1 푞푚푞푇 푚, 푃, linear constraints 퐴푥 ≤ 푏, and a tolerance 휀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' output : Global Solution of (2) 1 for 푛 ∈ {0, 1}푀 do 2 if 푅푛 ≠ ∅ then 3 Find an initial 푥푛 ∈ 푅푛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 4 repeat 5 Find stationary point 푥∗ of (2) via Algorithm 2 with initial point 푥푛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 6 Set 휆 = 퐹 (푥∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 7 Compute 휋 (휆) and 푥(휆);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 8 Update 푥푛 = 푥(휆);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 9 until | 휋 (휆) | < 휀;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 10 end 11 end 12 Determine max 퐹 (푥푛);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Algorithm 3: Region-Checking Algorithm for Convex-Convex Quadratic Fractional Programming We do, however, mention explicitly the case when 푄 is a totally nonnegative matrix and the optimization problem in question is max 퐹(푥) = 푥푇 푄푥 푥푇 푃푥 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 퐴푥 = 푏 0 ≤ 푥 ≤ 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In this case, we write 푄 = 퐿퐷퐿푇 for 퐷 a diagonal matrix and 퐿 a lower triangular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' This gives a decomposition of 푄 as 푄 = 푀 � 푚=1 푑푚ℓ푚ℓ푇 푚, where ℓ푚 and 푑푚 are, respectively, the 푚푡ℎ column of 퐿 and 푚푡ℎ diagonal entry of 퐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Since 푄 is totally nonnegative, the entries of each vector ℓ푚 are nonnegative [6], in which case ⟨ℓ푚, 푥⟩ ≥ 0 for all feasible 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Thus, there is only one subregion 푅푛 of the feasible region which is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' It is quite possible that Algorithm 3 is considerably slower than the algorithms proposed in [7]and [14]given thepotentiallylargenumberoftimes 휋(휆)iscomputed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' However, as we will show empirically in the next section, often the first local maximum found by Algorithm 3 in a given region is, in fact, the global maximum of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Thus, we will also compare the efficiency and accuracy of Algorithm 3 without computing 휋(휆) and assuming each 푥∗ found by Algorithm 2 is a global maximum of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' This modification of Algorithm 3 is Algorithm 4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' On Low-Rank Convex-Convex Quadratic Fractional Programming 9 input : Matrices 푄 = �푀 푚=1 푞푚푞푇 푚, 푃, linear constraints 퐴푥 ≤ 푏, and a tolerance 휀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' output : Local Solution of (2) 1 for 푛 ∈ {0, 1}푀 do 2 if 푅푛 ≠ ∅ then 3 Find an initial 푥푛 ∈ 푅푛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 4 Find stationary point 푥∗ of (2) via Algorithm 2 with initial point 푥푛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 5 Update 푥푛 = 푥∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 6 end 7 end 8 Determine max 퐹 (푥푛);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Algorithm 4: Fast Region-Checking Algorithm Remark 2 Suppose 푄 = 푞푞푇 is a rank-one matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We write the objective function as 퐹(푥) = ⟨푞, 푥⟩2 푥푇 푃푥 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In this case, there are only two subregions of the feasible region: 푅0 and 푅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Also, we may equivalently maximize the square-root of the objective, which is given by � 퐹(푥) = � −⟨푞, 푥⟩/ √ 푥푇 푃푥 푥 ∈ 푅0 ⟨푞, 푥⟩/ √ 푥푇 푃푥 푥 ∈ 푅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Thus, � 퐹(푥) is a concave-convexfractionalprogrammingproblemin each subregion of the feasible region, and hence can be solved by two applications of Algorithm 1 where computing 휋(휆) is a concave programming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' This method is superior to Algorithm 4 as it was shown in [13] that Algorithm 2 is slower than the standard Dinkelbach method for standard single-ratio concave-convex fractional programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 4 Numerical Experiments In this section we conduct several numerical experiments on the following optimiza- tion problem: max 퐹(푥) = 푥푇 푄푥 푥푇 푃푥 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 퐴푥 ≤ 푏, 퐴 ∈ R푇 ×푛, 푏 ∈ R푇 푛 � 푖=1 푥푖 = 1 0 ≤ 푥푖 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='1, 푖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=', 푛 10 Ilya Krishtal and Brendan Miller where 퐴 and 푏 have random entries in the interval [0, 1] so that the vector (1/푛, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=', 1/푛) is feasible, and 푄 = 푋푇 푋 and 푃 = 푌푇푌 where 푋 and 푌 are, re- spectively, an 푀 × 푛 and an 푛 × 푛 random matrix with entries in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Unless otherwise specified, we will always decompose 푄 as a sum of rank one matrices according to its eigendecomposition as noted in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We first demonstrate the efficiency and accuracy of Algorithms 3 and 4 against Algorithm 1 for various combinations of (푛, 푀,푇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Next, we examine the average number of nonempty subregions in Algorithms 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Finally, we demonstrate the accuracy of Algorithm 4 in a full-rank example (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 푟 = 푛, which has several local maxima) when there are a small number of subregions (see Remark 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' All computation was done in MATLAB (on AMD A6-7400K Radeon R5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='09 GHz processor), using Gurobi 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='5 interface to solve the nonconvex quadratic pro- gramming problems involved in Algorithms 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' The tolerance 휀 is always set as 휀 = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' All values in the forthcoming tables are averages of five tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='1 Algorithm Comparison We first give a demonstration of how differing combinations of (푛, 푀,푇) affect the computation time of Algorithm 3 in a single nonempty region of the feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' CPU Time (sec) Iterations 푇 = 1 푇 = 10 푇 = 30 푇 = 50 푇 = 1 푇 = 10 푇 = 30 푇 = 50 푛 = 25 푀 = 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='4657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='3689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='7372 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='7063 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='2 1 1 1 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='074 0.' metadata={'source': 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increasing complexity of solving the On Low-Rank Convex-Convex Quadratic Fractional Programming 11 nonconvex quadratic subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Likewise, the computation time increases with both the number of variables and the rank of 푄, albeit not as sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Second, none of the problems solved in this experiment took more than two iterations of Algorithm 2 to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In fact, 209 of the 240 problems solved in this experiment converged in just one iteration of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' This suggests that the expensive procedure of computing 휋(휆) can safely be dropped from the algorithm, if a low probability of missing the exact solution may be tolerated by the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In the next two experiments, we fix 푇 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We now compare Algorithm 3 with the Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We show the results for Algorithm 3 both converging to the global solution and forcing only one iteration per region without computing 휋(휆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' CPU Time (sec) Alg 4 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='05 Table 2: Comparison of Algorithms in CPU Time (sec) Error = (퐹(푥표푝푡) − 퐹(푥))/퐹(푥표푝푡) 12 Ilya Krishtal and Brendan Miller Table 2 shows the comparison of Algorithms 1, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' There are several things to note about these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' First, the error incurred fromperforming only one iteration per region is negligible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' it is always under one percent and quite often is under 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='1 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' This implies that one need only performAlgorithm 2 once in each subregion of the feasible region, making irrelevant the need to compute 휋(휆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Second, for a given value of 푀, the computation time needed to complete Algorithm 2 increases steadily as 푛 increases, but does so at a much slower rate for Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' In fact, Table 3 shows that Algorithm 4 can be used efficiently for large scale problems when the number of subregions is less than 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' For comparison, the authors in [14] state that the case when 푛 = 500 is within reach via their algorithm only by employing an elaborate local search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Algorithm 4 푛 = 250 푀 = 7 푇 = 10 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='32 500 7 10 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='34 750 7 10 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='42 1000 7 10 743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='24 Table 3: CPU Time (sec) for Algorithm 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='2 Accuracy of Algorithm 4 The numerical experiments above call into question the accuracy of Algorithm 4 in high-rank problems with a small number of nonempty regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Table 1 shows that there is typically only one local maximum per subregion, and this may seem to be attributableto the large number of subregionsrelative to the numberof local maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Thus, one might conclude that it may be disadvantageous to choose a decomposition of 푄 with fewer terms as this leads to fewer possible subregions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We aim to show that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' The number of subregions is, of course, determined by the linear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Table 4 shows that the numberof subregionsis usually 2푟푎푛푘(푄)−1, and this decreases only when the number of inequality constraints is much larger than the number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' This implies that the number of subregions checked by the algorithm increases exponentially with the rank of 푄, and thus the accuracy of Algorithm 4 could be a product of the brute-force nature of checking each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' For illustration we construct a full-rank example with a small number of subre- gions to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' For a given value of 푛, we generate 푛 random, linearly independent vectors (푞푖)푛 푖=1 with values in the unit interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We construct the matrix 푄 as 푄 = 푛 � 푖=1 푞푖푞푇 푖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' On Low-Rank Convex-Convex Quadratic Fractional Programming 13 Number of Subregions 푇 = 푛/2 푇 = 푛 푇 = 3푛/2 푇 = 2푛 푇 = 5푛/2 푛 = 30 푀 = 2 2 2 2 1 1 3 4 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='8 2 1 5 16 16 12 3 1 7 64 64 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='4 11 1 50 2 2 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='8 1 3 4 4 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='8 1 5 16 16 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='4 5 1 7 64 64 64 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='4 1 100 2 2 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='4 1 3 4 4 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='8 1 5 16 16 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='6 1 7 64 64 64 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='4 1 150 2 2 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='2 1 3 4 4 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='8 1 5 16 16 16 8 1 7 64 64 64 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='6 1 Table 4: Average number of subregions checked by Algorithm 4 Note that since the 푞푖 are constructed to be linearly independent, the matrix 푄 will be invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Using this decomposition of 푄 and the constraints as before, there will be only one non-empty subregion of the feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' This is because the entries of each 푞푖 are positive, and hence ⟨푞푖, 푥⟩ ≥ 0 for all feasible 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' All other matrices are constructed in the same manner as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' It is shown in Table 5 that the accuracy of Algorithm 4 remains quite high using this decomposition of 푄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' So, the accuracy of Algorithm 4 should not be attributed primarily to the number of subregions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Therefore, since choosing a decomposition of 푄 which results in few nonempty subregions yields a faster algorithm, it is advantageous to choose one which yields the fewest number of nonempty subregions of the feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' 5 Conclusion We have presented an efficient and accurate algorithm for globally maximizing low-rank convex-convex quadratic fractional programming problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' We have also demonstrated that this algorithm can be utilized in high-rank problems if the numer- 14 Ilya Krishtal and Brendan Miller CPU Time (sec) Alg 4 Error Alg 1 Alg 4 Error (%) 푛 = 20 푇 = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='2189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='01942 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='6094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='02111 0 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='9902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='02257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='02936 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='766 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='04667 0 35 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} 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+page_content='05105 50 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='05936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='02039 50 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='03352 0 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='753 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='03664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='02419 30 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='04325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='01203 50 189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='09199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='01428 Table 5: CPU Time (sec) and Error in Full Rank Example ator admits a decomposition which divides the feasible region into a small number of subregions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Although Algorithm 4 is only guaranteed to converge to a local maxi- mum of (2), we have shown heuristically that it almost always converges to the global solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' To guarantee global convergenceof Algorithm 4, one needs only to have a method to determine if there is a feasible 푥 ∈ 푋 ∩ 푅 such that 퐹(푥) > 퐹(푥∗) where 푥∗ is the local maximum found in the subregion 푅 by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' One way this may be done is by using a solver to maximize the nonconvex quadratic 퐺(푥) = 푥푇 (푄 − 퐹(푥∗)푃)푥 with the added quadratic constraint that 퐺(푥) > 0, and artificially terminating the solver once a feasible 푥 is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' If no such 푥 can be found, then 푥∗ is the global solution in the subregion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Finally, we remark that Algorithm 4 is also applicable to the quadratic sum-of- ratios problems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' in the case when the matrix 푃 in the denominator of (3) is allowed to vary with 푚 (in fact, the denominators need not be quadratic, they just need to be convex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Both authors of the paper were supported in part by the NSF grant DMS-2208031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' The paper is dedicated to the everlasting memory of Guido L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' Weiss whose research, teaching, and friendship has inspired generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} +page_content=' On Low-Rank Convex-Convex Quadratic Fractional Programming 15 References 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFIT4oBgHgl3EQfdSsx/content/2301.11269v1.pdf'} diff --git a/O9AyT4oBgHgl3EQfg_h-/vector_store/index.faiss b/O9AyT4oBgHgl3EQfg_h-/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..36cd23a9860051b4d0ab506dc8d0538e51daf170 --- /dev/null +++ b/O9AyT4oBgHgl3EQfg_h-/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4f6387283de14b2f9e67da5b79f5765d97e2f3fce420ecee76f7dd85b0fbc292 +size 5373997 diff --git a/O9E4T4oBgHgl3EQf-A5z/content/tmp_files/2301.05360v1.pdf.txt b/O9E4T4oBgHgl3EQf-A5z/content/tmp_files/2301.05360v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..53abea75f9e0afc40cf75eb362a63b55e2b1736e --- /dev/null +++ b/O9E4T4oBgHgl3EQf-A5z/content/tmp_files/2301.05360v1.pdf.txt @@ -0,0 +1,382 @@ +Second-Order SUSY-QM and zeroes of the +Riemann zeta function +Juan D Garc´ıa-Mu˜noz1, A Raya1,2 and Y Concha-S´anchez3 +1Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´as de Hidalgo, +Edificio C-3, Ciudad Universitaria, Francisco J. M´ujica S/N Col. Fel´ıcitas del R´ıo, 58040 +Morelia, Michoac´an, M´exico. +2 Centro de Ciencias Exactas - Universidad del Bio-Bio. Avda. Andr´es Bello 720, Casilla +447, Chill´an, Chile. +3Facultad de Ingenier´ıa Civil, Universidad Michoacana de San Nicol´as de Hidalgo, Edificio +C, Ciudad Universitaria. Francisco J. M´ujica s/n. Col. Fel´ıcitas del R´ıo. 58030, Morelia, +Michoac´an, Mexico. +Email: juan.domingo.garcia@umich.mx, alfredo.raya@umich.mx and yajaira.concha@umich.mx +Abstract +We build a quantum mechanical Hamiltonian whose spectrum is related to the Riemann zeta +function ζ(s) making use of the confluent algorithm of supersymmetric quantum mechanics +(SUSY-QM). Inspired by the first-order SUSY-QM model of Das and Kalauni [1], which +corresponds to this function in the strip 0 < Re[s] < 1, we use its ground state wave +function as a seed solution for our algorithm and take the factorization energy equal to zero. +We thus construct a pair of intertwined Hamiltonians by means of second-order differential +operators and upon demanding that the ground state corresponds to a zero mode, we locate +exactly the nontrivial zeroes of ζ(s) along the critical line Re[s] = 1/2 in the complex plane +from a entirely different algebra. We further find that unlike the first order case, where the +corresponding SUSY-partner potentials belong to the family of inverse squared distance +potentials with complex couplings, in the second order model the partner potentials exhibit +a more intricate behavior. +Keywords: Riemann zeta function, supersymmetric quantum mechanics, confluent algorithm +Quantum mechanics offers a fertile ground to explore the location of the zeroes of the Riemann +ζ(s) function [2–15] defined as [16] +ζ(s) = +∞ +� +n=1 +1 +sn, +s = σ + iλ, +σ, λ ∈ R, +Re[s] > 1. +(1) +In particular, Das and Kalani (DK) [1] constructed a first order supersymmetric quantum mechani- +cal (SUSY-QM) model closely related to this function from the observation that monomials of the +form x−s are eigenfunctions of the operators +O = +∞ +� +n=1 +(−1)n+1 exp +� +(ln n)x d +dx +� +, +O† = +∞ +� +n=1 +(−1)n+1 +n +exp +� +(ln n−1)x d +dx +� +, +(2) +1 +arXiv:2301.05360v1 [math-ph] 13 Jan 2023 + +namely, +Ox−s = (1 − 21−s)ζ(s)x−s, +Re[s] > 0, +O†x−s = (1 − 2s)ζ(1 − s)x−s, +Re[s] < 1. +(3) +Thus, defining the raising and lowering operators +A(ω) = |x|−i ω +2 O|x|,i ω +2 , +A†(ω) = |x|i ω +2 O†|x|−i ω +2 , +(4) +the DK model is defined as the pair of SUSY-partner Hamiltonians +HDK ++ += A(ω)A†(ω), +HDK +− += A†(ω)A(ω). +(5) +These share the same energy spectrum up to the additional ground state ψ0(x) of HDK +− +, defined +such that A(ω)ψ0(x) = 0, which is a zero mode. Explicitly, +ψ0(x) = |x|− 1 +2 +i( ω +2 −λ∗), +(6) +where the requirement of vanishing of the ground state energy fixes λ∗ = ω/2 − ρ to the position +of a zero of ζ(s) along the critical line, namely +ζ +�1 +2 + iλ∗ +� += 0. +(7) +Cast in the traditional Schr¨odinger form, +HDK +± += − d2 +dx2 + V DK +± +(x), +(8) +the superpotential that relates these two SUSY-Hamiltonians is +W(x) = −ψ′ +0(x) +ψ0(x) = 1 +|x| +�1 +2 − iρ +� +, +(9) +from which the partner potentials are +V DK +± +(x) = W 2 ± W ′ ≡ α± +x2 , +(10) +where the (complex) couplings are, respectively, +α− = − +� +ρ + i +2 +� � +ρ + 3i +2 +� +, +α+ = − +�1 +4 + ρ2 +� +. +(11) +We use these results to set up our model through the confluent algorithm of SUSY-QM. +Confluent second-order SUSY-QM is an algebraic method intertwining two Schr¨odinger-like +Hamiltonians by means of the relation +H+L− = L−H−, +(12) +where L± are second-order differential intertwined operators. Specifically, we have these operators +have the following form +L− = d2 +dx2 + η(x) d +dx + γ(x), +L+ = (L−)†, +H± = − d2 +dx2 + V ±(x), +(13) +2 + +with η(x) and γ(x) being functions to be determined [17] (see also [18–28]). Substituting the +expressions from Eq. (13) in Eq. (12), in a straightforward way it is obtained that +V + = V − + 2η′, +γ = η2 +2 − η′ +2 − V − + ϵ, +V − = η′′ +2η − +� η′ +2η +�2 +− η′ + η2 +4 + ϵ. +(14) +For simplicity, in equations we omit the dependence of the functions on x and write f ′ to denote +the derivative with respect to that variable. In the previous equations, the constant ϵ is the so-called +factorization energy associated to the seed solution u(x), which fulfills the stationary Schr¨odinger- +like equation for H−, i.e., +− u′′ + V −u = ϵu. +(15) +It is worth mentioning the confluent algorithm is defined by means of the function η. In this case, +that said function can be written as +η = −w′ +w , +w = w0 − +x +� +x0 +u2(y)dy, +(16) +where x0 is a point in the appropriate x-domain and w0 is a parameter that guarantees the function +w(x) remains nodeless. If the Hamiltonian H− is solvable, i.e., we know the eigenvalues ε− +n +and eigenfunctions ψ− +n (x) in advance, then, the eigenfunctions ψ+ +n (x) of the Hamiltonian H+ are +related with the functions ψ− +n (x) by means of the following expressions +ψ± +n = L∓ψ∓ +n +|ε− +n − ϵ|. +(17) +Furthermore, the eigenfunction ψ+ +ϵ (x) of the Hamiltonian H+ corresponding to the factorization +energy ϵ is directly proportional to +ψ+ +ϵ ∝ u +w. +(18) +Taking the seed solution as +u = |x|− 1 +2 +iρ, +(19) +and considering the integral +I = +x +� +x0 +u2(y)dy = −i|x|i2ρ − |x0|i2ρ +2ρ +, +(20) +we have that +w = w0 − I += +� +w0 + sin(2ρ ln |x0|) − sin(2ρ ln |x|) +2ρ +� ++ i +�cos(2ρ ln |x|) − cos(2ρ ln |x0|) +2ρ +� +. +(21) +In order to avoid zeroes in this function, we can see its imaginary part vanishes provided x = x0. +Nevertheless, there exist an infinite number of points where such an imaginary part would vanish, +given the periodic nature of the functions involved. For simplicity, we consider the interval +2ρ ln |x| = 2n + 1 +2 +π +(22) +3 + +Figure 1: First-order supersymmetric partner potentials V DK +± +(x) of DK model and the confluent supersymmetric +partner potentials V ±(x). Note that V DK +− +(x) = V −(x). +for n = 0, namely, +|x| = e +π +4ρ. +(23) +It becomes convenient to select the left corner of the interval such that x0 = −e +π +4ρ. Thus, +w(x) = w0 + 1 +2ρ + i|x|i2ρ +2ρ . +(24) +Therefore, the potentials that come from the confluent transformation are +V −(x) = −(ρ + i +2)(ρ + 3i +2 ) +x2 +, +V +(x) = +3 +4 − ρ2 + 2ρ (cot(ρ ln x) + ρ csc2(ρ ln x)) +x2 +. +(25) +Notice that (19) is a zero mode eigenstate of V −(x), and thus the vanishing of its energy eigenvalue +also implies that the location of a zero of the Riemann zeta function is found as in Eq. (7). Fur- +thermore, we must mention that being rigorous, in Eqs.(10) and (25) we should write |x|2 instead +of x2. However, this simplification is possible since x ∈ R. +In summary, we have obtained a second order SUSY-QM model with exact spectra of the +Hamiltonians in Eq. (13) where V −(x) in Eq. (25) has a zero mode fixing the location of the +zeroes of the function ζ(s) along the critical line as in Eq. (7). Although, the SUSY partner +potential V +(x) in this case exhibit a more intricate behavior as compared with the corresponding +to the DK model, Eq. (10) and Eq. (11) (see also Fig. 1). Remarkable features are that the V +(x) +potential remains real and its energy spectrum turns out to be equal to the spectrum of the potential +V DK ++ +(x). +JDGM and AR benefited from financial support from CONACYT Project FORDECYT-PRO- +NACES/61533/2020. YCS also acknowledges the CIC-UMSNH research grant 6976882/2022. +4 + +20 +15 - +10 - +5 +V(ar) +0 +-5 +10 +VDK (r) +究[V-(a)] +15 +3[V-(r)] +V+(r) +20 +-2 +-1 +0 +2References +[1] A. Das and P. Kalauni, “Supersymmetry and the Riemann zeros on the critical line,” Physics +Letters B, vol. 791, pp. 265–269, 2019. +[2] M. V. Berry and J. P. Keating, “The Riemann zeros and eigenvalue asymptotics,” SIAM Re- +view, vol. 41, no. 2, pp. 236–266, 1999. +[3] R. Aschheim, C. C. Perelman, and K. Irwin, “The search for a hamiltonian whose energy +spectrum coincides with the Riemann zeta zeroes,” International Journal of Geometric Meth- +ods in Modern Physics, vol. 14, no. 06, p. 1750109, 2017. +[4] J. Berra-Montiel and A. Molgado, “Polymeric quantum mechanics and the zeros of the Rie- +mann zeta function,” International Journal of Geometric Methods in Modern Physics, vol. 15, +no. 06, p. 1850095, 2018. +[5] M. V. Berry and J. P. Keating, “H= xp and the Riemann zeros,” in Supersymmetry and Trace +Formulae, pp. 355–367, Springer, 1999. +[6] C. M. Bender, D. C. Brody, and M. P. M¨uller, “Hamiltonian for the zeros of the Riemann zeta +function,” Phys. Rev. Lett., vol. 118, p. 130201, Mar 2017. +[7] R. He, M.-Z. Ai, J.-M. Cui, Y.-F. Huang, Y.-J. Han, C.-F. Li, T. Tu, C. E. Creffield, G. Sierra, +and G.-C. Guo, “Identifying the Riemann zeros by periodically driving a single qubit,” Phys. +Rev. A, vol. 101, p. 043402, Apr 2020. +[8] P. Betzios, N. Gaddam, and O. Papadoulaki, “Black holes, quantum chaos, and the Riemann +hypothesis,” SciPost Phys. Core, vol. 4, p. 032, 2021. +[9] C. M. Bender and D. C. Brody, “Asymptotic analysis on a pseudo-hermitian Riemann-zeta +hamiltonian,” Journal of Physics A: Mathematical and Theoretical, vol. 51, p. 135203, mar +2018. +[10] G. Sierra, “The Riemann zeros as spectrum and the Riemann hypothesis,” Symmetry, vol. 11, +no. 4, p. 494, 2019. +[11] F. Tamburini and I. Licata, “Majorana quanta, string scattering, curved spacetimes and the +Riemann hypothesis,” Physica Scripta, vol. 96, p. 125276, dec 2021. +[12] R. A. El-Nabulsi, “Quantization of non-standard hamiltonians and the Riemann zeros,” Qual- +itative Theory of Dynamical Systems, vol. 18, pp. 69–84, Apr 2019. +[13] M. McGuigan, “Riemann hypothesis, modified Morse potential and supersymmetric quantum +mechanics,” 2020. +[14] E. Yakaboylu, “Formally self-adjoint hamiltonian for the Hilbert-P´olya conjecture,” 2022. +[15] M. A. Gulas, Using Hilbert Space Theory and Quantum Mechanics to Examine the Zeros of +The Riemann-Zeta Function. PhD thesis, 2020. +5 + +[16] B. Riemann, “ ¨Uber die anzahl der primzahlen unter einer gegebenen gr¨osse,(on the number +of primes less than a given quantity), monatsberichte der berliner akademie 1859,” 1860. +[17] D. J. Fernandez C. and N. Fernandez-Garcia, “Higher-order supersymmetric quantum me- +chanics,” AIP Conference Proceedings, vol. 744, 02 2005. +[18] A. Andrianov, M. Ioffe, and V. Spiridonov, “Higher-derivative supersymmetry and the Witten +index,” Phys. Lett. A, vol. 174, p. 273, 1993. +[19] A. Andrianov, M. Ioffe, F. Cannata, and J. Dedonder, “Second order derivative supersym- +metry, q deformations and the scattering problem,” Int. J. Mod. Phys. A, vol. 10, p. 2683, +1995. +[20] D. J. Fern´andez C and E. Salinas-Hern´andez, “The confluent algorithm in second-order super- +symmetric quantum mechanics,” Journal of Physics A: Mathematical and General, vol. 36, +pp. 2537–2543, feb 2003. +[21] D. J. Fern´andez C. and E. Salinas-Hern´andez, “Wronskian formula for confluent second- +order supersymmetric quantum mechanics,” Physics Letters A, vol. 338, no. 1, pp. 13 – 18, +2005. +[22] D. J. Fern´andez C and B. Roy, “Confluent second-order supersymmetric quantum mechanics +and spectral design,” Physica Scripta, vol. 95, p. 055210, feb 2020. +[23] A. Contreras-Astorga and A. Schulze-Halberg, “The generalized zero-mode supersymmetry +scheme and the confluent algorithm,” Annals of Physics, vol. 354, pp. 353–364, mar 2015. +[24] A. Contreras-Astorga and A. Schulze-Halberg, “On integral and differential representations +of Jordan chains and the confluent supersymmetry algorithm,” Journal of Physics A: Mathe- +matical and Theoretical, vol. 48, p. 315202, jul 2015. +[25] D. Bermudez, “Wronskian differential formula for k-confluent SUSY-QM,” Annals of +Physics, vol. 364, pp. 35–52, jan 2016. +[26] A. Contreras-Astorga and A. Schulze-Halberg, “Recursive representation of Wronskians in +confluent supersymmetric quantum mechanics,” Journal of Physics A: Mathematical and +Theoretical, vol. 50, p. 105301, feb 2017. +[27] A. Schulze-Halberg and O. Yesiltas, “The generalized confluent supersymmetry algorithm: +Representations and integral formulas,” Journal of Mathematical Physics, vol. 59, p. 043508, +apr 2018. +[28] D. Bermudez, D. J. Fern´andez C, and N. Fern´andez-Garc´ıa, “Wronskian differential formula +for confluent supersymmetric quantum mechanics,” Physics Letters A, vol. 376, pp. 692–696, +jan 2012. +6 + diff --git a/O9E4T4oBgHgl3EQf-A5z/content/tmp_files/load_file.txt b/O9E4T4oBgHgl3EQf-A5z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..294b9f342e63387683b297d8208f46461de5369e --- /dev/null +++ b/O9E4T4oBgHgl3EQf-A5z/content/tmp_files/load_file.txt @@ -0,0 +1,255 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf,len=254 +page_content='Second-Order SUSY-QM and zeroes of the Riemann zeta function Juan D Garc´ıa-Mu˜noz1, A Raya1,2 and Y Concha-S´anchez3 1Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´as de Hidalgo, Edificio C-3, Ciudad Universitaria, Francisco J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' M´ujica S/N Col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Fel´ıcitas del R´ıo, 58040 Morelia, Michoac´an, M´exico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' 2 Centro de Ciencias Exactas - Universidad del Bio-Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Andr´es Bello 720, Casilla 447, Chill´an, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' 3Facultad de Ingenier´ıa Civil, Universidad Michoacana de San Nicol´as de Hidalgo, Edificio C, Ciudad Universitaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Francisco J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' M´ujica s/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Fel´ıcitas del R´ıo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' 58030, Morelia, Michoac´an, Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Email: juan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content='domingo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content='garcia@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content='mx, alfredo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content='raya@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content='mx and yajaira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content='concha@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content='mx Abstract We build a quantum mechanical Hamiltonian whose spectrum is related to the Riemann zeta function ζ(s) making use of the confluent algorithm of supersymmetric quantum mechanics (SUSY-QM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Inspired by the first-order SUSY-QM model of Das and Kalauni [1], which corresponds to this function in the strip 0 < Re[s] < 1, we use its ground state wave function as a seed solution for our algorithm and take the factorization energy equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' We thus construct a pair of intertwined Hamiltonians by means of second-order differential operators and upon demanding that the ground state corresponds to a zero mode, we locate exactly the nontrivial zeroes of ζ(s) along the critical line Re[s] = 1/2 in the complex plane from a entirely different algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' We further find that unlike the first order case, where the corresponding SUSY-partner potentials belong to the family of inverse squared distance potentials with complex couplings, in the second order model the partner potentials exhibit a more intricate behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Keywords: Riemann zeta function, supersymmetric quantum mechanics, confluent algorithm Quantum mechanics offers a fertile ground to explore the location of the zeroes of the Riemann ζ(s) function [2–15] defined as [16] ζ(s) = ∞ � n=1 1 sn, s = σ + iλ, σ, λ ∈ R, Re[s] > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (1) In particular, Das and Kalani (DK) [1] constructed a first order supersymmetric quantum mechani- cal (SUSY-QM) model closely related to this function from the observation that monomials of the form x−s are eigenfunctions of the operators O = ∞ � n=1 (−1)n+1 exp � (ln n)x d dx � , O† = ∞ � n=1 (−1)n+1 n exp � (ln n−1)x d dx � , (2) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content='05360v1 [math-ph] 13 Jan 2023 namely, Ox−s = (1 − 21−s)ζ(s)x−s, Re[s] > 0, O†x−s = (1 − 2s)ζ(1 − s)x−s, Re[s] < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (3) Thus, defining the raising and lowering operators A(ω) = |x|−i ω 2 O|x|,i ω 2 , A†(ω) = |x|i ω 2 O†|x|−i ω 2 , (4) the DK model is defined as the pair of SUSY-partner Hamiltonians HDK + = A(ω)A†(ω), HDK − = A†(ω)A(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (5) These share the same energy spectrum up to the additional ground state ψ0(x) of HDK − , defined such that A(ω)ψ0(x) = 0, which is a zero mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Explicitly, ψ0(x) = |x|− 1 2 +i( ω 2 −λ∗), (6) where the requirement of vanishing of the ground state energy fixes λ∗ = ω/2 − ρ to the position of a zero of ζ(s) along the critical line, namely ζ �1 2 + iλ∗ � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (7) Cast in the traditional Schr¨odinger form, HDK ± = − d2 dx2 + V DK ± (x), (8) the superpotential that relates these two SUSY-Hamiltonians is W(x) = −ψ′ 0(x) ψ0(x) = 1 |x| �1 2 − iρ � , (9) from which the partner potentials are V DK ± (x) = W 2 ± W ′ ≡ α± x2 , (10) where the (complex) couplings are, respectively, α− = − � ρ + i 2 � � ρ + 3i 2 � , α+ = − �1 4 + ρ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (11) We use these results to set up our model through the confluent algorithm of SUSY-QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Confluent second-order SUSY-QM is an algebraic method intertwining two Schr¨odinger-like Hamiltonians by means of the relation H+L− = L−H−, (12) where L± are second-order differential intertwined operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Specifically, we have these operators have the following form L− = d2 dx2 + η(x) d dx + γ(x), L+ = (L−)†, H± = − d2 dx2 + V ±(x), (13) 2 with η(x) and γ(x) being functions to be determined [17] (see also [18–28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Substituting the expressions from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (13) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (12), in a straightforward way it is obtained that V + = V − + 2η′, γ = η2 2 − η′ 2 − V − + ϵ, V − = η′′ 2η − � η′ 2η �2 − η′ + η2 4 + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (14) For simplicity, in equations we omit the dependence of the functions on x and write f ′ to denote the derivative with respect to that variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' In the previous equations, the constant ϵ is the so-called factorization energy associated to the seed solution u(x), which fulfills the stationary Schr¨odinger- like equation for H−, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=', − u′′ + V −u = ϵu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (15) It is worth mentioning the confluent algorithm is defined by means of the function η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' In this case, that said function can be written as η = −w′ w , w = w0 − x � x0 u2(y)dy, (16) where x0 is a point in the appropriate x-domain and w0 is a parameter that guarantees the function w(x) remains nodeless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' If the Hamiltonian H− is solvable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=', we know the eigenvalues ε− n and eigenfunctions ψ− n (x) in advance, then, the eigenfunctions ψ+ n (x) of the Hamiltonian H+ are related with the functions ψ− n (x) by means of the following expressions ψ± n = L∓ψ∓ n |ε− n − ϵ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (17) Furthermore, the eigenfunction ψ+ ϵ (x) of the Hamiltonian H+ corresponding to the factorization energy ϵ is directly proportional to ψ+ ϵ ∝ u w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (18) Taking the seed solution as u = |x|− 1 2 +iρ, (19) and considering the integral I = x � x0 u2(y)dy = −i|x|i2ρ − |x0|i2ρ 2ρ , (20) we have that w = w0 − I = � w0 + sin(2ρ ln |x0|) − sin(2ρ ln |x|) 2ρ � + i �cos(2ρ ln |x|) − cos(2ρ ln |x0|) 2ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (21) In order to avoid zeroes in this function, we can see its imaginary part vanishes provided x = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Nevertheless, there exist an infinite number of points where such an imaginary part would vanish, given the periodic nature of the functions involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' For simplicity, we consider the interval 2ρ ln |x| = 2n + 1 2 π (22) 3 Figure 1: First-order supersymmetric partner potentials V DK ± (x) of DK model and the confluent supersymmetric partner potentials V ±(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Note that V DK − (x) = V −(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' for n = 0, namely, |x| = e π 4ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (23) It becomes convenient to select the left corner of the interval such that x0 = −e π 4ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Thus, w(x) = w0 + 1 2ρ + i|x|i2ρ 2ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (24) Therefore, the potentials that come from the confluent transformation are V −(x) = −(ρ + i 2)(ρ + 3i 2 ) x2 , V +(x) = 3 4 − ρ2 + 2ρ (cot(ρ ln x) + ρ csc2(ρ ln x)) x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (25) Notice that (19) is a zero mode eigenstate of V −(x), and thus the vanishing of its energy eigenvalue also implies that the location of a zero of the Riemann zeta function is found as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Fur- thermore, we must mention that being rigorous, in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (10) and (25) we should write |x|2 instead of x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' However, this simplification is possible since x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' In summary, we have obtained a second order SUSY-QM model with exact spectra of the Hamiltonians in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (13) where V −(x) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (25) has a zero mode fixing the location of the zeroes of the function ζ(s) along the critical line as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Although, the SUSY partner potential V +(x) in this case exhibit a more intricate behavior as compared with the corresponding to the DK model, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (10) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' (11) (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Remarkable features are that the V +(x) potential remains real and its energy spectrum turns out to be equal to the spectrum of the potential V DK + (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' JDGM and AR benefited from financial support from CONACYT Project FORDECYT-PRO- NACES/61533/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' YCS also acknowledges the CIC-UMSNH research grant 6976882/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' 4 20 15 - 10 - 5 V(ar) 0 5 10 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Fern´andez C, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' Fern´andez-Garc´ıa, “Wronskian differential formula for confluent supersymmetric quantum mechanics,” Physics Letters A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' 376, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' 692–696, jan 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} +page_content=' 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E4T4oBgHgl3EQf-A5z/content/2301.05360v1.pdf'} diff --git a/OtE1T4oBgHgl3EQfuAVi/content/tmp_files/2301.03383v1.pdf.txt b/OtE1T4oBgHgl3EQfuAVi/content/tmp_files/2301.03383v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e69493cbbd83214e2cf628497692ba03af1e6c85 --- /dev/null +++ b/OtE1T4oBgHgl3EQfuAVi/content/tmp_files/2301.03383v1.pdf.txt @@ -0,0 +1,1197 @@ +arXiv:2301.03383v1 [math.AP] 9 Nov 2022 +On the continuity of the solution map of the Euler-Poincar´e +equations in Besov spaces +Min Li1 ∗ +1 Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang, 330032, China +Abstract: By constructing a series of perturbation functions through localization in the Fourier +domain and using a symmetric form of the system, we show that the data-to-solution map for the +Euler-Poincar´e equations is nowhere uniformly continuous in Bs +p,r(Rd) with s > max{1 + d +2, 3 +2} and +(p, r) ∈ (1, ∞) × [1, ∞). This improves our previous result which shows the data-to-solution map +for the Euler-Poincar´e equations is non-uniformly continuous on a bounded subset of Bs +p,r(Rd) near +the origin. +Keywords: Euler-Poincar´e equations; Nowhere uniformly continuous; Besov spaces; Data-to- +solution map. +MSC (2010): 35Q35, 35Q51, 35L30 +1 +Introduction +In this paper, we consider the Cauchy problem in Rd for Euler-Poincar´e equations + + + + + + + + + +∂tm + u · ∇m + ∇uT · m + (divu)m = 0, +(t, x) ∈ R+ × Rd, +m = (1 − ∆)u, +(t, x) ∈ R+ × Rd, +u(0, x) = u0, +x ∈ Rd. +(1.1) +The equations (1.1) were first introduced by Holm, Marsden, and Ratiu in [17, 18] as a high +dimensional generalization of the following Camassa-Holm equation for modeling and analyzing +the nonlinear shallow water waves : +mt + umx + 2uxm = 0, m = u − uxx. +(CH) +Indeed, when d = 1 the Euler-Poincar´e equations are the same as the Camassa-Holm equation +(CH). Also, the Euler-Poincar´e equations were investigated as the system describe geodesic motion +on the diffeomorphism group with respect to the kinetic energy norm in [16]. +∗E-mail: limin@jxufe.edu.cn +1 + +For d = 1, the equation (CH) was introduced by Camassa and Holm [5] as a bi-Hamiltonian +model for shallow water waves. Most importantly, CH equation has peakon solutions of the form +Ce−|x−Ct| which aroused a lot of interest in physics, see [8, 29]. There is an extensive literature +about the strong well-posedness, weak solutions and analytic or geometric properties of the CH +equation, here we name some. Local well-posedness and ill-posedness for the Cauchy problem +of the CH equation were investigated in [9, 12, 13]. Blow-up phenomena and global existence of +strong solutions were discussed in [7,9–11]. The existence of global weak solutions and dissipative +solutions were investigated in [3,4,30], more results can be found in the references therein. +The first rigorous analysis of the Euler-Poincar´e equations (1.1) was done by Chae and Liu [6], +they eatablished the local existence of weak solution in W 2,p(Rd), p > d and local existence of +unique classical solutions in Hs(Rd), s > +d +2 + 3. Yan and Yin [31] further discussed the local +existence and uniqueness of the solution to (1.1) in Besov spaces. On the other hand, Li, Yu and +Zhai [27] proved that the solutions to (1.1) with a large class of smooth initial data blows up in +finite time or exists globally in time, which settled an open problem raised by Chae and Liu [6]. +Later, Luo and Yin have obtained a new blow-up result in the periodic case by using the rotational +invariant properties of the equation [25]. For more results of Euler-Poincar´e equations, see [25,32]. +Recently, starting from the research of Himonas et al. [14,15], the continuity properties of the +data-to-solution maps of the Camassa-Holm type equations are gradually attracting interest of +many authors, see [21,23]. Most of the non-uniform constinuity results are established only on a +bounded set near the origin. To overcome this limitation, Inci obtained a series of nowhere uniform +continuity results including many Camassa-Holm type equations [19,20]. And for the incompress- +ible Euler equation, Bourgain and Li [2] showed that the data-to-solution map is nowhere-uniform +continuity in Hs(Rd) with s ≥ 0 by using an idea of localized Galilean boost, this method will +inspire us in this article. +As part of the well-posedness theory, the continuity properties of the data-to-solution map is +indeed very important. In fact, the non-uniform continuity of data-to-solution map suggests that +the local well-posedness cannot be established by the contraction mappings principle since this +would imply Lipschitz continuity for the solution map. On the other hand, in some critical spaces +the continuity of the data-to-solution maps are first broken before the existence and uniqueness of +the solution, which leads to ill-posedness [22]. +Most previous work on constinuity has focused on the spacial one-dimensional Camassa-Holm +type equations equations, for the multi-dimensional Euler-Poincar´e equations (1.1), the continu- +ity problem has not been thoroughly investigated. Until recently, Li et al. [24] shown that the +corresponding solution to (1.1) is not uniformly constinuous dependence for that the initial data +in Hs(Rd), s > 1 + d +2. Later, the non-uniformly constinuous result was extended to Besov space +Bs +p,r(Rd), s > max{1 + d +2, 3 +2} in [26]. +It is worth to mention that, the non-uniform constinuity results of (1.1) are established only +on a bounded set near the origin. +In this paper, we will remove the boundedness restriction +and prove that the data-to-solution map of the Euler-Poincar´e equations (1.1) is not uniformly +2 + +continuous on any open subset U ⊂ Bs +p,r(Rd), s > max{1 + d +2, 3 +2}. Technically, our proof based on +a symmetric form of the equation (1.1), and a translation method to construct perturbation data, +this method was introduced by Bourgain and Li [2] to proof the nowhere uniform constinuity of +the incompressible Euler equations. +For simplicity, we first transform Eq.(1.1) into a transport type system. According to Yan [31], +we can rewrite (1.1) to the following nonlocal form: +∂tu + u · ∇u = Q(u, u) + R(u, u), +(1.2) +where + + + +Q(u, v) = −(I − ∆)−1div +� +∇u∇v + ∇u∇vT − ∇uT∇v − ∇u(divv) + 1 +2(∇u : ∇v)I +� +, +R(u, v) = −(I − ∆)−1� +u divv + ∇uTv +� +. +(1.3) +We now define a symmetric bilinear operator T by +T (u, v) := 1 +2 +� +Q(u, v) + Q(v, u) + R(u, v) + R(v, u) +� += −(I − ∆)−1div +� +M(∇u, ∇v) +� +− (I − ∆)−1� +N(u, ∇u; v, ∇v) +� +, +(1.4) +here M, N are bilinear functions of (∇u, ∇v) and (u, ∇u; v, ∇v) respectively according to (1.3), +they are symmetric on u, v. Then, the Euler-Poincar´e equations becomes + + + +∂tu + u · ∇u = T (u, u), +(t, x) ∈ R+ × Rd, +u(0, x) = u0, +x ∈ Rd. +(E-P) +We first recall the non-uniform continuity results established in [26]. +Theorem 1.1 (Non-uniform continuity on a bounded set). Let d ≥ 2 and s > 2+max +� +1+ +d +p, 3 +2 +� +with 1 ≤ p, r ≤ ∞. The data-to-solution map St for Euler-Poincar´e equations (E-P) is +not uniformly continuous from any bounded subset ON = {u0 ∈ Bs +p,r(Rd) : ∥u0∥Bsp,r ≤ N} into +C([0, T]; Bs +p,r). More precisely, there exists two sequences of initial data fn + gn, fn such that +∥fn∥Bsp,r ≲ 1 +and +lim +n→∞ ∥gn∥Bsp,r = 0, +with the solutions St(fn + gn), St(fn) satisfy +lim inf +n→∞ ∥St(fn + gn) − St(fn)∥Bsp,r ≥ c0t, ∀t ∈ [0, T0], +for some constant c0 > 0 and small time T0. +The main result of this paper is the following theorem. +3 + +Theorem 1.2 (Nowhere uniform continuity). Assume that d ≥ 2, and +s > 2 + max +� +1 + d +p, 3 +2 +� +and +(p, r) ∈ (1, ∞) × [1, ∞). +(1.5) +Then the data-to-solution map St for Euler-Poincar´e equations for the Cauchy problem (E-P) +St : Bs +p,r(Rd) → C([0, T]; Bs +p,r), +u0 �→ St(u0), +is nowhere uniformly continuous from Bs +p,r into C([0, T]; Bs +p,r). More precisely, for any u0 ∈ Bs +p,r +and N > 0, there exists two sequences of functions fn(x), gn(x) such that +∥fn∥Bsp,r ≲ 2−N +and +lim +n→∞ ∥gn∥Bsp,r = 0, +the corresponding solutions St(fn + gn), St(fn) satisfy +lim inf +n→∞ ∥St(u0 + fn + gn) − St(u0 + fn)∥Bsp,r ≥ c0t, ∀t ∈ [0, T0], +for some constant c0 > 0 and small time T0. +Remark 1.1. As a comparison with Theorem 1.1, Theorem 1.2 avoids endpoints p = 1 and p = ∞, +this is because we need to use the boundedness of Riez transform in Lp(Rd) when doing gradient +estimate of T (see Lemma 3.2 blow), which is only available when p ∈ (1, ∞). +Remark 1.2. The non-uniform constinuity in Theorem 1.1 established only on a bounded set near +the origin, in Theorem 1.2 we have removed these restrictions and showed that for any u0 and any +neighbour U(u0) ⊂ Bs +p,r, the data-to-solution map restrict on U is not uniformly continuous. In +this sense, Theorem 1.2 improves the previous results in [26]. +The remainder of this paper is organized as follows. In Section 2, we list some notations and +recall basic results of the Littlewood-Paley theory. In Section 3, we present the proof of Theorem +1.2 by establishing some technical lemmas and propositions. +2 +Littlewood-Paley analysis +We first present some facts about the Littlewood-Paley decomposition, the nonhomogeneous Besov +spaces and their some useful properties (see [1] for more details). +Let B := {ξ ∈ Rd : |ξ| ≤ 4/3} and C := {ξ ∈ Rd : 3/4 ≤ |ξ| ≤ 8/3}. Choose a radial, non- +negative, smooth function χ : Rd �→ [0, 1] such that it is supported in B and χ ≡ 1 for |ξ| ≤ 3/4. +Setting ϕ(ξ) := χ(ξ/2) − χ(ξ), then we deduce that ϕ is supported in C. Moreover, +χ(ξ) + +� +j≥0 +ϕ(2−jξ) = 1 +for any ξ ∈ Rd. +We should emphasize that the fact ϕ(ξ) ≡ 1 for 4/3 ≤ |ξ| ≤ 3/2 will be used in the sequel. +4 + +For every u ∈ S′(Rd), the inhomogeneous dyadic blocks ∆j are defined as follows +∆ju = + + + + + + + + + +0, +if +j ≤ −2; +χ(D)u = F −1(χFu), +if +j = −1; +ϕ(2−jD)u = F −1� +ϕ(2−j·)Fu +� +, +if +j ≥ 0. +In the inhomogeneous case, the following Littlewood-Paley decomposition makes sense +u = +� +j≥−1 +∆ju +for any u ∈ S′(Rd). +Definition 2.1. labelbesov Let s ∈ R and (p, r) ∈ [1, ∞]2. The nonhomogeneous Besov space +Bs +p,r(Rd) is defined by +Bs +p,r(Rd) := +� +f ∈ S′(Rd) : ∥f∥Bsp,r(Rd) < ∞ +� +, +where +∥f∥Bsp,r(Rd) = + + + + + + + + + +� � +j≥−1 +2sjr∥∆jf∥r +Lp(Rd) +� 1 +r +, +if 1 ≤ r < ∞, +sup +j≥−1 +2sj∥∆jf∥Lp(Rd), +if r = ∞. +The following Bernstein’s inequalities will be used in the sequel. +Lemma 2.1. Let B be a Ball and C be an annulus. There exist constants C > 0 such that for all +k ∈ N ∪ {0}, any positive real number λ and any function f ∈ Lp(Rd) with 1 ≤ p ≤ q ≤ ∞, we +have +supp ˆf ⊂ λB ⇒ ∥Dkf∥Lq := sup +|α|=k +∥∂αf∥Lq ≤ Ck+1λk+( d +p − d +q )∥f∥Lp, +supp ˆf ⊂ λC ⇒ C−k−1λk∥f∥Lp ≤ ∥∆kf∥Lp ≤ Ck+1λk∥f∥Lp. +Lemma 2.2 (See [1]). Let (s1, s2, p, r) ∈ R2 × [1, ∞]2, and s1 < s2, 0 < θ < 1, then we have +∥u∥Bθs1+(1−θ)s2 +p,r +≤∥u∥θ +Bs1 +p,r∥u∥1−θ +Bs2 +p,r, +∥u∥Bθs1+(1−θ)s2 +p,1 +≤ +C +s2 − s1 +�1 +θ + +1 +1 − θ +� +∥u∥θ +Bs1 +p,∞∥u∥1−θ +Bs2 +p,∞. +Then, we give some important product estimates which will be used throughout the paper. +Lemma 2.3 (See [1]). For (p, r) ∈ [1, ∞]2 and s > 0, Bs +p,r(Rd) ∩ L∞(Rd) is an algebra. Moreover, +for any u, v ∈ Bs +p,r(Rd) ∩ L∞(Rd), we have +∥uv∥Bsp,r ≤ C(∥u∥Bsp,r∥v∥L∞ + ∥v∥Bsp,r∥u∥L∞). +In addition, if s > max +� +1 + d +p, 3 +2 +� +, then +∥uv∥Bs−2 +p,r (Rd) ≤ C∥u∥Bs−2 +p,r (Rd)∥v∥Bs−1 +p,r (Rd). +5 + +Lemma 2.4 (See [1,28]). Let (p, r) ∈ [1, ∞]2 and σ ≥ − min +�d +p, 1− d +p +� +. Assume that f0 ∈ Bσ +p,r(Rd), +g ∈ L1([0, T]; Bσ +p,r(Rd)) and ∇u ∈ L1([0, T]; Bσ−1 +p,r (Rd)) if σ > 1 + d +p or σ = 1 + d +p, r = 1. If +f ∈ L∞([0, T]; Bσ +p,r(Rd)) ∩ C([0, T]; S′(Rd)) solves the following linear transport equation: +∂tf + u · ∇f = g, +f|t=0 = f0. +1. There exists a constant C = C(σ, p, r) such that the following statement holds +∥f(t)∥Bσp,r ≤ eCV (t)� +∥f0∥Bσp,r + +� t +0 e−CV (τ)∥g(τ)∥Bσp,rdτ +� +, +where +V (t) = +� t +0 ∥∇u(τ)∥Bσ−1 +p,r dτ +if +σ > 1 + d +p +or +{σ = 1 + d +p, r = 1}. +2. If σ > 0, then there exists a constant C = C(σ, p, r) such that the following holds +∥f(t)∥Bσp,r ≤∥f0∥Bσp,r + +� t +0 ∥g(τ)∥Bσp,rdτ ++ +� t +0 +� +∥f(τ)∥Bσp,r∥∇u∥L∞ + ∥∇u∥Bσ−1 +p,r ∥∇f(τ)∥L∞ +� +dτ. +3 +Proof of the main theorem +We first recall the local existence and uniqueness theory of solutions for the Cauchy problem (1.1) +in Besov spaces [31], then provide some technical lemmas and propositions. +3.1 +Preparation and technical lemmas +Lemma 3.1 (See [31]). Assume that +d ∈ N+, 1 ≤ p, r ≤ ∞ and s > max{1 + d +p, 3 +2}. +(3.1) +Let u0 ∈ Bs +p,r(Rd), then there exists a time T = T(∥u0∥Bsp,r(Rd)) > 0 such that (1.1) has a unique +solution in + + + +C([0, T]; Bs +p,r(Rd)) ∩ C1([0, T]; Bs−1 +p,r (Rd)), +if +r < ∞, +L∞([0, T]; Bs +p,∞(Rd)) ∩ Lip([0, T]; Bs−1 +p,∞(Rd)), +if +r = ∞. +And the mapping u0 �→ u is continuous from Bs +p,r(Rd) into C([0, T]; Bs′ +p,r(Rd))∩C1([0, T]; Bs′−1 +p,r (Rd)) +for all s′ < s if r = ∞, and s′ = s otherwise. Moreover, for all t ∈ [0, T], there holds +∥u(t)∥Bsp,r(Rd) ≤ C∥u0∥Bsp,r(Rd). +Lemma 3.2. Let (s, p, r) satisfy (1.5), then for the symmetric bilinear operator T (f, g) defined by +(1.3) and (1.4), we have +∥T (f, g)∥Bsp,r ≤ C∥f∥Bsp,r∥g∥Bsp,r +(3.2) +6 + +If 0 < p < ∞, there holds +∥T (f, g)∥Lp ≤ ∥∇f∥Lp∥g, ∇g∥L∞ +(3.3) +∥T (f, g)∥Lp ≤ +� +0≤|a|,|b|≤1 +∥∂af∂bg∥Lp = W1,p(f, g) +(3.4) +And, for the gradient ∇T , we have +∥∇T (f, g)∥Lp ≤ ∥∇f∥Lp∥g, ∇g∥L∞ +(3.5) +∥∇T (f, g)∥Lp ≤ +� +0≤|a|,|b|≤2 +∥∂af∂bg∥Lp = W2,p(f, g) +(3.6) +Where we denote Wm,p(f, g) = +� +0≤|a|,|b|≤m +∥∂af∂bg∥Lp with the multiindex a = (a1, a2, · · · , ad), |a| = +a1 + · · · + ad and ∂a = +∂|a| +∂xa1 +1 ···∂x +ad +d . +Proof. As the operator (I − ∆)−1 is a Fourier S−2-multiplier, it’s easy to see that +∥T (f, g)∥Bsp,r ≤ C∥M(∇f, ∇g)∥Bs−1 +p,r + C∥N(f, ∇f, g, ∇g)∥Bs−2 +p,r ≤ C∥f∥Bsp,r∥g∥Bsp,r, +here we have use the Lemma 2.3. Then in Lp spaces, +∥T (f, g)∥Lp = ∥(I − ∆)−1div +� +M(∇f, ∇g) +� ++ (I − ∆)−1� +N(f, ∇f, g, ∇g) +� +∥Lp +≤ ∥M(∇f, ∇g)∥Lp + ∥N(f, ∇f, g, ∇g)∥Lp +≤ ∥∇f∥Lp∥∇g∥L∞ + ∥∇f∥Lp∥g∥L∞ +≤ ∥∇f∥Lp∥g, ∇g∥L∞ +we also have +∥T (f, g)∥Lp ≤ ∥M(∇f, ∇g)∥Lp + ∥N(f, ∇f, g, ∇g)∥Lp +≤ +� +0≤|a|,|b|≤1 +∥∂af∂bg∥Lp = W1,p(f, g) +For the gradient ∇T , noting that (I−∆)−1∂i∂j = −∆(I−∆)−1� +(−∆)−1∂i∂j +� += +� +(1−∆)−1+1 +� +RiRj +and the Riesz transform Ri is bounded in Lp → Lp, p ∈ (1, ∞), then we have +∥∇T (f, g)∥Lp = ∥∇(I − ∆)−1div +� +M(∇f, ∇g) +� ++ ∇(I − ∆)−1� +N(f, ∇f, g, ∇g) +� +∥Lp +≤ ∥M(∇f, ∇g)∥Lp + ∥N(f, ∇f, g, ∇g)∥Lp +≤ ∥∇f∥Lp∥∇g∥L∞ + ∥∇f∥Lp∥g∥L∞ +≤ ∥∇f∥Lp∥g, ∇g∥L∞ +and +∥∇T (f, g)∥Lp ≤ ∥divM(∇f, ∇g)∥Lp + ∥N(f, ∇f, g, ∇g)∥Lp +≤ +� +0≤|a|,|b|≤2 +∥∂af∂bg∥Lp = W2,p(f, g) +7 + +We’ll need the following estimates of the difference u(t) − v(t) in Besov spaces. +Proposition 3.1. Let 1 ≤ p, r ≤ ∞ and s > max{1 + d +p, 3 +2}. Assume that u(t), v(t) are solutions +of (E-P) with initial data (u0, v0) ∈ Bs +p,r(Rd), then δ(t) := u(t) − v(t) satisfies +∥δ(t)∥Bs−1 +p,r ≤ ∥δ0∥Bs−1 +p,r exp +� +C +� t +0 ∥u(τ), v(τ)∥Bsp,rdτ +� +and +∥δ(t)∥Bsp,r ≤ +� +∥δ0∥Bsp,r + C +� t +0 ∥δ∥Bs−1 +p,r ∥∇v∥Bsp,rdτ +� +exp +� +C +� t +0 ∥u(τ), v(τ)∥Bsp,rdτ +� +(3.7) +Proof. The first inequality has been proved in [31], it remains to prove (3.7). As T is a symmetric +bilinear operator, it’s easy to deduce that δ = u − v solves the transport equation +∂tδ + u · ∇δ = −δ · ∇v + T (δ, u + v). +(3.8) +Then, by Lemma 2.4 and 3.2 +∥δ(t)∥Bsp,r ≤∥δ0∥Bsp,r + C +� t +0 +� +∥u∥Bsp,r∥δ∥Bsp,r + ∥δ · ∇v∥Bsp,r + ∥T (δ, u + v)∥Bsp,r +� +dτ +≤∥δ0∥Bsp,r + C +� t +0 +� +∥u(τ), v(τ)∥Bsp,r∥δ∥Bsp,r + ∥δ∥Bs−1 +p,r ∥∇v∥Bsp,r +� +dτ. +now (3.7) is direct result from Gronwall’s inequality. +Proposition 3.2. Suppose �u(t), u(t), v(t) are the solutions of (E-P) of initial data u0 + v0, u0, v0 +respectively. Then, under the assumptions of (1.5), we have +∥�u − u − v∥Bsp,r ≤ C∥u0, v0∥1−θ +Bs+1 +p,∞ exp +� +∥u0, v0∥Bs+1 +p,r θ +�� � t +0 W2,p(u, v)dτ +�θ +, +where θ = +1 +s+1 and use the notation W2,p(u, v) = +� +0≤|a|,|b|≤2 +∥∂au∂bv∥Lp. +Proof. Since �u(t), u(t), v(t) are solutions of + + + + + + + + + +∂t�u + �u · ∇�u = T (�u, �u), +�u(0) = u0 + v0, +∂tu + u · ∇u = T (u, u), +u(0) = u0, +∂tv + v · ∇v = T (v, v), +v(0) = v0. +by the symmetry and linearity of T , we can deduce that w(t) = �u(t) − u(t) − v(t) satisfies + + + + + + + + + +∂tw + �u · ∇w = +−w · ∇(u + v) + T (w, �u + u + v) +−u · ∇v − v · ∇u − 2T (u, v), +w(0) = +0. +(3.9) +8 + +By the interpolation inequality (see Lemma 2.2 ), we obtain +∥w∥Bsp,r ≤ C∥w∥θ +B0p,∞∥w∥1−θ +Bs+1 +p,∞ ≤ ∥u0, v0∥1−θ +Bs+1 +p,∞∥w∥θ +Lp +(3.10) +The rest of the proof is to bound the Lp norm of w, taking the inner product of (3.9) with +�wp−1 := (|w1|p−2w1, |w2|p−2w2, · · · , |wd|p−2wd), we obtain +1 +p +d +dt∥w∥p +Lp = +d +� +i=1 +� +p−1|wi|p(div�u)dx − +� +i,j +� +�wp−1 +i +wj∂j(ui + vi)dx ++ +d +� +i=1 +� +�wp−1 +i +Ti(w, �u + u + v)dx − +d +� +i=1 +� +�wp−1 +i +(u · ∇vi + v · ∇ui + Ti(u, v)dx +≤1 +p∥div�u∥L∞∥w∥p +Lp + Cd(∥∇u∥L∞ + ∥∇v∥L∞)∥w∥p +Lp ++ ∥w∥p−1 +Lp ∥T (w, �u + u + v)∥Lp + ∥w∥p−1 +Lp ∥u · ∇v + v · ∇u + T (u, v)∥Lp +(3.11) +Thanks to the estimates of T in Lemma 3.2, in particular take (3.3), (3.4), into (3.11) we have +d +dt∥w∥Lp ≤ C(∥div�u∥L∞ + ∥∇u∥L∞ + ∥∇v∥L∞)∥w∥Lp ++ C(∥�u, ∇�u∥L∞ + ∥u, ∇u∥L∞ + ∥v, ∇v∥L∞)∥∇w∥Lp + W2,p(u, v) +≤ ∥u0, v0∥Bsp,r(∥w∥Lp + ∥∇w∥Lp) + W2,p(u, v) +Now, we should bound the gradient matrix ∇w, take the gradient to (3.9), then in components +∂t∂jwi = − �uk∂k∂jwi − ∂j�uk∂kwi + ∂jTi(w, �u + u + v) +− wk∂k(∂jui + ∂jvi) − ∂jwk∂k(ui + vi) − ∂j +� +uk∂kvi − vk∂kui − 2Ti(u, v) +� +Taking the L2 inner product with �wp−1 +i,j +:= |∂jwi|p−2∂jwi and sum the indices i, j, we get +1 +p +d +dt∥∇w∥p +Lp = +� +1≤i,j≤d +� +p−1|∂jwi|p(div�u)dx − +� +∇ �wp−1 : (∇w∇�u)dx + +� +∇ �wp−1 : ∇T (w, �u + u + v)dx +− +� +∇ �wp−1 : +� +w · ∇(∇u + ∇v) +� +dx − +� +∇ �wp−1 : +� +(∇u + ∇v)∇w +� +dx +− +� +∇ �wp−1 : ∇(u · ∇v + v · ∇u + 2T (u, v))dx +≤1 +p∥div�u∥L∞∥∇w∥p +Lp + Cd∥∇�u∥L∞∥∇w∥p +Lp + ∥∇w∥p−1 +Lp ∥∇T (w, �u + u + v)∥Lp ++ ∥∇w∥p−1 +Lp ∥w∥Lp� +∥∇2u∥L∞ + ∥∇2v∥L∞� ++ ∥∇w∥p +Lp +� +∥∇u∥L∞ + ∥∇v∥L∞� ++ |∇w∥p−1 +Lp ∥∇ +� +u · ∇v + v · ∇u + 2T (u, v) +� +∥Lp +(3.12) +where we denote ∇ �wp−1 = ( �wp−1 +i,j )d×d and A : B := � +i,j ai,jbi,j. Again using Proposition 3.2 for +9 + +the matrix operator ∇T , by plug (3.5),(3.6) into (3.12), we obtain +d +dt∥∇w∥Lp ≤ C(∥∇�u∥L∞ + ∥∇u∥L∞ + ∥∇v∥L∞)∥∇w∥Lp + ∥w∥Lp� +∥∇2u∥L∞ + ∥∇2v∥L∞� ++ ∥∇T (w, �u + u + v)∥Lp + ∥∇ +� +u · ∇v + v · ∇u + 2T (u, v) +� +∥Lp +≤ C(∥�u, ∇�u∥L∞ + ∥u, ∇u∥L∞ + ∥v, ∇v∥L∞)∥∇w∥Lp ++ +� +∥∇2u∥L∞ + ∥∇2v∥L∞� +∥w∥Lp + W2,p(u, v) +≤ C∥u0, v0∥Bs+1 +p,r (∥w∥Lp + ∥∇w∥Lp) + W2,p(u, v) +Combining (3.14) and (3.17) yields that +d +dt∥w, ∇w∥Lp ≤ C∥u0, v0∥Bs+1 +p,r (∥w∥Lp + ∥∇w∥Lp) + 2W2,p(u, v) +By Gronwall’s inequality and (3.10) we complete the proof. +Remark 3.1. The proofs of Proposition 3.1 and 3.2 rely on the symmetry of T , especially when it +comes to getting simplified equations (3.8) and (3.9). Most previous studies on the well-posedness of +Euler-Poincar´e equations use the bilinear form (1.2), the lack of symmetry makes the calculation +complicated. +Infact, when d = 1 namely the Camassa-Holm equation has the transport form +∂tu + u∂xu = P(u, u) with P(u, v) = −∂x(1 − ∂2 +x)−1� +uv + 1 +2(∂xu∂xv) +� +is symmetric by default. +In this respect, our new form (E-P) is a more natural high-dimensional generalization of the CH +equation. +3.2 +Construction of Perturbation Data +For localization in the Fourier domain, we introduce the following bump function in the frequency +space. Let �φ ∈ C∞ +0 (R) be a non-negative and even function satisfy +�φ(ξ) = +� +1, +if |ξ| ≤ 1 +4, +0, +if |ξ| ≥ 1 +2. +and let + + + +fn = 2−ns−N� +cos( 17 +122nx1)φ(x1)φ(x2) · · ·φ(xd), 0, · · · , 0 +� +gn = +� +2−nφ(x1)φ(x2) · · · φ(xd), 0, · · · , 0 +� +. +(3.13) +We define the perturbation data by adding a translation transform + + + +f m +n = fn(x1 − m, x2, · · · , xd) +gm +n = gn(x1 − m, x2, · · · , xd) +(3.14) +Noting that � +f m +n is supported in [−1 +2, 1 +2]d ± ( 17 +122n, 0, · · · , 0), this support set is completely covered +by the ring Cn = {ξ ∈ Rd : 4 +32n ≤ |ξ| ≤ 3 +22n}. Thus, by the definition of ∆j, we know +∆j(fn) = +�f m +n , +if j = n, +0, +if j ̸= n. +(3.15) +10 + +On account of above and the definition of Besov space, we can show that for k ∈ R +∥f m +n ∥Bs+k +p,r ≤ C2kn−N +and +∥gm +n ∥Bs+k +p,r → 0 +for n → ∞ +(3.16) +By the previous work [26] and translation invariance of the system (E-P), we know that, for the +corresponding solutions St(f m +n + gm +n ) and St(f m +n ) there is a positive constant c0 and a small time +T0, such that for any t ∈ [0, T0], +lim inf +n→∞ ∥St(f m +n + gm +n ) − St(f m +n )∥Bsp,r ≥ c0t. +(3.17) +3.3 +Proof of Theorem 1.2 +Roughly speaking, our proof of Theorem 1.2 based on the following approximation +St(u0 + f m +n + gm +n ) − St(u0 + f m +n ) +(I) += St(Snu0 + f m +n + gm +n ) − St(Snu0 + f m +n ) + Em +n +(II) += +� +St(Snu0) + St(f m +n + gm +n ) +� +− +� +St(Snu0) + St(f m +n ) +� ++ En,m += St(f m +n + gm +n ) − St(f m +n ) + En,m, +(III) +with some small error terms Em +n , En,m. More precisely, we devide (I) into three parts +St(u0 + f m +n + gm +n ) − St(u0 + f m +n ) = +� +St(u0 + f m +n + gm +n ) − St(Snu0 + f m +n + gm +n ) +� +− +� +St(u0 + f m +n ) − St(Snu0 + f m +n ) +� +� +�� +� +Em +n ++ +� +St(Snu0 + f m +n + gm +n ) − St(Snu0) − St(f m +n + gm +n ) +� +− +� +St(Snu0 + f m +n ) − St(Snu0) − St(f m +n ) +� +� +�� +� +En,m ++ St(f m +n + gm +n ) − St(f m +n ). +(3.18) +We proof the approximation (III) → (II) → (I) in the following sense. +Proposition 3.3. Let f m +n , gm +n be the perturbation data defined by (3.13) and (3.14), then for any +initial data u0 ∈ Bs +p,r with ∥u0∥Bsp,r = ρ, the error terms Em +n , En,m in (3.18) satisfy +sup +m,t ∥Em +n ∥Bsp,r ≤ Cρ∥(I − Sn)u0∥Bsp,r, +(3.19) +lim +m→∞ +� +sup +0≤t≤T +∥En,m∥Bsp,r +� += 0 +for any fixed n. +(3.20) +. +Proof. We first to handle (3.19). Using proposition 3.1 with δ(t) = St(u0 + f m +n ) − St(Snu0 + f m +n ), +as ∥u0 + f m +n ∥Bsp,r ≈ ∥Snu0 + f m +n ∥Bsp,r for m ∈ R and n ≫ 1, the solution sequences have a common +11 + +lifespan T ≈ T ∗(∥u0∥Bsp,r), then for any t ∈ [0, T) we have +∥δ(t)∥Bsp,r ≤ +� +∥(I − Sn)u0∥Bsp,r + +� t +0 ∥δ(τ)∥Bs−1 +p,r ∥∇St(Snu0 + f m +n )∥Bsp,rdτ +� +· exp +� � t +0 ∥St(u0 + f m +n ), St(Snu0 + f m +n )∥Bsp,rdτ +� +≤ +� +∥(I − Sn)u0∥Bsp,r + +� t +0 ∥δ(τ)∥Bs−1 +p,r ∥Snu0 + f m +n ∥Bs+1 +p,r dτ +� +· exp +� � t +0 ∥u0 + f m +n , Snu0 + f m +n ∥Bsp,rdτ +� +≤Cρ +� +∥(I − Sn)u0∥Bsp,r + +� t +0 ∥δ(τ)∥Bs−1 +p,r · 2ndτ +� +(3.21) +and +∥δ(t)∥Bs−1 +p,r ≤∥(I − Sn)u0∥Bs−1 +p,r exp +� � t +0 ∥St(u0 + f m +n ), St(Snu0 + f m +n )∥Bsp,rdτ +� +≤Cρ2−n∥(I − Sn)u0∥Bsp,r +(3.22) +take (3.22) into (3.21) we get +∥δ(t)∥Bsp,r ≤ Cρ∥(I − Sn)u0∥Bsp,r. +(3.23) +As in (3.23) the Cρ not depend on the translation parameter m and t ∈ [0, T], then we have +sup +m,t ∥St(u0 + f m +n ) − St(Snu0 + f m +n )∥Bsp,r = sup +m,t ∥δ(t)∥Bsp,r ≤ Cρ∥(I − Sn)u0∥Bsp,r. +(3.24) +With exactly the same argument, we can deduce that +sup +m,t ∥St(u0 + f m +n + gm +n ) − St(Snu0 + f m +n + gm +n )∥Bsp,r ≤ Cρ∥(I − Sn)u0∥Bsp,r, +along with (3.24), we complete the proof of (3.19). +In order to deduce (3.20), we should use Proposition 3.2 with the setting �u(t) = St(Snu0 + +f m +n ), u(t) = St(Snu0) and v(t) = St(f m +n ), and denote +w = �u − u − v = St(Snu0 + f m +n ) − St(Snu0) − St(f m +n ). +Since u0 ∈ Bs +p,r and ∥f m +n ∥Bsp,r ≈ 1, it’s easy to see that +∥Snu0, f m +n ∥Bs+1 +p,r ≤ Cρ2n, +(3.25) +with Cρ only depend on ρ := ∥u0∥Bsp,r, Then from Proposition 3.2 we know that +∥w(t)∥Bsp,r ≤ Cρ2neCρ2nθ� +� +0≤|a|,|b|≤2 +� t +0 ∥∂aSt(Snu0)∂bSt(f m +n )∥Lpdτ +�θ +. +(3.26) +Notice that, by definition ∂bSt(f m +n ) = ∂bSt(fn(x1 − m, · · · , xd)) = ∂bSt(fn)(x1 − m, · · · , xd), for +fixed n and any (t, x), considering that St(fn) is a smooth function decay at infinity, we have +lim +m→∞ ∂aSt(Snu0)(x)∂bSt(fn)(x1 − m, · · · , xd) = 0, +|∂aSt(Snu0)(x)∂bSt(fn)(x1 − m, · · · , xd)| ≤ M|∂aSt(Snu0)|(τ, x) ∈ L1� +[0, T], Lp(R) +� +. +12 + +By the Lebesgue Dominated Convergence Theorem, we have +lim +m→∞ +� T +0 ∥∂aSt(Snu0)∂bSt(f m +n )∥Lpdτ = 0 +(3.27) +Then from (3.26),(3.27) we know that, for the fixed n and any t ∈ [0, T] +lim +m→∞ sup +0≤t≤T +∥w(t)∥Bsp,r = lim +m→∞ sup +0≤t≤T +∥St(Snu0 + f m +n ) − St(Snu0) − St(f m +n )∥Bsp,r = 0, +(3.28) +with the same argument, we can also get that, for any fixed n and t ∈ [0, T] +lim +m→∞ sup +0≤t≤T +∥St(Snu0 + f m +n + gm +n ) − St(Snu0) − St(f m +n + gm +n )∥Bsp,r = 0. +(3.29) +Combining (3.28) and (3.29), this yields (3.20). +With (3.17), (3.18) and Propositions 3.3 in hand, we can complete our proof of Theorem 1.2. +First of all, from the identity (3.18) we know that for any time t ∈ [0, T] +∥St(u0 + f m +n + gm +n ) − St(u0 + f m +n )∥Bsp,r +≥ ∥St(f m +n + gm +n ) − St(f m +n )∥Bsp,r − sup +m,t ∥Em +n ∥Bsp,r − sup +0≤t≤T +∥En,m∥Bsp,r +(3.30) +Then, by (3.20) in Proposition 3.3, for the fixed n, we can find a sufficiently large mn such that +sup +0≤t≤T +∥En,mn∥Bsp,r ≤ 2−n. +combining this and (3.19) in Proposition 3.3, by (3.30) we get +∥St(u0 + f mn +n ++ gmn +n ) − St(u0 + f mn +n )∥Bsp,r +≥ ∥St(f mn +n ++ gmn +n ) − St(f mn +n )∥Bsp,r − Cρ∥(I − Sn)u0∥Bsp,r − 2−n. +(3.31) +As u0 ∈ Bs +p,r, that means ∥(I − Sn)u0∥Bsp,r → 0 when n → ∞. For the small t ∈ [0, T0], we already +have (3.17), it follows from (3.31) that +lim inf +n→∞ ∥St(u0 + f mn +n ++ gmn +n ) − St(u0 + f mn +n )∥Bsp,r ≥ c0t, +∀t ∈ [0, T0]. +(3.32) +And the sequences of initial data satisfy +lim +n→∞ ∥(u0 + f mn +n ++ gmn +n ) − (u0 + f mn +n )∥Bsp,r = lim +n→∞ ∥gmn +n ∥Bsp,r = 0. +(3.33) +This complete the proof Theorem 1.2. +Acknowledgements. M. Li was supported by Educational Commission Science Programm of +Jiangxi Province (No. +GJJ190284) and Natural Science Foundation of Jiangxi Province (No. +20212BAB211011 and 20212BAB201008). +13 + +References +[1] H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differen- +tial Equations, Grundlehren der Mathematischen Wissenschaften, vol. 343, Springer-Verlag, +Berlin, Heidelberg, 2011. +[2] J. Bourgain and D. Li, Galilean Boost and Non-uniform Continuity for Incompressible Euler, +Commun. Math. Phys., 372 (2019), 261–280. +[3] A. Bressan and A. 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Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, +Acta Math., 181 (1998), 229-243. +[11] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. +Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303-328. +[12] R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, +192 (2003), 429-444. +[13] Z. Guo, X. Liu, M. Luc and Z. Yin, Ill-posedness of the Camassa-Holm and related equations +in the critical space, J. Differential Equations, 266 (2019), 1698-1707. +[14] A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on +the line, Diff. Integral Eqns, 22 (2009), 201–224. +14 + +[15] A. Himonas, C. Kenig and Misio�lek Non-uniform dependence for the periodic CH equation, +Commun. Partial Diff. Eqns, 35 (2010), 1145–1162. +[16] D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolu- +tionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323–380. +[17] D. D. Holm, J.E. Marsden and T.S. Ratiu, Euler-Poincar´e models of ideal fluids with nonlinear +dispersion, Phys. Rev. Lett., 80 (2007), 4173–4177. +[18] D. D. Holm, J.E. Marsden and T.S. Ratiu, Euler-Poincar´e equations and semidirect products +with applications to continuum theories, Adv. Math., 137 (1998), 1–81. +[19] H. Inci, On the well-posedness of the Holm-Staley b-family of equations, Journal of Nonlinear +Mathematical Physics, 23 (2016), 213–233. +[20] H. Inci, On the local well-posedness of the two component b-family of equations, Monatsh. +Math., 197 (2022), 479–492. +[21] J. Li, Y. Yu and W. Zhu, Non-uniform dependence on initial data for the Camassa-Holm +equation in Besov spaces, J. Differ. Equ., 269 (2020), 8686–8700. +[22] J. Li, Y. Yu and W. Zhu, Ill-posedness for the Camassa-Holm and related equations in Besov +spaces, J. Differ. Equ., 306 (2022), 403–417. +[23] J. Li, X. Wu, Y. Yu and W. Zhu , Non-uniform dependence on initial data for the Camassa- +Holm equation in the critical Besov space, J. Math. Fluid Mech., 23 (2021), 1422-6928. +[24] J. Li, L. Dai and W. Zhu, Non-uniform continuous dependence on initial data of solutions to +the Euler-Poincar´e system, J. Math. Phys., 60 (2019), 111510, 9. +[25] W. Luo and Z. Yin, Blow-up phenomena, ill-posedness and peakon solutions for the periodic +Euler-Poincar´e equations, J. of Differential Equations, 268 (2020), 1307–1325. +[26] J. Li, W. Deng and M. Li , Non-uniform dependence for Euler-Poincar´e equations in Besov +spaces, Nonlinear Analysis: Real World Applications, 63 (2022), 103420. +[27] D. Li, X. Yu and Z. Zhai, On the Euler-Poincare equation with non-zero dispersion, Arch. +Ration. Mech. Anal., 210 (2013), 955-974. +[28] J. Li and Z. Yin, Remarks on the well-posedness of Camassa-Holm type equations in Besov +spaces, J. Differential Equations, 261 (2016), 6125-6143. +[29] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. +[30] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. +Math., 53 (2000), 1411-1433. +15 + +[31] K. Yan and Z. Yin, On the initial value problem for Euler-Poincar´e equations, Discrete Contin. +Dyn. Syst., 35 (2015), 1327-1358. +[32] Y. Zhao, M. Yang and Y. Li, Non-uniform dependence for the periodic Euler-Poincar´e equa- +tions, J. Math. Anal. Appl., 461 (2018), 59-73. +16 + diff --git a/OtE1T4oBgHgl3EQfuAVi/content/tmp_files/load_file.txt b/OtE1T4oBgHgl3EQfuAVi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1cf4d69bc9c43e47d6bf8423c2628ae6d80d0f82 --- /dev/null +++ b/OtE1T4oBgHgl3EQfuAVi/content/tmp_files/load_file.txt @@ -0,0 +1,586 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf,len=585 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='03383v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='AP] 9 Nov 2022 On the continuity of the solution map of the Euler-Poincar´e equations in Besov spaces Min Li1 ∗ 1 Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Jiangxi University of Finance and Economics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Nanchang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' 330032,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' China Abstract: By constructing a series of perturbation functions through localization in the Fourier domain and using a symmetric form of the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' we show that the data-to-solution map for the Euler-Poincar´e equations is nowhere uniformly continuous in Bs p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='r(Rd) with s > max{1 + d 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' 3 2} and (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' r) ∈ (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∞) × [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' This improves our previous result which shows the data-to-solution map for the Euler-Poincar´e equations is non-uniformly continuous on a bounded subset of Bs p,r(Rd) near the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Keywords: Euler-Poincar´e equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Nowhere uniformly continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Besov spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Data-to- solution map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' MSC (2010): 35Q35, 35Q51, 35L30 1 Introduction In this paper, we consider the Cauchy problem in Rd for Euler-Poincar´e equations \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∂tm + u · ∇m + ∇uT · m + (divu)m = 0, (t, x) ∈ R+ × Rd, m = (1 − ∆)u, (t, x) ∈ R+ × Rd, u(0, x) = u0, x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1) The equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1) were first introduced by Holm, Marsden, and Ratiu in [17, 18] as a high dimensional generalization of the following Camassa-Holm equation for modeling and analyzing the nonlinear shallow water waves : mt + umx + 2uxm = 0, m = u − uxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (CH) Indeed, when d = 1 the Euler-Poincar´e equations are the same as the Camassa-Holm equation (CH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Also, the Euler-Poincar´e equations were investigated as the system describe geodesic motion on the diffeomorphism group with respect to the kinetic energy norm in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∗E-mail: limin@jxufe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='cn 1 For d = 1, the equation (CH) was introduced by Camassa and Holm [5] as a bi-Hamiltonian model for shallow water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Most importantly, CH equation has peakon solutions of the form Ce−|x−Ct| which aroused a lot of interest in physics, see [8, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' There is an extensive literature about the strong well-posedness, weak solutions and analytic or geometric properties of the CH equation, here we name some.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Local well-posedness and ill-posedness for the Cauchy problem of the CH equation were investigated in [9, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Blow-up phenomena and global existence of strong solutions were discussed in [7,9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' The existence of global weak solutions and dissipative solutions were investigated in [3,4,30], more results can be found in the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' The first rigorous analysis of the Euler-Poincar´e equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1) was done by Chae and Liu [6], they eatablished the local existence of weak solution in W 2,p(Rd), p > d and local existence of unique classical solutions in Hs(Rd), s > d 2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Yan and Yin [31] further discussed the local existence and uniqueness of the solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1) in Besov spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' On the other hand, Li, Yu and Zhai [27] proved that the solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1) with a large class of smooth initial data blows up in finite time or exists globally in time, which settled an open problem raised by Chae and Liu [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Later, Luo and Yin have obtained a new blow-up result in the periodic case by using the rotational invariant properties of the equation [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' For more results of Euler-Poincar´e equations, see [25,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Recently, starting from the research of Himonas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' [14,15], the continuity properties of the data-to-solution maps of the Camassa-Holm type equations are gradually attracting interest of many authors, see [21,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Most of the non-uniform constinuity results are established only on a bounded set near the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' To overcome this limitation, Inci obtained a series of nowhere uniform continuity results including many Camassa-Holm type equations [19,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' And for the incompress- ible Euler equation, Bourgain and Li [2] showed that the data-to-solution map is nowhere-uniform continuity in Hs(Rd) with s ≥ 0 by using an idea of localized Galilean boost, this method will inspire us in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' As part of the well-posedness theory, the continuity properties of the data-to-solution map is indeed very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' In fact, the non-uniform continuity of data-to-solution map suggests that the local well-posedness cannot be established by the contraction mappings principle since this would imply Lipschitz continuity for the solution map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' On the other hand, in some critical spaces the continuity of the data-to-solution maps are first broken before the existence and uniqueness of the solution, which leads to ill-posedness [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Most previous work on constinuity has focused on the spacial one-dimensional Camassa-Holm type equations equations, for the multi-dimensional Euler-Poincar´e equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1), the continu- ity problem has not been thoroughly investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Until recently, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' [24] shown that the corresponding solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1) is not uniformly constinuous dependence for that the initial data in Hs(Rd), s > 1 + d 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Later, the non-uniformly constinuous result was extended to Besov space Bs p,r(Rd), s > max{1 + d 2, 3 2} in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' It is worth to mention that, the non-uniform constinuity results of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1) are established only on a bounded set near the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' In this paper, we will remove the boundedness restriction and prove that the data-to-solution map of the Euler-Poincar´e equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1) is not uniformly 2 continuous on any open subset U ⊂ Bs p,r(Rd), s > max{1 + d 2, 3 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Technically, our proof based on a symmetric form of the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1), and a translation method to construct perturbation data, this method was introduced by Bourgain and Li [2] to proof the nowhere uniform constinuity of the incompressible Euler equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' For simplicity, we first transform Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1) into a transport type system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' According to Yan [31], we can rewrite (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1) to the following nonlocal form: ∂tu + u · ∇u = Q(u, u) + R(u, u), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2) where \uf8f1 \uf8f2 \uf8f3 Q(u, v) = −(I − ∆)−1div � ∇u∇v + ∇u∇vT − ∇uT∇v − ∇u(divv) + 1 2(∇u : ∇v)I � , R(u, v) = −(I − ∆)−1� u divv + ∇uTv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='3) We now define a symmetric bilinear operator T by T (u, v) := 1 2 � Q(u, v) + Q(v, u) + R(u, v) + R(v, u) � = −(I − ∆)−1div � M(∇u, ∇v) � − (I − ∆)−1� N(u, ∇u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' v, ∇v) � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='4) here M, N are bilinear functions of (∇u, ∇v) and (u, ∇u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' v, ∇v) respectively according to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='3), they are symmetric on u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Then, the Euler-Poincar´e equations becomes \uf8f1 \uf8f2 \uf8f3 ∂tu + u · ∇u = T (u, u), (t, x) ∈ R+ × Rd, u(0, x) = u0, x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (E-P) We first recall the non-uniform continuity results established in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1 (Non-uniform continuity on a bounded set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Let d ≥ 2 and s > 2+max � 1+ d p, 3 2 � with 1 ≤ p, r ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' The data-to-solution map St for Euler-Poincar´e equations (E-P) is not uniformly continuous from any bounded subset ON = {u0 ∈ Bs p,r(Rd) : ∥u0∥Bsp,r ≤ N} into C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Bs p,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' More precisely, there exists two sequences of initial data fn + gn, fn such that ∥fn∥Bsp,r ≲ 1 and lim n→∞ ∥gn∥Bsp,r = 0, with the solutions St(fn + gn), St(fn) satisfy lim inf n→∞ ∥St(fn + gn) − St(fn)∥Bsp,r ≥ c0t, ∀t ∈ [0, T0], for some constant c0 > 0 and small time T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' The main result of this paper is the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 (Nowhere uniform continuity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Assume that d ≥ 2, and s > 2 + max � 1 + d p, 3 2 � and (p, r) ∈ (1, ∞) × [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='5) Then the data-to-solution map St for Euler-Poincar´e equations for the Cauchy problem (E-P) St : Bs p,r(Rd) → C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Bs p,r), u0 �→ St(u0), is nowhere uniformly continuous from Bs p,r into C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Bs p,r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' More precisely, for any u0 ∈ Bs p,r and N > 0, there exists two sequences of functions fn(x), gn(x) such that ∥fn∥Bsp,r ≲ 2−N and lim n→∞ ∥gn∥Bsp,r = 0, the corresponding solutions St(fn + gn), St(fn) satisfy lim inf n→∞ ∥St(u0 + fn + gn) − St(u0 + fn)∥Bsp,r ≥ c0t, ∀t ∈ [0, T0], for some constant c0 > 0 and small time T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' As a comparison with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 avoids endpoints p = 1 and p = ∞, this is because we need to use the boundedness of Riez transform in Lp(Rd) when doing gradient estimate of T (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 blow), which is only available when p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' The non-uniform constinuity in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1 established only on a bounded set near the origin, in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 we have removed these restrictions and showed that for any u0 and any neighbour U(u0) ⊂ Bs p,r, the data-to-solution map restrict on U is not uniformly continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' In this sense, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 improves the previous results in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' In Section 2, we list some notations and recall basic results of the Littlewood-Paley theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' In Section 3, we present the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 by establishing some technical lemmas and propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' 2 Littlewood-Paley analysis We first present some facts about the Littlewood-Paley decomposition, the nonhomogeneous Besov spaces and their some useful properties (see [1] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Let B := {ξ ∈ Rd : |ξ| ≤ 4/3} and C := {ξ ∈ Rd : 3/4 ≤ |ξ| ≤ 8/3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Choose a radial, non- negative, smooth function χ : Rd �→ [0, 1] such that it is supported in B and χ ≡ 1 for |ξ| ≤ 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Setting ϕ(ξ) := χ(ξ/2) − χ(ξ), then we deduce that ϕ is supported in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Moreover, χ(ξ) + � j≥0 ϕ(2−jξ) = 1 for any ξ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' We should emphasize that the fact ϕ(ξ) ≡ 1 for 4/3 ≤ |ξ| ≤ 3/2 will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' 4 For every u ∈ S′(Rd), the inhomogeneous dyadic blocks ∆j are defined as follows ∆ju = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 0, if j ≤ −2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' χ(D)u = F −1(χFu), if j = −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ϕ(2−jD)u = F −1� ϕ(2−j·)Fu � , if j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' In the inhomogeneous case, the following Littlewood-Paley decomposition makes sense u = � j≥−1 ∆ju for any u ∈ S′(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' labelbesov Let s ∈ R and (p, r) ∈ [1, ∞]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' The nonhomogeneous Besov space Bs p,r(Rd) is defined by Bs p,r(Rd) := � f ∈ S′(Rd) : ∥f∥Bsp,r(Rd) < ∞ � , where ∥f∥Bsp,r(Rd) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 � � j≥−1 2sjr∥∆jf∥r Lp(Rd) � 1 r , if 1 ≤ r < ∞, sup j≥−1 2sj∥∆jf∥Lp(Rd), if r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' The following Bernstein’s inequalities will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Let B be a Ball and C be an annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' There exist constants C > 0 such that for all k ∈ N ∪ {0}, any positive real number λ and any function f ∈ Lp(Rd) with 1 ≤ p ≤ q ≤ ∞, we have supp ˆf ⊂ λB ⇒ ∥Dkf∥Lq := sup |α|=k ∥∂αf∥Lq ≤ Ck+1λk+( d p − d q )∥f∥Lp, supp ˆf ⊂ λC ⇒ C−k−1λk∥f∥Lp ≤ ∥∆kf∥Lp ≤ Ck+1λk∥f∥Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 (See [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Let (s1, s2, p, r) ∈ R2 × [1, ∞]2, and s1 < s2, 0 < θ < 1, then we have ∥u∥Bθs1+(1−θ)s2 p,r ≤∥u∥θ Bs1 p,r∥u∥1−θ Bs2 p,r, ∥u∥Bθs1+(1−θ)s2 p,1 ≤ C s2 − s1 �1 θ + 1 1 − θ � ∥u∥θ Bs1 p,∞∥u∥1−θ Bs2 p,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Then, we give some important product estimates which will be used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='3 (See [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' For (p, r) ∈ [1, ∞]2 and s > 0, Bs p,r(Rd) ∩ L∞(Rd) is an algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Moreover, for any u, v ∈ Bs p,r(Rd) ∩ L∞(Rd), we have ∥uv∥Bsp,r ≤ C(∥u∥Bsp,r∥v∥L∞ + ∥v∥Bsp,r∥u∥L∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' In addition, if s > max � 1 + d p, 3 2 � , then ∥uv∥Bs−2 p,r (Rd) ≤ C∥u∥Bs−2 p,r (Rd)∥v∥Bs−1 p,r (Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' 5 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='4 (See [1,28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Let (p, r) ∈ [1, ∞]2 and σ ≥ − min �d p, 1− d p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Assume that f0 ∈ Bσ p,r(Rd), g ∈ L1([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Bσ p,r(Rd)) and ∇u ∈ L1([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Bσ−1 p,r (Rd)) if σ > 1 + d p or σ = 1 + d p, r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' If f ∈ L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Bσ p,r(Rd)) ∩ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' S′(Rd)) solves the following linear transport equation: ∂tf + u · ∇f = g, f|t=0 = f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' There exists a constant C = C(σ, p, r) such that the following statement holds ∥f(t)∥Bσp,r ≤ eCV (t)� ∥f0∥Bσp,r + � t 0 e−CV (τ)∥g(τ)∥Bσp,rdτ � , where V (t) = � t 0 ∥∇u(τ)∥Bσ−1 p,r dτ if σ > 1 + d p or {σ = 1 + d p, r = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' If σ > 0, then there exists a constant C = C(σ, p, r) such that the following holds ∥f(t)∥Bσp,r ≤∥f0∥Bσp,r + � t 0 ∥g(τ)∥Bσp,rdτ + � t 0 � ∥f(τ)∥Bσp,r∥∇u∥L∞ + ∥∇u∥Bσ−1 p,r ∥∇f(τ)∥L∞ � dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' 3 Proof of the main theorem We first recall the local existence and uniqueness theory of solutions for the Cauchy problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1) in Besov spaces [31], then provide some technical lemmas and propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1 Preparation and technical lemmas Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1 (See [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Assume that d ∈ N+, 1 ≤ p, r ≤ ∞ and s > max{1 + d p, 3 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1) Let u0 ∈ Bs p,r(Rd), then there exists a time T = T(∥u0∥Bsp,r(Rd)) > 0 such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1) has a unique solution in \uf8f1 \uf8f2 \uf8f3 C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Bs p,r(Rd)) ∩ C1([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Bs−1 p,r (Rd)), if r < ∞, L∞([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Bs p,∞(Rd)) ∩ Lip([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Bs−1 p,∞(Rd)), if r = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' And the mapping u0 �→ u is continuous from Bs p,r(Rd) into C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Bs′ p,r(Rd))∩C1([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Bs′−1 p,r (Rd)) for all s′ < s if r = ∞, and s′ = s otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Moreover, for all t ∈ [0, T], there holds ∥u(t)∥Bsp,r(Rd) ≤ C∥u0∥Bsp,r(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Let (s, p, r) satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='5), then for the symmetric bilinear operator T (f, g) defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='4), we have ∥T (f, g)∥Bsp,r ≤ C∥f∥Bsp,r∥g∥Bsp,r (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2) 6 If 0 < p < ∞, there holds ∥T (f, g)∥Lp ≤ ∥∇f∥Lp∥g, ∇g∥L∞ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='3) ∥T (f, g)∥Lp ≤ � 0≤|a|,|b|≤1 ∥∂af∂bg∥Lp = W1,p(f, g) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='4) And, for the gradient ∇T , we have ∥∇T (f, g)∥Lp ≤ ∥∇f∥Lp∥g, ∇g∥L∞ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='5) ∥∇T (f, g)∥Lp ≤ � 0≤|a|,|b|≤2 ∥∂af∂bg∥Lp = W2,p(f, g) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='6) Where we denote Wm,p(f, g) = � 0≤|a|,|b|≤m ∥∂af∂bg∥Lp with the multiindex a = (a1, a2, · · · , ad), |a| = a1 + · · · + ad and ∂a = ∂|a| ∂xa1 1 ···∂x ad d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' As the operator (I − ∆)−1 is a Fourier S−2-multiplier, it’s easy to see that ∥T (f, g)∥Bsp,r ≤ C∥M(∇f, ∇g)∥Bs−1 p,r + C∥N(f, ∇f, g, ∇g)∥Bs−2 p,r ≤ C∥f∥Bsp,r∥g∥Bsp,r, here we have use the Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Then in Lp spaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∥T (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' g)∥Lp = ∥(I − ∆)−1div � M(∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g) � + (I − ∆)−1� N(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g) � ∥Lp ≤ ∥M(∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g)∥Lp + ∥N(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g)∥Lp ≤ ∥∇f∥Lp∥∇g∥L∞ + ∥∇f∥Lp∥g∥L∞ ≤ ∥∇f∥Lp∥g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g∥L∞ we also have ∥T (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' g)∥Lp ≤ ∥M(∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g)∥Lp + ∥N(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g)∥Lp ≤ � 0≤|a|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='|b|≤1 ∥∂af∂bg∥Lp = W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='p(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' g) For the gradient ∇T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' noting that (I−∆)−1∂i∂j = −∆(I−∆)−1� (−∆)−1∂i∂j � = � (1−∆)−1+1 � RiRj and the Riesz transform Ri is bounded in Lp → Lp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' p ∈ (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' then we have ∥∇T (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' g)∥Lp = ∥∇(I − ∆)−1div � M(∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g) � + ∇(I − ∆)−1� N(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g) � ∥Lp ≤ ∥M(∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g)∥Lp + ∥N(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g)∥Lp ≤ ∥∇f∥Lp∥∇g∥L∞ + ∥∇f∥Lp∥g∥L∞ ≤ ∥∇f∥Lp∥g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g∥L∞ and ∥∇T (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' g)∥Lp ≤ ∥divM(∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g)∥Lp + ∥N(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' ∇g)∥Lp ≤ � 0≤|a|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='|b|≤2 ∥∂af∂bg∥Lp = W2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='p(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' g) 7 We’ll need the following estimates of the difference u(t) − v(t) in Besov spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Let 1 ≤ p, r ≤ ∞ and s > max{1 + d p, 3 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Assume that u(t), v(t) are solutions of (E-P) with initial data (u0, v0) ∈ Bs p,r(Rd), then δ(t) := u(t) − v(t) satisfies ∥δ(t)∥Bs−1 p,r ≤ ∥δ0∥Bs−1 p,r exp � C � t 0 ∥u(τ), v(τ)∥Bsp,rdτ � and ∥δ(t)∥Bsp,r ≤ � ∥δ0∥Bsp,r + C � t 0 ∥δ∥Bs−1 p,r ∥∇v∥Bsp,rdτ � exp � C � t 0 ∥u(τ), v(τ)∥Bsp,rdτ � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' The first inequality has been proved in [31], it remains to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' As T is a symmetric bilinear operator, it’s easy to deduce that δ = u − v solves the transport equation ∂tδ + u · ∇δ = −δ · ∇v + T (δ, u + v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='8) Then, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 ∥δ(t)∥Bsp,r ≤∥δ0∥Bsp,r + C � t 0 � ∥u∥Bsp,r∥δ∥Bsp,r + ∥δ · ∇v∥Bsp,r + ∥T (δ, u + v)∥Bsp,r � dτ ≤∥δ0∥Bsp,r + C � t 0 � ∥u(τ), v(τ)∥Bsp,r∥δ∥Bsp,r + ∥δ∥Bs−1 p,r ∥∇v∥Bsp,r � dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' now (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='7) is direct result from Gronwall’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Suppose �u(t), u(t), v(t) are the solutions of (E-P) of initial data u0 + v0, u0, v0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Then, under the assumptions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='5), we have ∥�u − u − v∥Bsp,r ≤ C∥u0, v0∥1−θ Bs+1 p,∞ exp � ∥u0, v0∥Bs+1 p,r θ �� � t 0 W2,p(u, v)dτ �θ , where θ = 1 s+1 and use the notation W2,p(u, v) = � 0≤|a|,|b|≤2 ∥∂au∂bv∥Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Since �u(t), u(t), v(t) are solutions of \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∂t�u + �u · ∇�u = T (�u, �u), �u(0) = u0 + v0, ∂tu + u · ∇u = T (u, u), u(0) = u0, ∂tv + v · ∇v = T (v, v), v(0) = v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' by the symmetry and linearity of T , we can deduce that w(t) = �u(t) − u(t) − v(t) satisfies \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∂tw + �u · ∇w = −w · ∇(u + v) + T (w, �u + u + v) −u · ∇v − v · ∇u − 2T (u, v), w(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='9) 8 By the interpolation inequality (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 ), we obtain ∥w∥Bsp,r ≤ C∥w∥θ B0p,∞∥w∥1−θ Bs+1 p,∞ ≤ ∥u0, v0∥1−θ Bs+1 p,∞∥w∥θ Lp (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='10) The rest of the proof is to bound the Lp norm of w, taking the inner product of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='9) with �wp−1 := (|w1|p−2w1, |w2|p−2w2, · · · , |wd|p−2wd), we obtain 1 p d dt∥w∥p Lp = d � i=1 � p−1|wi|p(div�u)dx − � i,j � �wp−1 i wj∂j(ui + vi)dx + d � i=1 � �wp−1 i Ti(w, �u + u + v)dx − d � i=1 � �wp−1 i (u · ∇vi + v · ∇ui + Ti(u, v)dx ≤1 p∥div�u∥L∞∥w∥p Lp + Cd(∥∇u∥L∞ + ∥∇v∥L∞)∥w∥p Lp + ∥w∥p−1 Lp ∥T (w, �u + u + v)∥Lp + ∥w∥p−1 Lp ∥u · ∇v + v · ∇u + T (u, v)∥Lp (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='11) Thanks to the estimates of T in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2, in particular take (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='4), into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='11) we have d dt∥w∥Lp ≤ C(∥div�u∥L∞ + ∥∇u∥L∞ + ∥∇v∥L∞)∥w∥Lp + C(∥�u, ∇�u∥L∞ + ∥u, ∇u∥L∞ + ∥v, ∇v∥L∞)∥∇w∥Lp + W2,p(u, v) ≤ ∥u0, v0∥Bsp,r(∥w∥Lp + ∥∇w∥Lp) + W2,p(u, v) Now, we should bound the gradient matrix ∇w, take the gradient to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' then in components ∂t∂jwi = − �uk∂k∂jwi − ∂j�uk∂kwi + ∂jTi(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' �u + u + v) − wk∂k(∂jui + ∂jvi) − ∂jwk∂k(ui + vi) − ∂j � uk∂kvi − vk∂kui − 2Ti(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' v) � Taking the L2 inner product with �wp−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='j := |∂jwi|p−2∂jwi and sum the indices i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' we get 1 p d dt∥∇w∥p Lp = � 1≤i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='j≤d � p−1|∂jwi|p(div�u)dx − � ∇ �wp−1 : (∇w∇�u)dx + � ∇ �wp−1 : ∇T (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' �u + u + v)dx − � ∇ �wp−1 : � w · ∇(∇u + ∇v) � dx − � ∇ �wp−1 : � (∇u + ∇v)∇w � dx − � ∇ �wp−1 : ∇(u · ∇v + v · ∇u + 2T (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' v))dx ≤1 p∥div�u∥L∞∥∇w∥p Lp + Cd∥∇�u∥L∞∥∇w∥p Lp + ∥∇w∥p−1 Lp ∥∇T (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' �u + u + v)∥Lp + ∥∇w∥p−1 Lp ∥w∥Lp� ∥∇2u∥L∞ + ∥∇2v∥L∞� + ∥∇w∥p Lp � ∥∇u∥L∞ + ∥∇v∥L∞� + |∇w∥p−1 Lp ∥∇ � u · ∇v + v · ∇u + 2T (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' v) � ∥Lp (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='12) where we denote ∇ �wp−1 = ( �wp−1 i,j )d×d and A : B := � i,j ai,jbi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Again using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 for 9 the matrix operator ∇T , by plug (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='5),(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='6) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='12), we obtain d dt∥∇w∥Lp ≤ C(∥∇�u∥L∞ + ∥∇u∥L∞ + ∥∇v∥L∞)∥∇w∥Lp + ∥w∥Lp� ∥∇2u∥L∞ + ∥∇2v∥L∞� + ∥∇T (w, �u + u + v)∥Lp + ∥∇ � u · ∇v + v · ∇u + 2T (u, v) � ∥Lp ≤ C(∥�u, ∇�u∥L∞ + ∥u, ∇u∥L∞ + ∥v, ∇v∥L∞)∥∇w∥Lp + � ∥∇2u∥L∞ + ∥∇2v∥L∞� ∥w∥Lp + W2,p(u, v) ≤ C∥u0, v0∥Bs+1 p,r (∥w∥Lp + ∥∇w∥Lp) + W2,p(u, v) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='17) yields that d dt∥w, ∇w∥Lp ≤ C∥u0, v0∥Bs+1 p,r (∥w∥Lp + ∥∇w∥Lp) + 2W2,p(u, v) By Gronwall’s inequality and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='10) we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' The proofs of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 rely on the symmetry of T , especially when it comes to getting simplified equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Most previous studies on the well-posedness of Euler-Poincar´e equations use the bilinear form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2), the lack of symmetry makes the calculation complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Infact, when d = 1 namely the Camassa-Holm equation has the transport form ∂tu + u∂xu = P(u, u) with P(u, v) = −∂x(1 − ∂2 x)−1� uv + 1 2(∂xu∂xv) � is symmetric by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' In this respect, our new form (E-P) is a more natural high-dimensional generalization of the CH equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 Construction of Perturbation Data For localization in the Fourier domain, we introduce the following bump function in the frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Let �φ ∈ C∞ 0 (R) be a non-negative and even function satisfy �φ(ξ) = � 1, if |ξ| ≤ 1 4, 0, if |ξ| ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' and let \uf8f1 \uf8f2 \uf8f3 fn = 2−ns−N� cos( 17 122nx1)φ(x1)φ(x2) · · ·φ(xd), 0, · · · , 0 � gn = � 2−nφ(x1)φ(x2) · · · φ(xd), 0, · · · , 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='13) We define the perturbation data by adding a translation transform \uf8f1 \uf8f2 \uf8f3 f m n = fn(x1 − m, x2, · · · , xd) gm n = gn(x1 − m, x2, · · · , xd) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='14) Noting that � f m n is supported in [−1 2, 1 2]d ± ( 17 122n, 0, · · · , 0), this support set is completely covered by the ring Cn = {ξ ∈ Rd : 4 32n ≤ |ξ| ≤ 3 22n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Thus, by the definition of ∆j, we know ∆j(fn) = �f m n , if j = n, 0, if j ̸= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='15) 10 On account of above and the definition of Besov space, we can show that for k ∈ R ∥f m n ∥Bs+k p,r ≤ C2kn−N and ∥gm n ∥Bs+k p,r → 0 for n → ∞ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='16) By the previous work [26] and translation invariance of the system (E-P), we know that, for the corresponding solutions St(f m n + gm n ) and St(f m n ) there is a positive constant c0 and a small time T0, such that for any t ∈ [0, T0], lim inf n→∞ ∥St(f m n + gm n ) − St(f m n )∥Bsp,r ≥ c0t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='17) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='3 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 Roughly speaking, our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 based on the following approximation St(u0 + f m n + gm n ) − St(u0 + f m n ) (I) = St(Snu0 + f m n + gm n ) − St(Snu0 + f m n ) + Em n (II) = � St(Snu0) + St(f m n + gm n ) � − � St(Snu0) + St(f m n ) � + En,m = St(f m n + gm n ) − St(f m n ) + En,m, (III) with some small error terms Em n , En,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' More precisely, we devide (I) into three parts St(u0 + f m n + gm n ) − St(u0 + f m n ) = � St(u0 + f m n + gm n ) − St(Snu0 + f m n + gm n ) � − � St(u0 + f m n ) − St(Snu0 + f m n ) � � �� � Em n + � St(Snu0 + f m n + gm n ) − St(Snu0) − St(f m n + gm n ) � − � St(Snu0 + f m n ) − St(Snu0) − St(f m n ) � � �� � En,m + St(f m n + gm n ) − St(f m n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='18) We proof the approximation (III) → (II) → (I) in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Let f m n , gm n be the perturbation data defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='14), then for any initial data u0 ∈ Bs p,r with ∥u0∥Bsp,r = ρ, the error terms Em n , En,m in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='18) satisfy sup m,t ∥Em n ∥Bsp,r ≤ Cρ∥(I − Sn)u0∥Bsp,r, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='19) lim m→∞ � sup 0≤t≤T ∥En,m∥Bsp,r � = 0 for any fixed n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='20) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' We first to handle (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Using proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='1 with δ(t) = St(u0 + f m n ) − St(Snu0 + f m n ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' as ∥u0 + f m n ∥Bsp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='r ≈ ∥Snu0 + f m n ∥Bsp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='r for m ∈ R and n ≫ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' the solution sequences have a common 11 lifespan T ≈ T ∗(∥u0∥Bsp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' then for any t ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' T) we have ∥δ(t)∥Bsp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='r ≤ � ∥(I − Sn)u0∥Bsp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='r + � t 0 ∥δ(τ)∥Bs−1 p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='r ∥∇St(Snu0 + f m n )∥Bsp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='rdτ � exp � � t 0 ∥St(u0 + f m n ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' St(Snu0 + f m n )∥Bsp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='rdτ � ≤ � ∥(I − Sn)u0∥Bsp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='r + � t 0 ∥δ(τ)∥Bs−1 p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='r ∥Snu0 + f m n ∥Bs+1 p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='r dτ � exp � � t 0 ∥u0 + f m n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Snu0 + f m n ∥Bsp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='rdτ � ≤Cρ � ∥(I − Sn)u0∥Bsp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='r + � t 0 ∥δ(τ)∥Bs−1 p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='r · 2ndτ � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='21) and ∥δ(t)∥Bs−1 p,r ≤∥(I − Sn)u0∥Bs−1 p,r exp � � t 0 ∥St(u0 + f m n ), St(Snu0 + f m n )∥Bsp,rdτ � ≤Cρ2−n∥(I − Sn)u0∥Bsp,r (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='22) take (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='22) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='21) we get ∥δ(t)∥Bsp,r ≤ Cρ∥(I − Sn)u0∥Bsp,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='23) As in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='23) the Cρ not depend on the translation parameter m and t ∈ [0, T], then we have sup m,t ∥St(u0 + f m n ) − St(Snu0 + f m n )∥Bsp,r = sup m,t ∥δ(t)∥Bsp,r ≤ Cρ∥(I − Sn)u0∥Bsp,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='24) With exactly the same argument, we can deduce that sup m,t ∥St(u0 + f m n + gm n ) − St(Snu0 + f m n + gm n )∥Bsp,r ≤ Cρ∥(I − Sn)u0∥Bsp,r, along with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='24), we complete the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' In order to deduce (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='20), we should use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 with the setting �u(t) = St(Snu0 + f m n ), u(t) = St(Snu0) and v(t) = St(f m n ), and denote w = �u − u − v = St(Snu0 + f m n ) − St(Snu0) − St(f m n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Since u0 ∈ Bs p,r and ∥f m n ∥Bsp,r ≈ 1, it’s easy to see that ∥Snu0, f m n ∥Bs+1 p,r ≤ Cρ2n, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='25) with Cρ only depend on ρ := ∥u0∥Bsp,r, Then from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2 we know that ∥w(t)∥Bsp,r ≤ Cρ2neCρ2nθ� � 0≤|a|,|b|≤2 � t 0 ∥∂aSt(Snu0)∂bSt(f m n )∥Lpdτ �θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='26) Notice that, by definition ∂bSt(f m n ) = ∂bSt(fn(x1 − m, · · · , xd)) = ∂bSt(fn)(x1 − m, · · · , xd), for fixed n and any (t, x), considering that St(fn) is a smooth function decay at infinity, we have lim m→∞ ∂aSt(Snu0)(x)∂bSt(fn)(x1 − m, · · · , xd) = 0, |∂aSt(Snu0)(x)∂bSt(fn)(x1 − m, · · · , xd)| ≤ M|∂aSt(Snu0)|(τ, x) ∈ L1� [0, T], Lp(R) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' 12 By the Lebesgue Dominated Convergence Theorem, we have lim m→∞ � T 0 ∥∂aSt(Snu0)∂bSt(f m n )∥Lpdτ = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='27) Then from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='26),(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='27) we know that, for the fixed n and any t ∈ [0, T] lim m→∞ sup 0≤t≤T ∥w(t)∥Bsp,r = lim m→∞ sup 0≤t≤T ∥St(Snu0 + f m n ) − St(Snu0) − St(f m n )∥Bsp,r = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='28) with the same argument, we can also get that, for any fixed n and t ∈ [0, T] lim m→∞ sup 0≤t≤T ∥St(Snu0 + f m n + gm n ) − St(Snu0) − St(f m n + gm n )∥Bsp,r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='29) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='29), this yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' With (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='17), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='18) and Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='3 in hand, we can complete our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' First of all, from the identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='18) we know that for any time t ∈ [0, T] ∥St(u0 + f m n + gm n ) − St(u0 + f m n )∥Bsp,r ≥ ∥St(f m n + gm n ) − St(f m n )∥Bsp,r − sup m,t ∥Em n ∥Bsp,r − sup 0≤t≤T ∥En,m∥Bsp,r (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='30) Then, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='20) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='3, for the fixed n, we can find a sufficiently large mn such that sup 0≤t≤T ∥En,mn∥Bsp,r ≤ 2−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' combining this and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='19) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='3, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='30) we get ∥St(u0 + f mn n + gmn n ) − St(u0 + f mn n )∥Bsp,r ≥ ∥St(f mn n + gmn n ) − St(f mn n )∥Bsp,r − Cρ∥(I − Sn)u0∥Bsp,r − 2−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='31) As u0 ∈ Bs p,r, that means ∥(I − Sn)u0∥Bsp,r → 0 when n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' For the small t ∈ [0, T0], we already have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='17), it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='31) that lim inf n→∞ ∥St(u0 + f mn n + gmn n ) − St(u0 + f mn n )∥Bsp,r ≥ c0t, ∀t ∈ [0, T0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='32) And the sequences of initial data satisfy lim n→∞ ∥(u0 + f mn n + gmn n ) − (u0 + f mn n )∥Bsp,r = lim n→∞ ∥gmn n ∥Bsp,r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='33) This complete the proof Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfuAVi/content/2301.03383v1.pdf'} +page_content=' Li was supported by Educational Commission Science Programm of Jiangxi Province (No.' metadata={'source': 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self-energy +H. R. de Assis∗1 and B. F. Rizzuti†2 +1Departamento de Matemática, ICE, Universidade Federal de Juiz de Fora, MG, +Brazil +2Departamento de Física, ICE, Universidade Federal de Juiz de Fora, MG, Brazil +Abstract +This paper is devoted to analyze the divergence of the electron self- +energy in classical electrodynamics. To do so, we develop the basics on +the theory of distributions and a method for obtaining corresponding ex- +tensions. At first sight, electrostatics implies a divergence once we treat +the electron as a charged point particle. However, our construction shows +that its self-energy turns out to be an undetermined constant upon renor- +malization. Appealing to empirical results we may fix it, demanding, for +example, that all its mass comes from an electrostatic origin. +Keywords: Theory of distributions. Extension of distributions. Elec- +tron self-energy. +1 +Introduction +One of the most compatible matches between theory and experiment in +physics is devoted to Quantum Electrodynamics (QED), as the precision on +the electron magnetic moment goes far from expected [1]. Since its early days +on QED computations, it became clear that the behavior of fields was more +singular than usual functions. In turn, this has led the community to stare at +fields not as maps, but as distributions. In fact, for the case of the electric field +⃗E(⃗x, t) originated from a point particle, for instance, one would expect an ultra- +violet divergence at origin, while +� +d3⃗xdt ⃗E(⃗x, t)f(⃗x) = ⃗E(f) is well behaved [2]. +Here, f(·) is a smooth function of compact support. We are interested, in this +manuscript, on the self-energy of the electron. While it has a fascinating history +so depicted in [3], involving an entire war and a new generation of physicists +developing regularization and renormalization techniques, the classical counter- +part is often subdue, justifying our approach here. +∗Corresponding author: heitor.ribeiro@estudante.ufjf.br +†Contact: brunorizzuti@ice.ufjf.br +1 + +Simply put, the self-energy of a charged particle, such as the electron, is the +measure of the energy which it has when freed from any other interaction, be it +with other particles or with given fields. One finds in the study of classical elec- +trodynamics that, summed to the kinetic and potential energies given particles +might have, a system composed of charged particles has a quantity of energy +related to the electromagnetic field it generates [4]. +The electromagnetic system we wish to examine could be seen, at first, as +the simplest one: that of an electron, stationary, free from any other interaction. +Seen as a point particle - that is, supposing it has no internal structure and can +be solely described by the position in which its whole charge is stored - which +is the standard way one encounters at first [4,5], the electric field and potential +are given by +E(r) = +1 +4πǫ0 +e +r2 ˆr, +V (r) = +1 +4πǫ0 +e +r , +(1) +where e denotes the strength of its charge. +Meanwhile, the expression for the self-energy for a system with electric field +E and magnetic field B is +E = ǫ0 +2 +� +R3 +� +E2 + c2B2� +dτ, +(2) +so that, using (1), we obtain +E0 = ǫ0 +2 +� +R3 +� +1 +4πǫ0 +�2 � e +r2 +�2 +dτ. +We denote E0 the self-energy of interest here, as we are allegedly neglecting +the magnetic field due to our interest only in the static case. Using spherical +coordinates,1 +E0 = +e2 +(4πǫ0)2 +�� 2π +0 +dφ +� �� π +0 +sen(θ)dθ +� �� +∞ +0 +1 +r2 dr +� += +e2 +8πǫ0 +�1 +r +�0 ++∞ += +∞. +(3) +The conclusion we arrive then is that there is an infinite amount of energy +stored in the field of a simple electron positioned at the origin of our system, +if one considers it to be a stationary point particle. Needless to say, any sat- +isfactory field theory, both classical or quantum, must resolve such type of di- +vergences. Should we discard the assumption that the electron has no spatial +1Since different materials might use different notations concerning the polar coordinates θ +and φ, we make explicit that we are considering here + + + + + +x = r senθ cos φ, +y = r senθ senφ, +z = r cos θ, +where θ ∈ [0, π], +φ ∈ [0, 2π). +2 + +extension? Both theoretical and experimental results seem to point in an oppo- +site way [6,7], indicating that we should seek a improvement in the theory and +in the conception of self-energy itself. +Therefore, here we present one of the available methods for the “removal” +of such infinite quantities, a process known as renormalization [8]. The main +idea of renormalization is that these infinities can be justified by attributing +them to quantities which we cannot directly measure (something that can be +seen as a parallel with the acceptance of complex numbers in the formalism of +quantum mechanics). Take, for example, our case of the electron and its infi- +nite self-energy. In calculating E0, we are, simultaneously, calculating its mass +melec arising from the electric field of such particle, since Einstein’s relativity +theory affirms that mass and energy are but two manifestations of the same +phenomenon. In light of this, we conclude that the divergence of E0 implies +that the electron possesses infinite inertia. If, however, we assume there exists +another contribution for the effective mass (that is, the one we can actually mea- +sure), originated from some unknown effect other than electromagnetism, then +we might conceive that this new contribution is negative enough to oppose the +infinite appearing from melec. The hypothesis of another source contributing +for the effective mass is not something difficultly justified, since we know that +neutral bodies are also provided with mass and generate no electric or magnetic +field. Thus, assuming this new contribution for the mass of the particle, inde- +pendent of where it comes from, we can “erase” the infinite we have just found, +obtaining the so called mass renormalization of the electron. +Such initial method of renormalization gave rise to new and more advanced +approaches which came to be used in the renormalization of other infinite quan- +tities, specially in the Quantum Field Theory (QFT), which advanced quite +a lot in the decades following the emergence of quantum mechanics and pre- +sented similar problems with divergent integrals in its equations [9]. For this +new theory, the methods had to be refined in its mathematical formulation and, +meanwhile, the development of approaches such as constructive quantum field +theory [2] and causal perturbation theory [10, 11] made clear that distribution +theory is a crucial subject for the understanding of such new ideas. It was also +through the latter that it was perceived the connection between the appearance +of divergent quantities and the product of distributions, whereas distribution +theory is strictly linear, as we shall see. This obstacle can be overcome using +the idea of extension of distributions, which we will encounter in Section 3. +What we shall present here is a direct consequence of the work of Epstein and +Glaser [11] and texts which adapted their ideas [12–15]. +In view of this, we seek, in the present work, to develop the theory of dis- +tributions, elaborating next a method for extension certain distributions and, +finally, exemplifying how it can be used in the renormalization of the self-energy +of the electron which we have just introduced. All this chain was written to be +both self-consistent and also, a pedestrian introduction to the theme. +3 + +2 +Distributions +Looking through the available literature, we may find different approaches to +the theory of distributions, ranging from superficial ones [2,4] to more advanced +and detailed studies on the subject [16–18]. Here, we shall confine ourselves +to a more peripheral point of view, since the reader interested in our work is +expected to be, at a certain level, familiar with these ideas. +Moreover, the +available literature is rich enough to permit the search for the missing gaps we +might leave along the way. In any case, the manuscript contains the standard +definitions/results on the subject and a couple of examples. They are intended +to make the text as self-consistent as possible. +2.1 +Space of test functions +First of all, we must make clear our notation for derivations, since we shall +deal quite often with multi-variable functions. For the euclidean space Rn, we +will consider the norm ∥·∥ : Rn → R+ to be +∥x∥ = +� +x2 +1 + x2 +2 + · · · + x2n, +arising from the scalar product +⟨x, y⟩ = x1y1 + x2y2 + · · · + xnyn, +and the topology will be the one derived from the metric our norm provides. +We define a multi-index β as an n-tuple (β1, · · · , βn) of natural numbers +(also written β ∈ Nn +0 ). Given a multi-index β and a function ϕ : Rn → C, we +shall denote the series of partial derivations as follows +Dβϕ(x) := +∂|β|ϕ +∂xβ1 +1 · · · ∂xβn +n +(x), +when ϕ is such that this new function exists. Here, |β| is the order of β, given by +|β| = β1 + · · · + βn. Furthermore, we define the factorial of β as β! = β1! · · · βn!, +notation that will come at hand later. +As elements of the n-dimensional space of natural numbers, β ∈ Nn +0, two +multi-index can be added and produce a new one, α+β = (α1+β1, · · · , αn+βn), +from which we shall obtain the operation Dα(Dβϕ) = D(α+β)ϕ. For functions +which are smooth enough to permit changes in the order of derivations, and +this will be always the case here, this operation is guaranteed to be associative. +Moreover, we are able to define, over Nn +0, a notion of partial order, establishing +α ≤ β when αi ≤ βi for all i ∈ {1, · · · , n}. From two multi-index, α and β, we +set +min{α, β} = (min{α1, β1}, · · · , min{αn, βn}) +and analogously for max{α, β}. +4 + +Remark 2.1. As is set by the literature, for the multi-index α = (0, · · · , 0), we +appoint Dαϕ = ϕ. +Remark 2.2. It is possible to demonstrate the interesting parallel between +Leibniz rule applied to multi-variable derivations and the Newton’s binomial +expansion +Dβ(ϕψ) = +� +0≤α≤β +β! +α!(β − α)!Dα(ϕ)Dβ−α(ψ). +Example 2.1. As an example of the use of the multi-index notation in dealing +with smooth functions, consider two multi-indices β, α and let f ∈ C∞(Rn) be +the function +f(x) = xβ = xβ1 +1 · · · xβn +n . +In this case, +Dαf(x) = +� ∂α1 +∂xα1 +1 +xβ1 +1 +� � ∂α2 +∂xα2 +2 +xβ2 +2 +� +· · · +� ∂αn +∂xαn +n xβn +n +� +. +We can thus see that, if αi > βi for some i ∈ {1, 2, · · · , n}, then the deivation +of order αi of the xβi +i +term will result in the null function, meaning Dβf = 0. +If, on the other hand, we have α ≤ β, then +Dβf(x) = +� +β1! +(β1 − α1)!xβ1−α1 +1 +� � +β2! +(β2 − α2)!xβ2−α2 +2 +� +· · · +� +βn! +(βn − αn)!xβn−αn +n +� +, +or, in other words, +Dβf(x) = +β! +(β − α)!xβ−α. +In particular, +(Dαf)(0) = +� +α!, +se α = β, +0, +se α ̸= β. +(4) +As we mentioned briefly in the last paragraph, we will always be considering +complex functions which are smooth enough. More precisely, we shall work in +a subspace of the vector space of infinitely differentiable functions, C∞(Rn), +namely the subspace of functions which vanish outside some compact K ⊂ Rn. +To make our words more exact and mathematically grounded, we define first +the concept of the support of a function ϕ ∈ C∞(Rn). That is the smallest +closed set containing all the points where ϕ does not vanish, in other words, the +closure of {x ∈ Rn ; ϕ(x) ̸= 0}. We shall denote the support of ϕ by supp ϕ. +We may now define in precise terms the space of functions we deal with when +introducing distributions. +Definition 2.1. We define D(Rn), the space of test functions, as the space +containing elements ϕ : Rn → C of C∞(Rn), meaning infinitely diferentiable +functions, whose support is compact. Thus, we can write +D(Rn) := {ϕ ∈ C∞(Rn) ; +supp ϕ is a compact set}. +5 + +We encounter no difficulties when trying to prove that D(Rn) is indeed +a vector space. The sum of two functions ϕ, ψ ∈ D(Rn), with support K1 = +supp ϕ and K2 = supp ψ, will have its support contained in the set K = K1∪K2, +which will again be compact. Besides that, it is trivial the fact that the support +of the function zϕ is equal to supp ϕ, for every z ̸= 0, from which we see that +the multiplication by a scalar will be closed in D(Rn). +The reader familiar with the area of functional analysis may recognize D(Rn) +by the name of C∞ +0 (Rn) and might question our choice of symbols. Our notation +is justified, however, by the notion of convergence we shall impose over D(Rn), +something that is not touched upon when the focus is other than the theory +of distributions. Here, a sequence (ϕk)k∈N of functions ϕk ∈ D(Rn) is said to +converge to a function ϕ ∈ D(Rn) when +• There exists a compact set K ⊂ Rn and a natural number k0 such that, +for every k ≥ k0, we have supp ϕk ⊂ K. +• For every multi-index β, Dβϕk converges uniformly to Dβϕ in K. +Example 2.2. Consider h > 0 and let ψh : (−h, h) −→ R be given by ψh(x) = +1 +x2−h2 . First of all, we observe that +lim +x→h− ψh(x) = lim +x→h− +x2 +(h/x)2 − 1 = +∞ +(5) +and the same will happen limx→−h+ ψh(x) = +∞, by reasons of symmetry. +Moreover, in the interval of definition, ψh is smooth, since it is the composi- +tion of smooth functions. For that reason, ξh(x) = eψh(x) shall also belong to +C∞((−h, h)) and, by (5), limx→h− ξh(x) = limx→−h+ ξh(x) = 0. +Taking the first derivative of ξ, we obtain +ξ′ +h(x) = ψ′ +h(x)eψh(x) = − +� +2x +(h2 − x2)2 +� +eψh(x), +whose limits in −h and h will both be zero,2 +lim +x→h− ξ′(x) = − lim +x→h− +� +2x +(h2 − x2)2 +� +eψh(x) = 0. +Taking one more derivative, we find +ξ′′ +h(x) = +� +4x2 +(h2 − x2)4 − +8x2 +(h2 − x2)3 − +2 +(h2 − x2)2 +� +eψh(x). +(6) +Once again, taking the limits x → h− e x → −h+, we shall have again zero, +since the exponential term will dominate any polynomial term in the denomi- +nator. Generally speaking, we can extend this result to any derivative of ξh(x), +lim +x→−h+ ξ(k) +h (x) = (−1)k lim +x→h− ξ(k) +h (x) = 0, +(7) +2From the fact that ψh is an even function, ψ′ +h will be odd, so that limx→−h+ ξ′(x) = +− limx→h− ξ′(x) = 0. +6 + +which permits us to define the following test function ηh ∈ D(R) +ηh(x) = +� +0, +if |x| ≥ h, +ξh(x), +if |x| ≤ h, +(8) +whose support is, by definition, +supp ηh = [−h, h]. +By itself, Example 2.2 should be interesting enough to give us the general +look of test functions. Besides that, we can use ηh to construct new test functions +which will be even more useful ahead. +Example 2.3. Let M, h > 0 be any two positive constants. We define the test +function ηM,h ∈ D(R) by simply translating and stretching of the function given +in (8), +ηM,h(x) = +� +0, +if x /∈ [M, M + h], +1 +Ch ξh(2(x − M − h)), +if x ∈ [M, M + h], +(9) +where Ch is a constant of normalization, i.e., +Ch = +� ∞ +−∞ +ξh(2(t − M − h))dt, +which implies +� ∞ +−∞ +ηM,h(t)dt = 1. +As we said, we have translated and stretched the support of our test function, +so that now we have supp ηM,h = [M, M + h]. With this, when considering the +integral +� x +−∞ +ηM,h(t)dt, +we obtain zero if x ≤ M and unity when x ≥ M + h. Another fact easily seen is +that this function is also infinitely differentiable. We cannot say, however, that +ηM,h belongs to D(R), since it does not possesses compact support, the same +happening to 1 − ηM,h. If, on the other hand, we consider Rn and take the +radial coordinate ∥x∥ = +� +x2 +1 + · · · + x2n as input, we can define ζM,h : Rn → R +which will be a test function. Subsequently, we define ζM,h by +ζM,h(x) = 1 − +� ∥x∥ +−∞ +ηM,h(t)dt, +(10) +obtaining ζM,h infinitely differentiable and with support +supp ζM,h = BM+h(0), +where BM+h(0) denotes the closed ball centered at the origin and with radius +M + h. From that, it follows that ζM,h ∈ D(Rn). +7 + +The importance of this example is better appreciated when we consider the +product ζM,hg, where we can, for now, consider g to be only locally integrable. +In this case, we obtain a function which is equal to g in BM(0) and zero outside +the ball BM+h(0), also satisfying |ζM,h(x)g(x)| ≤ |g(x)| for all x ∈ Rn. Since, +in our definition of ζM,h, M and h were any positive real numbers, we can make +the domain of coincidence of ζM,hg and g as big as we want to and the ring +BM+h(0)\BM(0) as narrow as we wish. Therefore, if we integrate this product +and take the limit h → 0, we have, by the Lebesgue Dominated Convergence +Theorem (see [19]), +lim +h→0+ +� +Rn ζM,h(x)g(x)dnx = +� +Rn +� +lim +h→0 ζM,h(x)g(x) +� +dnx += +� +Rn g(x)χBM (0)(x)dnx, +(11) +where χBM(0)(x) is the characteristic function of the ball BM(0), defined by +χBM(0)(x) = +� +1 , +if x ∈ BM(0), +0 , +if x /∈ BM(0). +(12) +In other words, +lim +h→0+ +� +Rn ζM,h(x)g(x)dnx = +� +BM(0) +g(x)dnx. +This fact will shall be important later ahead. +We have now all the necessary ingredients to define our main objects of +study. +Definition 2.2. A distribution is an element of D′(Rn), meaning it is a linear +continuous functional, in the sense of convergence defined in D(Rn).3 +Remark 2.3. The action of a given distribution T ∈ D′(Rn) over an element +ϕ of D(Rn) may be written in different ways and here we shall denote it by +ϕ �−→ ⟨T, ϕ⟩ = T (ϕ), +referencing the inner product notation. This convention will be justified in more +details below. +2.2 +Distributions +Given a first look at Definition 2.2, it may not be clear how one can say +that distributions are the generalization of locally integrable functions. We can, +3As usual, D′(Rn) stands for the dual topological space of D(Rn). +8 + +however, demonstrate that it is so by working on our first example of an element +of D′(Rn). Given a function f ∈ L1(Rn), let us define the following operation, +for each ϕ ∈ D(Rn) , +Tf(ϕ) = +� +Rn f(x)ϕ(x) dnx. +(13) +The condition that f be locally integrable is clearly seen to be necessary for +this application to be well defined. Since ϕ vanishes outside some compact set, +the integration is only performed in this set and since it is infinitely differentiable, +ϕf will also be locally integrable. +We affirm that expression (13) defines a distribution. Indeed, the operations +of multiplication by f and integration are well known to be linear, resulting in +the linearity of the composition of both. For the continuity, taking a sequence +(ϕk)k∈N ⊂ D(Rn) converging to ϕ ∈ D(Rn), it follows from our notion of +convergence in D(Rn) that there exists a compact set K containing supp (ϕk−ϕ), +for all k sufficiently big. Furthermore, since this convergence is uniform, we have, +for every ε > 0 and every x ∈ K, there exists k0 ∈ N such that, for k ≥ k0, +|ϕk(x)| ≤ |ϕ(x)| + |ϕk(x) − ϕ(x)| ≤ C + ε, +where C = sup {|ϕ(x)| , x ∈ K}. Thus, |fϕk| is bounded by (C + ε) |f| for all +k ≥ k0 and, since this is an uniform bound in K, we can apply the Dominated +Convergence Theorem to conclude that +lim +k→∞⟨Tf, ϕk⟩ = lim +k→∞ +� +Rn f(x)ϕk(x)dnx = +� +Rn +� +lim +k→∞ f(x)ϕk(x) +� +dnx += +� +Rn f(x)ϕ(x)dnx. +In other words, we have obtained +lim +k→∞⟨Tf, ϕk⟩ = ⟨Tf, ϕ⟩, +proving that Tf is indeed a linear continuous functional over D(Rn). +Such distributions, characterized by a locally integrable function f, are called +regular distributions. +This gives us the idea that we can construct, from a +function f ∈ L1(Rn), a distribution Tf ∈ D′(Rn). Still, it does not tells yet how +we can view such functions as proper elements of D′(Rn). Could we, for instance, +have different functions representing the same regular distribution? The answer +is yes and is justified by the fact that changing the quantity inside an integral +in a discrete set does not alter its value. More generally, any change in a null +measure set leaves the integral unaltered. This fact implies that two functions +which are equal almost everywhere4 define the same regular distribution. We +can prove, however, that this is the only way we get this coincidence. +Lemma 2.1. Two functions f, g ∈ L1 +loc(Rn) define the same distribution, in +the sense that Tf = Tg, if and only if f is equal to g almost everywhere. +4With that we mean that they are equal outside a null measure set. +9 + +Proof. (⇐) Firstly, if A is the null set where f differs from g and B = Rn\A = +Rn ∩ Ac, then for each ϕ ∈ D(Rn), +⟨Tf, ϕ⟩ = +� +Rn f(x)ϕ(x)dnx = +� +A +f(x)ϕ(x)dnx + +� +B +f(x)ϕ(x)dnx += 0 + +� +B +f(x)ϕ(x)dnx += +� +B +g(x)ϕ(x)dnx = ⟨Tg, ϕ⟩, +(14) +where we have used f = g in B. Since this is valid for every ϕ ∈ D(Rn), we +achieve Tf = Tg. +(⇒) If we suppose now that f and g define the same distribution, then +⟨f, ϕ⟩ = ⟨g, ϕ⟩ for every test function ϕ ∈ D(Rn). In particular, we can take +ϕ = ζM,h ∈ D(Rn) as constructed in Example 2.3, being define by the following: +0 ≤ ζM,h(x) ≤ 1 for all x ∈ Rn; ζM,h(x) = 1 for x in the ball centered at x0 +and of radius M > 0; ζM,h(x) = 0 for x outside the ball centered at x0 and of +radius M + h > 0, with h > 0. Therefore, we have +⟨f, ζM,h⟩ = ⟨g, ζM,h⟩ ⇒ +� +Rn f(x)ζM,h(x)dnx = +� +Rn g(x)ζM,h(x)dnx. +By the conclusion of Example 2.3, taking the limit h → 0 we obtain at last +� +BM(x0) +f(x)dnx = +� +BM(x0) +g(x)dnx, +which in turn implies that f and g are equal outside a null measure set in +BM(x0) (see, for example, Corollary 4.10 in [19]). Since x0 is arbitrary and +we can cover Rn with a countable number of balls of radius M, we obtain the +desired result. +In view of Lemma 2.1, if we consider elements of L1 +loc(Rn) to be the equiv- +alence classes,5 defined by the requirement that two functions are equivalent +if, and only if, they are equal almost everywhere, then we shall have an one- +to-one correspondence between regular distributions and elements f ∈ L1(Rn). +For this reason, it does not clouds our understanding when we refer to a reg- +ular distribution the function which characterizes it, justifying also the name +generalized functions, sometimes attributed to distributions. +Regular distributions are not, however, the only class of elements in D′(Rn). +Those which are not defined as in (13) by some f ∈ L1(Rn) are called singular +distributions. +Example 2.4. The first example of a singular distribution we can give is the +famous Dirac delta distribution δ, which can finally put a well defined meaning +to symbols such as δ(x). We define the distribution δ ∈ D′(Rn) by the expression +δ(ϕ) = ϕ(0), +∀ ϕ ∈ D(Rn). +(15) +5Still denoting them by f ∈ L1(Rn), meaning we identify each class with one of its repre- +sentatives. +10 + +If ϕ, ψ are functions in D(Rn) and z ∈ C is a complex number, then +δ(zϕ + ψ) = (zϕ + ψ)(0) = zϕ(0) + ψ(0) = zδ(ϕ) + δ(ψ). +Besides that, if ϕk → ϕ in D(Rn), then the point convergence ϕk(0) → ϕ(0) +is promptly assured, so that ⟨δ, ϕk⟩ → ⟨δ, ϕ⟩. With this, we prove the continuity +of δ. +Remark 2.4. To leave things clear, the notation δ(x) or expressions of the +form +δ(ϕ) = +� +∞ +−∞ +δ(x)ϕ(x)dx +(16) +become only abuses of notation. It is not possible to define a legitimate function +δ : R −→ R such that δ = Tδ(x), making (16) nothing more than a sometimes +useful convention on notation. Indeed, if we suppose there exists such function +δ(x), then we can show that, in R\{0}, it must be equal to the null function +outside a null measure set. This is given by the fact that, for test functions ϕ +such that supp ϕ ⊂ R\{0}, +� +A +δ(x)ϕ(x)dx = ϕ(0) = 0, +where A is a subset of R\{0} containing supp ϕ. Thus, if N is the mentioned +null measure set, N ∪ {0} remains of null measure. Therefore, we would have +δ(ϕ) = +� +∞ +−∞ +δ(x)ϕ(x)dx = 0, +now for ϕ in D(R). This is a contradiction with definition (15) of δ. +Remark 2.5. We shall utilize the symbol δx0, with x0 ∈ Rn to represent the +singular distribution +δx0(ϕ) = ⟨δx0, ϕ⟩ = ϕ(x0) , +∀ ϕ ∈ D(Rn). +With this, the delta distribution as defined in (15) is nothing more than δ0. +We will, however, write the more compact form, for convenience, only explicitly +writing the point x0 of evaluation when it is different from the origin. +Continuing on the topic of the delta distribution, we may find materials +where it is introduced as the limit of a sequence of smooth functions [4, 20], +providing but an intuition of the meaning of δ(x). Now that we have seen the +rigorous definition of δ and expressed the space in which it lives, we can in fact +solidify this idea of convergence. As a consequence, we also have, as expected, +a generalization of the convergence in the function spaces we have defined. +Definition 2.3. Let (Tk)k∈N be a sequence of distributions in D′(Rn) and let +T ∈ D′(Rn). We say that Tk converges to T if, for every ϕ ∈ D(Rn), we have +the convergence (in C) limk→∞⟨Tk, ϕ⟩ = ⟨T, ϕ⟩. We denote this convergence +simply by Tk → T or Tk +D′(Rn) +−−−−−→ T . +11 + +With this definition, the idea of defining the Dirac delta distribution as a +limit of bona fide functions becomes rigorous. This convergence cannot happen +point wise, but only if we view the sequence fk as a sequence in D′(R). With this +in mind, we next prove a result which allows us to obtain an infinity of sequences +of regular distributions converging to δ. These are called Dirac sequences. +Theorem 2.1. Let f : Rn → R be an integrable function such that +� +Rn f(x)dnx = 1. +Then, the sequence fk defined by fk(x) = knf(kx) is such that fk → δ, as a +sequence of distributions. +Proof. If ϕ is an element of D(Rn), then +� +Rn knf(kx)ϕ(x)dnx +y=kx += +� +Rn f(y)ϕ(y/k)dny += +� +Rn f(y) (ϕ(y/k) − ϕ(0)) dny + +� +Rn f(y)ϕ(0)dny. +(17) +Now, since ϕ is continuous by definition, it follows that +lim +k→∞ f(y) (ϕ(y/k) − ϕ(0)) = 0, +∀ y ∈ Rn +and, since +|f(y) (ϕ(y/k) − ϕ(0))| ≤ 2 ∥ϕ∥ |f(y)| , +we may apply the Lebesgue Dominated Convergence Theorem, obtaining +lim +k→∞ +� +Rn f(y) (ϕ(y/k) − ϕ(0)) dny = +� +Rn lim +k→∞ f(y) (ϕ(y/k) − ϕ(0)) dny = 0. +Thus, by (17), we have +lim +k→∞ +� +Rn knf(kx)ϕ(x)dnx = lim +k→∞ ϕ(0) +� +Rn f(y)dny = ϕ(0) = ⟨δ, ϕ⟩, +as we wished. +Remark 2.6. Note that Theorem 2.1 claims we can actually construct a Dirac +sequence from any integrable function f whose integral is different from zero. +To see this, we need only to take g = Cf, where C ∈ R is the constant +�� +Rn f(x)dnx +�−1, and use then gk(x) = kng(kx) as the sequence contained +in Theorem 2.1. +Example 2.5. With Theorem 2.1 and our last remark, we can now easily obtain +some examples of Dirac sequences: +fk(x) = +k +√ +2πe−k2x2 , +gk(x) = 1 +π +k +1 + k2x2 , +hk(x) = 1 +πk +sen2(kx) +x2 +. +12 + +2.3 +New distributions from old ones +What we wish to do now is construct some of the operators that take elements +of D′(Rn) and return new elements of the same space, much like the operations +of sum and product by a scalar z ∈ C. We are used to such operations acting on +function spaces, with derivations, products or convolutions being the main cases. +Here we are going to translate some of these into the language of distributions. +It must be clear that, since we are dealing with a generalization of functions, +the operations we want to define should also be generalization of the ones we +already know and this, in reality, gives us the insights we need to construct said +operations. +We begin with the concept of the derivative of a distribution T ∈ D′(Rn). +Given a function f ∈ L1 +loc(Rn) which admits a first derivation with respect to +some variable xi and whose derivative +∂f +∂xi also belongs to L1 +loc(Rn), we have, +for ϕ ∈ D(Rn), +� ∂f +∂xi +, ϕ +� += +� +Rn +∂f +∂xi +(x)ϕ(x) dnx = +� +∞ +−∞ +dx1 · · · +� +∞ +−∞ +dxn +�� +∞ +−∞ +∂f +∂xi +(x)ϕ(x)dxi +� +and, using integration by parts, +� +∞ +−∞ +∂f +∂xi +(x)ϕ(x)dxi = f(x)ϕ(x) +��� +xi=∞ +xi=−∞ − +� +∞ +−∞ +f(x) ∂ϕ +∂xi +(x) dxi += − +� +∞ +−∞ +f(x) ∂ϕ +∂xi +(x) dxi. +(18) +Since +∂ϕ +∂xi is again a test function, we can write +� ∂f +∂xi +, ϕ +� += − +� +f, ∂ϕ +∂xi +� +. +By induction, we can easily extend this result to any multi-index β, since any +derivation of a test function will again be a test function. Paying attention to +the factor of (−1) we must insert at each step, we have at last +� +Dβf, ϕ +� += (−1)|β|� +f, Dβϕ +� +. +(19) +Thus, the logical extension of this result to a general distribution T ∈ D′(Rn) +is given by the following +Definition 2.4. For any multi-index β, the derivative Dβ of a distribution +T ∈ D′(Rn) is a new distribution DβT ∈ D′(Rn) defined by +� +DβT, ϕ +� += (−1)|β|� +T, Dβϕ +� +, +∀ ϕ ∈ D(Rn). +(20) +The fact that equation (20) indeed represents a new distribution is a straight- +forward consequence of the facts that the operation Dβ is linear over D(Rn) and +that the convergence ϕn → ϕ in D(Rn) implies, by construction, in the conver- +gence Dβϕn → Dβϕ, again in D(Rn), whatever β we may have. +13 + +Remark 2.7. Perhaps the most interesting aspect of Definition 2.4 - something +that can actually be seen as the most interesting aspect of the theory of distri- +butions itself - is that, being ϕ ∈ D(Rn) infinitely differentiable, DβT is well +defined for every β. In other words, distributions are also infinitely differen- +tiable objects, and this is given from the start. We have no need in restraining +the space we work with to obtain infinite smoothness, something which is very +much desired in almost any theory. +Example 2.6. Consider the uni dimensional case where f is sectionally dif- +ferentiable, that is, f ′ exists for every point of R outside a finite set, say +{x1, x2, · · · , xm}. +Suppose, further, that f is such that the limits f(x+ +i ) = +limx→x+ +i f(x) and f(x− +i ) = limx→x− +i f(x) exist. The derivative of the regular +distribution Tf is thus given by +Tf +′ = Tf ′ + +m +� +i=1 +σiδxi, +(21) +where σi = f(x+ +i ) − f(x− +i ). +Indeed, we know that +� +Tf +′, ϕ +� += − +� ∞ +−∞ +f(x)ϕ′(x)dx += − +� x1 +−∞ +f(x)ϕ′(x)dx − +m−1 +� +i=1 +� xi+1 +xi +f(x)ϕ′(x)dx − +� ∞ +xn +f(x)ϕ′(x)dx += − lim +ε→0 +�� x1−ε +−∞ +f(x)ϕ′(x)dx + +m−1 +� +i=1 +� xi+1−ε +xi+ε +f(x)ϕ′(x)dx ++ +� ∞ +xn+ε +f(x)ϕ′(x)dx +� +. +Applying integration by parts, we obtain +� x1−ε +−∞ +f(x)ϕ′(x)dx = f(x)ϕ(x) +��� +x1−ε +−∞ − +� x1−ε +−∞ +f ′(x)ϕ(x)dx, +� ∞ +xn+ε +f(x)ϕ′(x)dx = f(x)ϕ(x) +��� +∞ +xn+ε − +� ∞ +xn+ε +f ′(x)ϕ(x)dx, +� xi+1−ε +xi+ε +f(x)ϕ′(x)dx = f(x)ϕ(x) +��� +xi+1−ε +xi+ε +− +� xi+1−ε +xi+ε +f(x)ϕ′(x)dx, +which implies, passing the limit ε → 0, +� +Tf +′, ϕ +� += +� ∞ +−∞ +f ′(x)ϕ(x)dx + lim +ε→0 +�m−1 +� +i=1 +f(xi+1 + ε)ϕ(xi+1 + ε) − f(xi − ε)ϕ(xi − ε) +� += +� ∞ +−∞ +f ′(x)ϕ(x)dx + +m−1 +� +i=1 +σiϕ(xi). +14 + +This equation is exactly the equality of distributions we wished to prove, (21). +On the other hand, we have no reason to believe that general results such as +(21) may be obtained for singular distributions. The only information we have +for DβT in this case is expression (20), defining it as a continuous functional +over D(Rn). +Example 2.7. The derivative of order m of the Dirac delta distribution δx0 ∈ +D(R), for example, is given by +� +δ(m), ϕ +� += (−1)mϕ(m)(0). +(22) +This follows directly from equation (20). +Now, the next operation we define over D′(Rn) is the multiplication of dis- +tributions by smooth functions. We develop these new elements like we have +done for DβT , dealing first with regular distributions and then generalizing for +the general case. +If T = Tf and h is an infinitely differentiable function, we can write +⟨hf, ϕ⟩ = +� +Rn h(x)f(x)ϕ(x) dx = ⟨f, hϕ⟩, +since hϕ is again a infinitely differentiable function which has support contained +in the support of ϕ, i.e., hϕ ∈ D(Rn). If this is the relation which we seek to +preserve when working with regular distributions, our definition for the product +of a distribution with a function must clearly be the following. +Definition 2.5. Given a distribution T and a function h ∈ C∞(Rn), the prod- +uct hT ∈ D′(Rn) is defined by +⟨hT, ϕ⟩ = ⟨T, hϕ⟩, +∀ ϕ ∈ D(Rn). +(23) +Alike the case of derivatives of distributions, the linearity of hT follows from +the linearity of the operation ϕ �→ hϕ, whereas the continuity is assured by the +preservation of the convergence in D(Rn) by this operation. This last assertion +requires a little more mathematical rigor, which we give now. +Lemma 2.2. For any T ∈ D′(Rn) and h ∈ C∞(Rn), expression (23) defines a +new distribution. +Proof. Indeed, if ϕk → ϕ in D(Rn), +��Dβ {h(ϕk − ϕ)} (x) +�� ≤ +� +α≤β +β! +α!(β − α)! +��Dβ−αh(x) +�� |Dα(ϕk − ϕ)(x)| . +Thus, if K ⊂ Rn is the compact set such that supp ϕk ⊂ K for k sufficiently +big, then we have +��Dβ {h(ϕk − ϕ)} (x) +�� ≤ +� +α≤β +β! +α!(β − α)!Cα ∥ϕk − ϕ∥β , +15 + +where Cα = sup{ +��(Dβ−αh)(x) +�� , x ∈ Rn} are constants independent of k. There- +fore, +lim +k→+∞ ∥h(ϕk − ϕ)∥β ≤ +� +α≤β +β! +α!(β − α)!Cα +� +lim +k→+∞ ∥ϕk − ϕ∥β +� += 0, +proving that hϕk converges in D(Rn) for hϕ and, consequently, ⟨hT, ϕk⟩ con- +verges to ⟨hT, ϕ⟩. We conslude from this that hT is, indeed, a linear and con- +tinuous functional over D(Rn). +The main example is the one related to the Dirac delta. +Example 2.8. Given any h ∈ C∞(Rn) and a test function ϕ ∈ D(Rn), we have +⟨hδx0, ϕ⟩ = ⟨δx0, hϕ⟩ = h(x0)ϕ(x0) = h(x0)⟨δx0, ϕ⟩. +We see then that, the only value necessary to define hδx0 is h(x0), that +means, +hδx0 = h(x0)δx0. +A particular result, and one of great importance, is when n = 1, x0 = 0 and +h(x) = x. In this case, we obtain +xδ = 0. +3 +Extension of distributions +As we have hinted in Section 1, the developments of QFT showed that renor- +malization in causal perturbation theory depends heavily on distribution the- +ory, more specifically on the procedures for obtaining extensions of distributions +whose behavior at the origin (this nomenclature will become clear later) does +not allow that we apply these over functions whose support contains the origin. +Our main problem then becomes: +Given a distribution T0 which is only well defined when we apply it on test +functions ϕ ∈ D(Rn) for which 0 /∈ supp ϕ, how can we construct a new distribu- +tion T ∈ D′(Rn) which is the extension of T0, that is, such that ⟨T0, ϕ⟩ = ⟨T, ϕ⟩ +whenever 0 /∈ supp ϕ? +The method for constructing such extensions is our goal in this section. The +ideas here presented originate mainly from [12, 13, 21] and references therein. +Some passages in these papers, however, may be too straightforward for an +unfamiliar public, perhaps in virtue of the public they are directed to, this +being researchers more familiar with the area. For that reason, we wish here +to perhaps fill some blanks one could find during those reads, providing then a +more accessible text to less experienced audience. +16 + +3.1 +Distributions with dependence in one parameter +Here, we deviate slightly from the common approach adopted by most of +the literature, which takes what we will do now as known, giving more space to +certain results crucial in later moments. +Our main focus now will be distributions with dependence on a real param- +eter µ. With this, we mean to say that we will study a family of distributions +in D′(Rn) of the form +{Tµ ∈ D′(Rn) ; +µ ∈ R}, +and explore the analytic properties of this dependence over µ. +Firstly, let us see that, for any distinct µ1, µ2 ∈ R, +Fµ1,µ2 = Tµ1 − Tµ2 +µ1 − µ2 +is well defined as an element of D′(Rn), with action given by +⟨Fµ1,µ2, ϕ⟩ = +1 +µ1 − µ2 +[⟨Tµ1, ϕ⟩ − ⟨Tµ2, ϕ⟩] ∈ C. +If, moreover, for all test function ϕ and for all µ ∈ R, the limit lim +δµ→0⟨Fµ+δµ,µ, ϕ⟩ +exists, then we are capable of defining the distribution +� d +dµTµ, ϕ +� +:= lim +δµ→0 +1 +δµ⟨Tµ+δµ − Tµ, ϕ⟩, ∀ µ ∈ R , ∀ ϕ ∈ D(Rn), +which will be the limit, in the sense of distributions, of Fµ+δµ,µ. +Is should be clear that this new distribution appears itself to be equivalent +to the derivation of the application µ �→ Tµ and should not be confused with +our previous definition of derivative DαT of T . Both represent new objects +belonging to D′(Rn) and are constructed from T , but whereas the former is +given by a differentiation in relation to the parameter µ of the family Tµ, without +any mention to test functions, the latter is dependent entirely upon derivations +on each ϕ and its variables. +Let us consider now the following: given the application µ �→ Tµ, we can +construct a new, complex, function for each ϕ ∈ D(Rn). For this, we take +η: R −→ C +µ �−→ η(µ) = ⟨Tµ, ϕ⟩. +(24) +It follows from this composition that +dη +dµ(µ) = lim +δµ→0 +1 +δµ (⟨Tµ+δµ, ϕ⟩ − ⟨Tµ, ϕ⟩) = lim +δµ→0⟨Fµ+δµ,µ, ϕ⟩, +that means, +dη +dµ(µ) = +� d +dµTµ, ϕ +� +. +(25) +17 + +This is an important result, justifying then its reiteration in an alternative form: +For any ϕ ∈ D(Rn) and µ ∈ R, we have +� d +dµTµ, ϕ +� += d +dµ⟨Tµ, ϕ⟩. +(26) +Besides that, we can, through η, define a new distribution. If η is continuous, +we know that, for a ∈ R, +H(µ) = +� µ +a +η(µ)d µ +is a well defined complex function. We have thus obtained, from a test function +ϕ and a fixed µ ∈ R, a new element I ∈ D′(Rn), characterized by +⟨I, ϕ⟩ = H(µ) = +� µ +a +η(µ)d µ ∈ C. +In reality, we shall utilize another symbol to reference I, defining +� µ +a +d µ Tµ = I. +This nomenclature permits us to write +�� µ +a +d µ Tµ, ϕ +� += +� µ +a +d µ ⟨Tµ, ϕ⟩. +In particular, if we take the integration of +d +dµTµ, we obtain, due to (26), +�� µ +a +d µ +� d +dµTµ +� +, ϕ +� += +� µ +a +d µ +� d +dµTµ, ϕ +� += +� µ +a +d µ d +dµ⟨Tµ, ϕ⟩ += η(µ) − η(a). +(27) +We see then that we have obtained a version of the Fundamental Theorem +of Calculus, +� µ +a +d µ d +dµTµ = Tµ − Ta, +(28) +for some mapping from R to D′(Rn). +This result will be an important piece for our main theorems in the next +section. +3.2 +Extensions of distributions +Let us consider a regular distribution Tf = f ∈ D(Rn) such that f(x) = 0 +for every x belonging to a subset U ⊂ Rn. +It is then evident that, for all +ϕ ∈ D(Rn) such that supp ϕ ⊂ U (we write, in this case, ϕ ∈ D(U)), we have +18 + +⟨f, ϕ⟩ = 0. This can be, therefore, a form of characterizing the support supp f +of the function f. +With the intent of extending this notion to distributions +T ∈ D′(Rn), we say that T is zero in a subset U ⊂ Rn when +⟨T, ϕ⟩ = 0 , +∀ ϕ ∈ D(U). +We are thus able to define the support of a distribution T . +Definition 3.1. For a given T ∈ D′(Rn), the support of T is the subset +supp T ⊂ Rn given by +supp T := {x ∈ Rn ; x does not contain a neighborhood in which T is zero}. +In an equivalent manner, supp T can be seen as the complement of the biggest +subset in which T is zero. +This definition, in turn, allows us to define an important subspace of D′(Rn). +For any subset U ⊂ Rn, the subspace of D′(Rn) given by the distributions such +that supp T ⊂ U will be denoted by D′(U). This notation comes from the clear +idea that we can associate this subset with the space of distributions whose +arguments are test functions in D(U). +Example 3.1. Given the distribution δx0 ∈ D′(Rn), we know that, for any +ϕ ∈ D(Rn) such that ϕ(x0) = 0, that is, such that x0 /∈ supp ϕ, we have +⟨δx0, ϕ⟩ = ϕ(x0) = 0. +It follows from this that supp δx0 = {x0}. +We can obtain a reciprocate from this last result with the following lemma +(for the proof, see for example [17]). +Lemma 3.1. If T ∈ D′(Rn) is such that supp T = {x0}, then there exists +m ∈ N and constants cν, |ν| ≤ m, such that +T = +� +|ν|≤m +cνDνδx0. +(29) +In what follows, we define a quantity which probes the behavior of a dis- +tribution T in the origin, in terms of singularities. +Since we have drawn a +clear distinction between bona fide functions (regular distributions) and general +(singular) distributions, we must develop for the later the idea of studying the +behavior of T at some point in Rn. It is in this ground that we introduce the +concept of pull-back. +This definition is characterized by a transformation Φ : Rn → Rn, which we +consider here to be invertible for simplification. For a function f ∈ L1 +loc(Rn), +the pull-back Φ∗f : Rn → C of f over Φ is a new complex function given by +Φ∗f(x) = f(Φ(x)). +19 + +Therefore, if viewing f as a regular distribution, we will have, for every +ϕ ∈ D(Rn),6 +⟨Φ∗f, ϕ⟩ = +� +Rn f(Φ(x))ϕ(x)dnx = +� +Rn f(y)ϕ(Φ−1(y)) +��DΦ−1(y) +�� dny +and, finally, +⟨Φ∗f, ϕ⟩ = ⟨f, |DΘ(y)| Θ∗ϕ⟩ , +Θ = Φ−1. +With this in mind, we can finally define the pull-back of a distribution. +Definition 3.2. The pull-back of T ∈ D′(Rn) over a invertible transformation +Φ : Rn → Rn is a new distribution Φ∗T , defined by +⟨Φ∗T, ϕ⟩ = ⟨T, |DΘ(y)| Θ∗ϕ⟩ , ∀ ϕ ∈ D(Rn), +(30) +where Θ = Φ−1. It is common the notation T (Φ(x)) (which we shall employ +from here on, for conformity) for the distribution Φ∗T , making reference to the +definition of the pull-back of a function. +Remark 3.1. We reiterate that the use of Φ invertible is a particular case of a +definition that can be made more general. For the generalization, it is defined +Φ∗T as the limit, in the distribution sense, of the regular distributions Φ∗fn if +fn is a sequence converging to T . The reader can utilize [17] for a source of +deeper reading in the matter. +Remark 3.2. Let us take here some more lines to achieve a better understand- +ing of Definition 3.2. Despite the indication given by the notation T (Φ(x)), the +pull-back of a distribution should not be read as an ordinary function. If T is +singular, then Φ∗T is only well defined as a new distribution, which will again +be singular. Thus, we should not interpret T (Φ(x)) as an object which varies +with x ∈ Rn, but actually as a distribution dependent on the transformation +Φ. Nonetheless, when regarding simple cases, such as Φ(x) = λx (λ > 0), it is +easier and perhaps more didactic to express the transformation directly as the +argument of T . +For regular functions, Tf(Φ(x)) will be indeed a bona fide function, just like +Tf itself. Moreover, let us see that, if ϕ ∈ D(Rn) is a test function such that +supp ϕ ⊂ BM(0), then supp ϕ(λ−1x) ⊂ BλM(0), so that +⟨f(λx), ϕ⟩ = +� +Rn f(λx)ϕ(x)dnx = +� +Rn f(x)ϕ(λ−1x)λ−ndnx += λ−n +� +BλM(0) +f(x)ϕ(λ−1x)dnx. +(31) +As we take the limit λ → 0+, the integration is performed over a ball with +ever decreasing radius, that means, we evaluate the behavior of f in smaller and +smaller neighborhoods of the origin, as indicated in our discussion preceding +Definition 3.2. +6Here the symbol |DF (y)| represents the Jacobian of F : Rn −→ Rn. +20 + +After such remarks, we present the following definition. +Definition 3.3. Let T ∈ D′(Rn) be a distribution. The scaling degree of T +is the real number (or ±∞), denoted here by σ(T ), such that +σ(T ) = inf{s ∈ R ; λsT (λx) +λ→0+ +−−−−→ 0}.7 +The singular order of T , denoted by ω(T ), is the value +ω(T ) = [σ(T )] − n, +where [m] denotes the biggest integer smaller (or equal) to m. +Example 3.2. Let us see some examples of distributions and their respective +scaling degrees, making our definitions clearer. The examples will be useful to +our further discussions as well. +1. If T = δ ∈ D′(R), then σ(δ) = 1. This is given by the result δ(λx) = +λ−1δ(x), which follows directly from (30). More generally, dealing with +the n-dimensional case, δ ∈ D′(Rn), we have +��DΦ−1(y) +�� = λ−n, if Φ(x) = λx, +implying that +⟨δ(λx), ϕ⟩ = λ−n⟨δ, ϕ⟩, +that means, σ(δ) = n. +2. If T = P 1 +x, then +λs⟨T (λx), ϕ⟩ = λs−1P +� +∞ +−∞ +1 +xϕ(λ−1x)dx = λs−1P +� +∞ +−∞ +ϕ(y)λdy +λy . +Therefore, λsT (λx) +λ→0+ +−−−−→ 0 if, and only if, s > 1, that is, σ(T ) = 1, +again. +3. If f is a continuous function, of one variable, homogeneous with degree m, +meaning f(λx) = λmf(x), then +⟨λsf(λx), ϕ⟩ = λs +� +∞ +−∞ +λmf(x)ϕ(x)dx = λs+m⟨f, ϕ⟩. +It follows immediately from this that σ(f) = −m. Thus, the scaling degree +of a homogeneous function is the (additive) inverse of its homogeneity +degree. +7Here the convergence is in the sense of distributions. +21 + +Example 3.3. We have seen that σ(δ) = n and, therefore, ω(δ) = 0. Let us +see now what occurs for a derivation T = Dαδ of the delta distribution. We +know that, for any ϕ ∈ D(Rn), +⟨Dαδ, ϕ⟩ = (−1)|α|⟨δ, Dαϕ⟩. +(32) +With this, we can verify that, for λ > 0, ⟨T (λx), ϕ⟩ = (1/λn+|α|)⟨δ, ϕ⟩. +Indeed, from (32) and given Example 3.2, +λs⟨Dαδ(λx), ϕ⟩ = (−1)|α|λs−n� +δ, Dαϕ(λ−1x) +� +. +From this, and from the equality +Dαϕ(λ−1x) = λ−|α|Dαϕ(x), +we conclude that +λs⟨Dαδ(λx), ϕ⟩ = λs−(n+|α|)⟨Dαδ, ϕ⟩. +This implies that σ(T ) = n + |α| and, thus, ω(T ) = |α|. In other words, the +derivative Dα of the delta distribution increases its initial scaling degree by a +factor ∥α∥. +Now that we have gone through some examples to fixate this new concepts +and definitions, we shall cite a lemma which gather their main properties. De- +spite its importance, we feel it is not necessary that we give here the complete +demonstration of this result. For the idea of its demonstration, we refer the +reader to [22]. +Lemma 3.2. Consider T, S ∈ D′(Rn), c ∈ C and β a multi-index. Then, we +have the following +1. σ(xβT ) = σ(T ) − |β|. +2. σ(DβT ) = σ(T ) + |β|. +3. σ(cT ) = σ(T ). +4. σ(ϕ) ≤ 0 and σ(ϕT ) ≤ σ(T ), for every ϕ ∈ D(Rn). +5. σ(T + S) ≤ max{σ(T ), σ(S)}. +Example 3.4. The proof of (5.) of Lemma 3.2 comes from the evident fact that, +if s ∈ R is such that λsT (λx) and λsS(λx) both converge to zero, the we cannot +have anything other than λs(T +S)(λx) → 0, in the sense of distributions. The +reciprocal, however, is not necessarily true, which gives rise to the inequality. +However, for the particular case when T and S are both derivatives of the Dirac +delta, say T = Dα1δ and S = Dα2δ, then we obtain the equality. +22 + +Indeed, let us suppose, without loss of generality, that |α1| > |α2|, from +which max{σ(T ), σ(S)} = σ(T ) = n + |α1|, by Example 3.3. +Now, taking +s < n + |α1|, we have, for ϕ ∈ D(Rn), +λs⟨(T + S)(λx), ϕ⟩ = λs [⟨Dα1δ(λx), ϕ⟩ + ⟨Dα2δ(λx), ϕ⟩] += λs � +(−1)|α1|λ−n−|α1|⟨δ, Dα1ϕ⟩ + (−1)|α2|λ−n−|α2|⟨δ, Dα2ϕ⟩ +� += (−1)|α1|λs−n−|α1|Dαϕ(0) + (−1)|α2|λs−n−|α2|Dαϕ(0). +(33) +Hence, if ϕ is such that Dα1ϕ(0) ̸= 0, then +lim +λ→0+ λs−n−|α1|Dαϕ(0) = ±∞, +that means, λs⟨(T + S)(λx), ϕ⟩ diverges. This proves that n + |α1| must be a +lower bound to {s ∈ R ; λs(T + S)(λx) +λ→0+ +−−−−→ 0}, which implies +max{σ(T ), σ(S)} ≤ σ(T + S) +and, by Lemma 3.2, +max{σ(T ), σ(S)} = σ(T + S). +(34) +We now posses the appropriate tools for proving the results that concern +the proper extension of distributions. As we shall see, we must separate our +problem into two cases, differentiated by a condition over the singular order of +T0. The biggest difference between the two cases, which are characterized by +ω(T ) < 0 and ω(T ) ≥ 0, is in the uniqueness of our extension. +Theorem 3.1. Let T0 ∈ D′(Rn\{0}) such that σ(T0) = s < n. Then there +exists a unique distribution T ∈ D′(Rn) such that σ(T ) = s and +⟨T, ϕ⟩ = ⟨T0, ϕ⟩ , +∀ ϕ ∈ D(Rn\{0}). +Proof. We first prove uniqueness. Indeed, if T1, T2 both are extensions of T0 +in D′(Rn), then ⟨T1 − T2, ϕ⟩ = 0 for any function ϕ ∈ D(Rn\{0}). On one +hand, supp (T1 − T2) = {0} and, according to Lemma 3.1, we have T1 − T2 = +� +|ν|≤m cνDνδ, for some set of complex constants {cν ; |ν| ≤ m}. Supposing +that some of these constants are not zero, we can use Example 3.4 to conclude +that σ(T1 − T2) ≥ n. On the other hand, Lemma 3.2 affirms that +σ(T1 − T2) ≤ max{σ(T1), σ(T1)} < n. +This is a contradiction and from that we take that T1 − T2 ≡ 0. +For the existence, we first take χ ∈ D(Rn) a test function such that χ(x) = 1 +in a neighborhood of the origin.8 +For every µ > 0 and ϕ ∈ D(Rn), (1 − +8A clear possibility would be to take χ as in Example 2.3. +23 + +χ(µx))ϕ(x) = 0 is also a neighborhood of the origin, from which (1−χ(µx))ϕ(x) ∈ +D(Rn\{0}) and, therefore, the distribution +Tµ = (1 − χ(µx))T0 +(35) +is well defined as an element of D′(Rn). Moreover, for every ϕ ∈ D(Rn\{0}), +taking µ0 > 0 big enough, we have +χ(µx)ϕ(x) = 0 , +∀ x ∈ Rn and for µ ≥ µ0, +thus the limit T = (Tµ)µ→∞ is defined for every argument in D′(Rn\{0}) and +lim +µ→∞⟨Tµ, ϕ⟩ = ⟨T0, ϕ⟩ , +∀ ϕ ∈ D′(Rn\{0}). +For the proof that T is defined over the whole space of test functions, take +any ϕ ∈ D(Rn). By the result (28) of subsection 3.1, we can rewrite Tµ as +Tµ = T1 + +� µ +1 +d µ d +dµTµ. +On the other side, we know from (26) that +� d +dµTµ, ϕ +� += d +dµ⟨Tµ, ϕ⟩ = d +dµ (⟨T0, (1 − χ(µx))ϕ⟩) . +(36) +The dependence of this term on µ is now entirely within the argument and, +since T0 is a continuous functional, we are able to transfer the derivation to the +inside of the brackets, giving +d +dµ (⟨T0, (1 − χ(µx))ϕ⟩) = − +� +T0, d +dµ(χ(µx)ϕ(x)) +� += − +n +� +i=1 +⟨T0, xi(∂iχ)(µx)ϕ(x)⟩, +(37) +where it was used that +d +dµ(χ(µx)) = +n +� +i=1 +d +dµ(µxi) ∂χ +∂xi +(µx). +Now, for each term of the sum (37), we see that +⟨T0, xi(∂iχ)(µx)ϕ(x)⟩ = µ−1⟨ϕT0, (µxi)(∂iχ)(µx)⟩ += µ−(n+1)� +(ϕT0)(µ−1x), xi(∂iχ)(x) +� +. +(38) +Thus, from (36), (37) and (38), we obtain +� d +dµTµ, ϕ +� += −µ−(n+1) +n +� +i=1 +� +(ϕT0)(µ−1x), xi(∂iχ)(x) +� +. +(39) +24 + +Let us then take ε such that σ(T0) < ε < n, λ = µ−1 and ψi(x) = xi(∂iχ)(x). +By Lemma 3.2, we know that σ(ϕT0) ≤ σ(T0) and, by the definition of scaling +degree of T0, +lim +µ→∞ µ−ε� +(ϕT0)(µ−1x), xi(∂iχ)(x) +� += lim +λ→0+ λε⟨(ϕT0)(λx), ψi⟩ = 0, +(40) +for all i ∈ {1, · · · , n}. +Therefore, for λ small enough, that is, for µ big enough, it follows from (39) +and (40) that +µn+1−ε +���� +� d +dµTµ, ϕ +����� ≤ 1, +that means, +���� +� µ +1 +d µ +� d +dµTµ, ϕ +����� ≤ +� µ +1 +(µ)ε−n−1d µ = +1 +ε − n(µε−n − 1). +We have just proven that T = (Tµ)µ→∞ exists as an element of D′(Rn) +and, by definition of Tµ in (35), T will also have scaling degree equal to s. +Furthermore, we have already seen that T will be the only extension of T0 with +such scaling degree. +Theorem 3.1 will also be used to help us prove our next result, which has +the same objective as our last, but now to distributions such that σ(T ) ≥ n. +Before that, however, we shall need to define some more concepts which will be +an important part of our proof. +We will denote by Dω(Rn) the subspace of D(Rn) composed by functions +such that their derivatives up to order ω vanish in the origin. Thus, for some +natural number ω > 0, +Dω(Rn) = {ϕ ∈ D(Rn) ; Dαϕ(0) = 0 , |α| ≤ ω}. +Now, for every function ϕ ∈ D(Rn), its Taylor expansion will be given by +ϕ(x) = +� +ν +xν +ν! Dνϕ(0), +(41) +from which, separating the terms whose multi-index have norm |ν| > ω, we have +ϕ(x) = +� +|ν|≤ω +xν +ν! Dνϕ(0) + +� +|ν|>ω +xν +ν! Dνϕ(0). +(42) +Therefore, the inclusion ϕ ∈ Dω(Rn) is equivalent to saying that the first +summation in (42) is zero. Meanwhile, the second summation term will then +belong to Dω(Rn), being actually equal to ϕ. +Furthermore, using Lagrange +Remainder formula, we can affirm the following result. +25 + +Lemma 3.3. Any function ϕ ∈ Dω(Rn) can be written as +ϕ(x) = +� +|α|=ω+1 +xαgα(x), +where gα ∈ D(Rn) for all multi-index in this summation. +Proof. To prove this, apply, for every ϕ ∈ Dω(Rn), the Taylor Theorem with +Remainder in the multi variable case,9 so that we have +ϕ(x) = +� +|β|≤ω +Dβϕ(0) +β! +xβ + +� +|α|=ω+1 +xα +α! fα(x), +where +fα(x) = (ω + 1) +� 1 +0 +(1 − t)ωDαϕ(tx)dt. +Since ϕ ∈ Dω(Rn), the first summation term vanishes for any x ∈ Rn. +Furthermore, being ϕ a infinitely differentiable function with compact support, +each fα(x) must also be infinitely differentiable. They may, however, not be +of compact support. Nonetheless, we can take a function ψ(x) ∈ D(Rn) such +that ψ(x) = 1 for x ∈ supp ϕ. We shall have ψ(x)ϕ(x) = ϕ(x) and the product +gα = ψfα ∈ Dω(Rn) will satisfy +ϕ(x) = +� +|α|=ω+1 +xαgα(x), +as desired. +Not only the subspace Dω(Rn) plays a crucial role to us, but also the projec- +tion of D(Rn) on Dω(Rn). Let W = Dω(Rn)⊥ be the orthogonal complement +of Dω(Rn) so that D(Rn) = Dω(Rn) ⊕ W.10 The direct sum of both subspaces +guarantees that functionals in D′(Rn) may be written as +T = Tω ⊕ l , +Tω ∈ D′ +ω(Rn) , +l ∈ W′. +We ask, then, what are the functionals in W′. Well, they are characterized +as elements l ∈ D′(Rn) such that ⟨l, ϕ⟩ = 0 for any ϕ ∈ Dω(Rn). Hence, we +have, in particular, +⟨l, ϕ⟩ = 0 for every ϕ ∈ D(Rn\{0}) ⊂ Dω(Rn) +and, therefore, supp l ⊂ {0}. According to the Lemma 3.1, it also follows that +l = +� +|α|≤m +cαDαδ , +m ∈ N, cα ∈ C. +9For the reader interested in the proof of this version of the Taylor Theorem, see, for +example, [23]. +10Dω(Rn) is evidently closed due to the continuity of any Dαϕ. +26 + +We state that m ≤ ω. In fact, if we suppose that m > ω, then we may take +ϕ ∈ Dω(Rn) such that Dβϕ(0) ̸= 0, with |β| = m, which implies ⟨l, ϕ⟩ ̸= 0. +This contradicts l ∈ W′. +Reciprocally, if +l = +� +|α|≤ω +cαDαδ, +then ⟨l, ϕ⟩ will only possess derivations of ϕ in x = 0 with order lesser or equal +to ω. Thus, if ϕ ∈ Dω(Rn), ⟨l, ϕ⟩ = 0, that is, l ∈ W′. +Thereby we have just proved the following +Lemma 3.4. For an arbitrary real number ω > 0, we have +W′ = {l ∈ D′(Rn) ; l = +� +|α|≤ω +cαDαδ , +cα ∈ C}. +Once W′ is also a linear space, it is a consequence of the last Lemma that +B = {Dαδ ; |α| ≤ ω} is a basis to the latter. Actually, we won’t use this set +of distributions to represent the orthogonal projection of D′(Rn) on D′ +ω(Rn), +but another one that depends upon a particular test function w ∈ D(Rn). We +will show that, for any function w ∈ D(Rn) such that w(0) ̸= 0, the set11 +C = {Dαδ(w−1·) ; |α| ≤ ω} is a basis to W′ as well. The number of elements of +C equals the number of elements in B, thus, it remains to show that the elements +of the latter may be generated by C. We use the Leibniz rule so pointed out in +Observation 2.2, which implies that, for any multi-index α and any ψ ∈ D(Rn) +that does not vanish in x ∈ Rn, +Dαϕ(x) = +1 +ψ(x)Dα(ψϕ)(x) − +1 +ψ(x) +� +0≤β≤α +β̸=α +α! +β!(α − β)!(Dβϕ)(x)(Dα−βψ)(x). +Hence, if ⟨Dγδ, ϕ⟩ = (−1)|γ|Dγϕ(0) may be written as a linear combination +of terms such as +� +Dβδ(w−1·), ϕ +� += (−1)|β|Dβ(w−1ϕ)(0) for arbitray12 γ < α, +then the same holds for Dαϕ(0). We conclude our proof if we note that the +result is valid for α = (0, · · · , 0), +⟨Dαδ, ϕ⟩ = ϕ(0) = w(0) ϕ(0) +w(0) = w(0) +� +Dαδ(w−1·), ϕ +� +. +Just as any element in B may be written as a linear combination of elements +in C, the same is true for W′. In addition, the set E = { (−1)|α| +α! +w(x)xα ; |α| ≤ ω} +generates W and will be a basis to the dual of C, once13 +� +Dαδ(w−1·), (−1)|β| +β! +w(x)xβ +� += (−1)|α|+|β| +β! +(Dαxβ)(0) = δα,β. +11This means the action of an element of C over a ϕ ∈ D(Rn) is +� +Dαδ(w−1·), ϕ +� += +� +Dαδ, w−1ϕ +� +. +12This inequality means that γ ≤ α. However γ ̸= α, that is, at least one of its coordinates +γi is strictly lesser than αi. +13We have used the generalization of the Kroenecker delta for multiple variables, which is 1 +whenever α = β, and zero otherwise. +27 + +Then, we can write the projection operator of D(Rn) on Dω(Rn) in the form +W(ω;w) : D(Rn) −→ Dω(Rn) +ϕ(x) �−→ ϕ(x) − w(x) +� +|α|≤ω +xα +α! +� +Dα ϕ +w +� +(0), +(43) +for any w ∈ D(Rn) such that w(0) ̸= 0. +In fact, if ϕ ∈ D(Rn) is arbitrary, then fixing γ a multi-index such that +|γ| ≤ ω, we have +(DγW(ω;w)ϕ)(0) = Dγϕ(0) − +� +|α|≤ω +� +Dγwxα +α! +� +(0) +� +Dα ϕ +w +� +(0) +and we know, due to Observation 2.2, that +� +Dγw(x)xα +α! +� +(0) = +� +β≤γ +γ! +β!(γ − β)!(Dγ−βw)(0) +� +Dβ xα +α! +� +(0) += +� +β≤γ +γ! +β!(γ − β)!(Dγ−βw)(0)δβ,α += +γ! +α!(γ − α)!(Dγ−αw)(0). +(44) +That way, we obtain +(DγW(ω;w)ϕ)(0) = Dγϕ(0) − +� +|α|≤ω +γ! +α!(γ − α)!(Dγ−αw)(0) +� +Dα ϕ +w +� +(0) += Dγϕ(0) − Dγ(w ϕ +w )(0) += Dγϕ(0) − Dγϕ(0) = 0. +(45) +Once again we have utilized, in the second equality, what we obtained in the +Observation 2.2. Thus, we have just proved that (W(ω;w)ϕ)(x) is indeed a test +function whose derivatives up to order ω vanish at the origin. +We may check that W is indeed a projection by showing that it is idempotent, +W 2 = W. For that, we observe that in Dω(Rn), W is the identity operator. +In fact, for a function ϕ ∈ Dω(Rn), we have +� +Dα ϕ +w +� +(0) = 0 for any |α| ≤ ω, +in a way that +(W(ω;w)ϕ)(x) = ϕ(x) − +� +|α|≤ω +xα +α! +� +Dα ϕ +w +� +(0) = ϕ(x). +We chose w−1 instead of the function w itself because the former allows +us to write an interesting and useful property, to be used ahead. Namely, the +operator W satisfies +W(ω;w)(wϕ) = wW(ω;1)(ϕ). +(46) +28 + +We point out that there is no problem at all when using W(ω;ψ), with ψ being +the constant function equals the unity. Although W(ω;1)ϕ is not a test function, +once its support is not compact, it is infinitely differentiable and, hence, its +product with w will be, in fact, in D(Rn). +In accordance with (46), we will have, if |α| ≤ ω, +W(ω;w)(wxα) = wW(ω;1)xα, +that is, +W(ω;w)(wxα)(x) = w(x) + +xα − +� +|β|≤ω +xβ +β! +� +Dβxα� +(0) + + . +By the Example 2.1, the sum reduces to xα, +W(ω;w)(wxα) ≡ 0. +(47) +Our next result is the following +Theorem 3.2. Let T0 ∈ D′(Rn\{0}) such that σ(T0) = s ≥ n and ω = ω(T0) = +s − n. Moreover, given w ∈ D(Rn), with w(0) ̸= 0, and constants Cα ∈ C for +all multi-index α, with |α| ≤ ω. There exists one, and only one, distribution +T ∈ D′(Rn) such that σ(T ) = s and satisfying +1. ⟨T, ϕ⟩ = ⟨T0, ϕ⟩ , +∀ ϕ ∈ D(Rn\{0}). +2. ⟨T, wxα⟩ = Cα. +Specifically, T is given by +⟨T, ϕ⟩ = +� +Tω, W(ω;w)ϕ +� ++ +� +|α|≤ω +Cα +α! +� +Dα ϕ +w +� +(0), +(48) +where Tω is the only extension guaranteed by the Theorem 3.1 and W(ω;w) is the +operator W, defined in (43). +Proof. We shall begin, once again, with the uniqueness first, supposing the +existence prior to its proof. If T1 and T2 are both extensions of T0 in D(Rn), +then, for ϕ ∈ D(Rn\{0}), +⟨T1 − T2, ϕ⟩ = ⟨T1, ϕ⟩ − ⟨T2, ϕ⟩ = 0, +which implies supp (T1 − T2) ⊂ {0}. +Again, according to the Lemma 3.1, +T1 − T2 = � +|ν|≤m cνDνδ, so that σ(T1 − T2) = n + |ν| and we must have, +by hypothesis, m ≤ ω. Thus, we have, for any α such that |α| ≤ ω, +⟨T1 − T2, wxα⟩ = ⟨T1, wxα⟩ − ⟨T2, wxα⟩ = Cα − Cα = 0. +On the other hand, if we first take |α| = m, then +⟨T1 − T2, wxα⟩ = +� +|ν|≤m +cν(Dνwxα)(0) +29 + +and, with the Example 2.1, we have +⟨T1 − T2, wxα⟩ = +� +|ν|≤m +cν + +� +β≤ν +ν! +β!(ν − β)!(Dβxα)(0)(Dν−βw)(0) + + += +� +|ν|=m +cν + +� +β≤ν +ν! +β!(ν − β)!(Dβxα)(0)(Dν−βw)(0) + + , +(49) +where we have used that all derivatives (Dβxα)(0) will be zero whenever |ν| < +|α| = m. Furthermore, the only non-vanishing term appears when β = ν = α, +so that the we are left with only +⟨T1 − T2, wxα⟩ = cα(Dαxα)(0)w(0) = α! cα w(0) = 0. +We have obtained that cν = 0 once |ν| = m. +Analogously, if we consider +|α| = m − i, 1 ≤ i ≤ m, then we will find the same result, canceling all the +constants cα, finally getting T1 = T2. +For the existence, we first restrict T0 to the subspace D′ +ω(Rn\{0}) of D′(Rn\{0}) +and, denoting this restriction by ˜T0, we have, by the Lemma 3.3, +� ˜T0, ϕ +� += +� +|α|=ω+1 +⟨xαT0, gα⟩. +Using now (1.), (4.) and (5.) from the Lemma 3.2, we obtain σ( ˜T0) ≤ σ(T0) − +ω − 1 < n. +Therefore, the restriction of T0 in D′ +ω(Rn\{0}) has, due to Theorem 3.1, an +extension14 in D′ +ω(Rn), which we denote by Tω. However, we seek an extension +over the whole space D′(Rn), so that we still need to extend Tω to general +elements of D(Rn). If we obtain such T ∈ D′(Rn), it is evident that it will be +an extension of T0, since D(Rn\{0}) ⊂ Dω(Rn). +Now, since we are dealing with a closed orthogonal subspace of D′(Rn), +extensions of Tω will be simply characterized by +T = Tω ⊕ l , +l ∈ W′. +It is through the operator W, which projects functions from D(Rn) onto the +subspace Dω(Rn), that we are capable of applying Tω over any ϕ ∈ D(Rn), +placing between then the action of W(ω;w). +Since the projection operator is +unique (once we have chosen w), each extension +T = Tω ◦ W(ω;w) ⊕ l +14The more careful reader may ponder if this extension will in fact belong to D′ +ω(Rn), since +Theorem 3.1 affirms only that the extension will be an element of D′(Rn). For that, we note +that our construction made no reference to the behavior of ϕ ∈ D(Rn). From that, we see that +the restrictions over the application of ˜ +T0 ∈ D′ +ω(Rn\{0}) will be inherited by Tω ∈ D′ +ω(Rn). +30 + +will be unique except by the change of l. Since C = {Dαδ(w−1·) ; |α| ≤ ω} is a +basis to W′, the constants Cα must define such functional l, which implies that +T = Tω ◦ W(ω;w) ⊕ +� +|α|≤ω +Cα +α! +� +Dα ϕ +w +� +(0) +is the only extension of T0 satisfying ⟨T, wxα⟩ = Cα. +Remark 3.3. We can simplify even further our calculations of the extension of +T0 if we restrict the class of functions permitted to w. More specifically, if we +take w such that (Dαw)(0) = δα,0, which is equivalent to taking w equal to 1 +in a neighborhood of the origin, we have +(Dα ϕ +w )(0) = +� +0≤β≤α +β̸=α +α! +β!(α − β)!(Dβϕ)(0)(Dα−βw−1)(0) += Dαϕ(0), +(50) +since any derivation of w−1 will also carry some derivative of w itself. From +that, it follows that W reduces to +(W(ω;w)ϕ)(x) = ϕ(x) − w(x) +� +|α|≤ω +xα +α! Dαϕ(0) +and the extension T given by Theorem 3.2 can, therefore, be written as +⟨T, ϕ⟩ = +� +Tω, W(ω;w)ϕ +� ++ +� +|α|≤ω +Cα +α! (Dαϕ) (0). +(51) +Such functions, satisfying (Dαw)(0) = δα,0, are called Epstein-Glaser func- +tions (see, for example, [14]). +3.3 +Dependence of the extension on the test function w(x) +We have seen that, for the case when σ(T0) ≥ n, it does not seem possible +to get rid of the dependence of the extension T ∈ D′(Rn) on the test function +w ∈ D(Rn) chosen to construct the projection of D(Rn) over Dω(Rn). We can, +nonetheless, observe the behavior of this dependence, mainly by studying the +term +� +Tω, W(ω;w)ϕ +� +, which we denote, following [21], by the integral kernel of the +extension. Here, we will stick with the supposition that w is an Epstein-Glaser +function, in the sense defined above. +31 + +Thus, choosing two Epstein-Glaser test functions w1, w2 ∈ D(Rn), we have +(W(ω;w1)ϕ)(x) = ϕ(x) − w1(x) +� +|α|≤ω +xα +α! Dαϕ(0) += ϕ(x) − w2(x) +� +|α|≤ω +xα +α! Dαϕ(0) + (w2(x) − w1(x)) +� +|α|≤ω +xα +α! Dαϕ(0) += (W(ω;w2)ϕ)(x) + (w2(x) − w1(x)) +� +|α|≤ω +xα +α! Dαϕ(0) +(52) +and, applying Tω over this expression, we obtain (bearing in mind that Tω is +unique, by Theorem 3.1) +� +Tω, W(ω;w1)ϕ +� += +� +Tω, W(ω;w2)ϕ +� ++ +� +|α|≤ω +� +Tω, (w2(x) − w1(x))xα +α! +� +Dαϕ(0). +(53) +We therefore conclude that the application of Tω over different projections +differ only by a linear combination of terms of the form ⟨Dαδ, ϕ⟩, that is, by +application, over the test function ϕ, of operators belonging to W′. +Our goal ahead will be, however, to get rid of the restriction that w be a test +function. As we have already mentioned, this seems to be a crucial condition for +defining W as a projection operator, since it is responsible for the fact that the +term w(x) � +|α|≤ω +xα +α! Dαϕ(0) has compact support. Nonetheless, very often (as +will be the case ahead) a distribution which is not well behaved at the origin will +behave nicely at infinity. To be more precise, we mean that such distributions +will be well defined when applied to functions ϕ ∈ C∞(Rn) whose support may +not be compact. In that sense, taking a sequence (wk) ⊂ D(Rn) whose pointwise +limit w is a function15 in C∞(Rn) and T0 is such that +lim +k→∞ +� +Tω, W(ω;wk)ϕ +� +∈ C, +∀ +ϕ ∈ D(Rn), +then there is no motive to not consider w in our renormalization scheme. We +shall see how this is an important part for our application of this method of +extension of distributions. +4 +Application to the electron self-energy +In this section, we finally attack the electron self-energy problem, seen as +a point particle. We posses now sufficient machinery to see it as a pathology +to be faced with extension of distributions defined over D(Rn\{0}). To clar- +ify our aims, we will translate the issues of electrostatics to the language of +distributions. +15Some works, such as [21], go even further as to only ask that w ∈ D′(Rn) be a distribution. +We will not need this generality, so that we have preferred to omit this possibility. +32 + +The first and default example is the charge distribution of an electron, seen +as a charged point particle, which is represented by the Dirac delta δ(·), cen- +tered where we suppose the whole electric charge of the particle should be +concentrated. Actually, this particular case is not the only one where we con- +sider the charge distribution ρ as a generalized function. After all, it would be +an incredible coincidence in nomenclature if the charge distributions presented +in the electrodynamics realm were not presented by distributions ρ ∈ D′(R3). +This is one of the main contributions of the theory of generalized functions to +electrodynamics. It not only incorporates more general charge distributions ρ +(which could not be defined just as real functions in R3), but also eases its +manipulations, employing the properties so described in Section 2. +Example 4.1. To illustrate our last paragraph, let us analyze the representa- +tion of an electric dipole as a distribution in D(R3). The pure dipole, considered +an idealization such as point charges, is constructed as follows: start by setting +two charges, q and −q, separated by a fixed distance ε. Suppose that they lie +in the x coordinate, with −q at the origin and q at x = ε, accordingly. That +way, the distribution ρ ∈ D(R) will be +ρ = −qδ0 + qδε = q(δε − δ0). +If we set q = 1/ε, then, in the limit ε → 0+ we obtain a distribution whose +total charge vanishes. +This result is not correct nonetheless. +A null charge +distribution would lead to a zero electric field, which is not the case for a dipole +(see [4] for the expression of ⃗E in this case). +This simple argument indicates that the distribution formalism is more than +necessary to an accurate description of even the basics of electrostatics. +Now, we apply ρε = ε−1(δε − δ0) over any ϕ ∈ D(R). We have +⟨ρε, ϕ⟩ = 1 +ε⟨δε − δ0, ϕ⟩ = ϕ(ε) − ϕ(0) +ε +and, therefore, we shall obtain, in the limit ε → 0+, +⟨ρε, ϕ⟩ → ϕ′(0) = −⟨δ′, ϕ⟩, +that is, +ρdip = −δ′. +The generalization to the three-dimensional case is straightforward. In this +case, the dipole moment p := q⃗ε, where ⃗ε is the displacement vector that con- +nects the negative to the positive charge. Once again we take the limit |⃗ε| → 0. +The way we defined q above, we guaranteed that the dipole moment was kept +constant, even when the charges are close enough. With this premise, the charge +distribution will be given by +⟨ρε, ϕ⟩ → − ∂ +∂pϕ(0) , +∀ ϕ ∈ D(R3), +that we denote by +ρdip = −p · ∇δ ∈ D(R3). +33 + +We return to the main problem we intend to attack, namely, the divergence +of the electron self-energy. +In this context, the fact that the charge is fully +concentrated in the origin is seen by the application of qδ on different test +functions ϕ ∈ D(R3). For any test function whose support does not contain the +origin, we have +� +supp ϕ +ρ(x)ϕ(x)d3x = 0. +(54) +At the same time, the distribution possess a finite charge once the integration +of ρ over R3 is equivalent to apply qδ on a test function ϕ ∈ D(R3) whose support +contains the origin such that ϕ(0) = 1. +� +R3 ρ(x)ϕ(x) d3x = +� +supp ϕ +ρ(x)ϕ(x) d3x = ⟨qδ, ϕ⟩ = q. +(55) +This charge distribution produces both a potential V and an electric field +E, which are also new distributions. +We haven’t considered vector fields as +distributions so far, just like E, but this generalization is quite natural and E +acts on an element of D(R3) according to +⟨E, ϕ⟩ = (⟨Ex, ϕ⟩, ⟨Ey, ϕ⟩, ⟨Ez, ϕ⟩). +The application of vector fields as distributions were mentioned here just for +completeness. It will be no longer necessary hence forth. +We can see that, in fact, V represents an element of D′(R3). The explicit +formula for the potential is given by V (r) = e +r, where e is the electron charge +and we are using unities in which 4πǫ0 = 1, with no further implications to +the final results whatsoever. As usual, r represents the radial coordinate of a +spherical coordinate system centered on the charge. V (·) is a smooth function +for any r ̸= 0. Hence, we just have to be concerned to the convergence of ⟨V, ϕ⟩, +for an arbitrary ϕ ∈ D(R3). In effect, if R > 0 is such that K = BR(0) ⊃ supp ϕ +and M = maxx∈R3 ϕ(x), then +���� +� +R3 V (x)ϕ(x)d3x +���� ≤ M +� +K +V (x)d3x += M +�� 2π +0 +dφ +� π +0 +senθdθ +� � R +0 +e +r r2dr += (2πMe)R2 < ∞. +(56) +An analogous consideration may be done for the electric field, since E ∼ +1 +r2 . +In view of that, the radial integral will converge as well. We can thus turn our +attention to the stored self-energy of a charged system. As we have already seen, +for the particular case of an electron there is a divergence at the origin. We are +considering here only the static case, so we don’t have to worry about magnetic +fields. We can consider, however, other cases where such divergence does not +appear and we are thus able to calculate the system self-energy. If we consider, +34 + +for example, the electron as a uniformly charged spherical shell of radius a, then +E(r) = +� +0 +, se r ≤ a, +(e/r2) ˆr +, se r > a. +(57) +Therefore, the self-energy, see eq. (2), will be given by +W = 1 +8π +� +R3 E2dτ = 1 +8π (4π) +� ∞ +a +e2 +r2 dr = 1 +2 +e2 +a . +(58) +In distribution parlance, the last equation is but the application of the reg- +ular distribution +1 +8πE2 over the function ϕ(x) ≡ 1. We point out that, even +with ϕ /∈ D(R3), +� +E2, ϕ +� +does exist. It is a consequence of the behavior of E2 +at infinity, which is good enough that we do not need to restrict the range of +integration to a compact set. For general distributions, with unknown behavior +at infinity, this restriction is considered by supposing that ϕ is a test function. +We have already mentioned, see Sec. 3.3, that it is often advantageous (or +even necessary) to work with functions that are not compactly supported. This +is allowed once our distribution possesses the necessary conditions so that its +application over this larger class of functions is well behaved. For instance, the +Dirac delta may be applied on any function ϕ continuous at the origin. In our +specific case, we will see how the electron self-energy (∼ E2) behavior far from +the origin permits such loosening of the conditions over ϕ. More precisely, +E2 = e2 +r4 , +r > 0 +(59) +is not well defined as a distribution. In effect, for any test function obeying +ϕ(x) = 1 for x in a neighborhood V of the origin, say, a ball, we have +� +E2, ϕ +� += +� +V +e2 +r4 d3x + +� +R3\V +e2 +r4 ϕ(x)d3x = +∞, +once the first term is linear divergent due to the fourth-order homogeneity of +E2.16 +Outside the origin, however, E2 is a smooth function, and as such, locally +integrable. Hence, we have, at least, E2 ∈ D′(R3\{0}). For this reason, we +may extend (renormalize) the distribution E2 to a new distribution U ∈ D′(Rn) +with the methods exposed previously in Sec. 3. In that way, we expect that the +electron self-energy E0 will be well defined as the application of +1 +8πU over the +function ϕ ≡ 1, +E0 = 1 +8π ⟨U, 1⟩. +(60) +According to what we have made so far, let us first determine the scaling +degree and the singular order of E2, which are key to Theorems 3.1 and 3.2. +16The classification as a linear divergence may be justified in polar coordinates. +Taking +R = 1/r, we find � +∞ +0 +dr +r2 = � +∞ +0 +dR. See [3] for details. +35 + +We observe that it is a homogeneous function of order −4, so that, as expressed +through the Example 3.2, +λsE2(λx) = λs +e2 +(λr)4 = λs−4E2, +which implies, +λsE2(λx) +λ→0+ +−−−−→ 0 ⇐⇒ s > 4. +That is, σ(E2) = 4, and also ω(E2) = σ(E2) − n = 1. +Therefore, for a test function w ∈ D(R3) and constants C0, C(1,0,0) ≡ C1, +C(0,1,0) ≡ C2, C(0,0,1) ≡ C3, we obtain, according to the Theorem 3.2, a distri- +bution U ∈ D′(R3) defined by +⟨U, ϕ⟩ = +� +E2 +1, W(1;w)ϕ +� ++ +� +|α|≤1 +Cα +α! +� +Dα ϕ +w +� +(0) += +� +E2 +1, W(1;w)ϕ +� ++ C0 ϕ(0) +w(0) + +3 +� +i=1 +Ci∂xi +� ϕ +w +� +(0) +(61) +satisfying +⟨U, ϕ⟩ = +� +E2, ϕ +� +, +∀ +ϕ ∈ D(R3\{0}), +(62) +⟨U, wxα⟩ = Cα , +|α| ≤ 1. +(63) +Moreover, if w is chosen as an Epstein-Glaser function, we have +⟨U, ϕ⟩ = +� +E2 +1, W(1,w)ϕ +� ++ C0ϕ(0) + +� +i=3 +Ci(∂xiϕ)(0). +(64) +Before moving on, there is a comment in order. The charge distribution of +a (charged) point particle possesses spherical symmetry. It is not only due to +the concentration of charge in a single point. In fact, a electric dipole has also +a distribution whose support is contained in the origin, although the allegedly +spherical symmetry is broken once the moment p defines a privileged direction. +The additional fact that our point particle model admits no internal structure +imposes the constraint of having no special direction. +Since we would like to preserve such symmetry when extending E2, we must +choose Ci = 0, for i = 1, 2, 3. This is justified because the last term in (64) +does not behave like a scalar under rotations of our coordinate system, unless +the three constants vanish. We can promptly see this by writing +� +i=3 +Ci(∂xiϕ)(0) = C · (∇ϕ)(0) , +C = (C1, C2, C3). +Now, ∇ϕ does behave as a vector, however we cannot say the same for C. For +this reason, the only way to keep this sum inert under rotations is to set C = 0. +36 + +From this, we may rewrite U as +⟨U, ϕ⟩ = +� +E2 +1, W(1;w)ϕ +� ++ C0ϕ(0). +(65) +Then, we seek, pretty much like what has been done in Sec. 3.3, to relax +the conditions under w employed in the renormalization of E2. Our path will +be to take a sequence of test functions wM that converges pointwise to w(x) = +1 ∈ C∞(R3), obtaining a well defined distribution given by +⟨U, ϕ⟩ = +lim +M→∞ +� +E2 +1, W(1;wM)ϕ +� ++ C0ϕ(0). +(66) +The last equation suggests that we will take all wM as Epstein-Glaser func- +tions. Specifically, each wM shall be taken as in the Example 2.3, +wM(x) = ζM,1(x) = 1 − +� r +−∞ +ηM,h(t)dt, +where r = ∥x∥ is a radial coordinate and ηM,h(t) is given in (9). We can write +this sequence of functions in a convenient manner that will be useful ahead. +The comments in the Example 2.3 suggest that wM equals 1 within the ball +BM(0) and 0 outside the ball BM+1(0). In the ring BM+1(0)\BM(0), i. e., for +M ≤ r ≤ M +1, we may write wM as a radial smooth function17 χ(r −M) such +that |χ(s)| ≤ 1, s ∈ [0, 1], χ(0) = 1 and χ(1) = 0, +wM(x) = + + + + + +1, +if x ∈ BM(0), +χ(r − M), +if x ∈ BM+1(0)\BM(0), +0, +if x /∈ BM+1(0). +(67) +Therefore, we have (DαwM)(0) = δα,0 and wM(x) → 1 for any x ∈ R3 +in the limit M → ∞. Moreover, due to the result (53), for any two naturals +M1, M2 ∈ N (say, M1 < M2), we have +� +E2 +1, W(1;wM2 )ϕ +� += +� +E2 +1, W(1;wM1 )ϕ +� ++ +� +|α|≤1 +� +E2 +1, (wM1(x) − wM2(x))xα +α! +� +Dαϕ(0), +wherein +wM1(x) − wM2(x) = + + + + + + + + + + + + + + + +0, +if x ∈ BM1(0), +χ(r − M1) − 1, +if x ∈ BM1+1(0)\BM1(0), +−1, +if x ∈ BM2(0)\BM1+1(0), +−χ(r − M2), +if x ∈ BM2+1(0)\BM2(0), +0, +if x /∈ BM2+1(0). +(68) +17which shall not be confused with the characteristic function χA on a set A ⊂ R3, also +present in the Example 2.3. +37 + +That way, if we denote (aM) the real sequence whose elements are +� +E2 +1, W(1;wM)ϕ +� +, +then we will show that it converges, proving that (aM) is a Cauchy sequence. +In fact, we have just seen that18 +aM1 − aM2 = +� +E2 +1, W(1;wM2 )ϕ +� +− +� +E2 +1, W(1;wM1 )ϕ +� += +� +|α|≤1 +� +E2, (wM1(x) − wM2(x))xα +α! +� +Dαϕ(0), +(69) +so that, if we limit this sum, then we will also limit the difference aM1 − aM2. +Now, given ε > 0, we take M ∈ N such that +1 +M < +ε +8πe2 and M1, M2 ∈ N, with +M ≤ M1 < M2. Since E2 acts on test functions whose support does not contain +the origin, we can employ the formula (59), obtaining +� +E2, (wM1(x) − wM2(x)) +� += +� +R3 +e2 +r4 (wM1(x) − wM2(x))d3x, +� +E2, (wM1(x) − wM2(x))xi +� += +� +R3 +e2 +r4 (wM1(x) − wM2(x))xid3x. +Now, once i. wM1 − wM2 is radial, ii. has support in BM2+1(0)\BM1(0) and +iii. assumes values only in [0, 1], we have +��� +E2, (wM1(x) − wM2(x)) +��� ≤ (4πe2) +� M2+1 +M1 +1 +r2 dr = (4πe2) +�1 +r +�M1 +M2+1 +≤ (8πe2) 1 +M1 +, +that is, +��� +E2, (wM1(x) − wM2(x)) +��� ≤ 8πe2 +M +< ε. +(70) +Meanwhile, +� +E2, (wM1(x) − wM2(x))xi +� += +� +R3 +e2 +r4 (wM1(r) − wM2(r))xid3x = 0, +(71) +because for any i = 1, 2, 3, corresponding to the three Euclidean axis x, y, z, +respectively, the integration in ϕ or in θ will vanish.19 In fact, for i = 1 and +i = 2, the integrals in ϕ vanish, +� 2π +0 +dφ cos φ = +� 2π +0 +dφ senφ = 0. Now, for i = 3, +we find +� π +0 dθ cos θ senθ = 0. +Hence, with the help of (70) and (71), we may conclude that +|aM1 − aM2| ≤ εϕ(0), +(72) +which, in turn, implies that (aM) will be a Cauchy sequence, that is, a convergent +one. +18Since wM1(x) − wM2(x) ∈ D(Rn\{0}), the action of E2 +1 may be replaced by E2. +19This is only possible because wM is a sequence of radial functions and as such, the +integration in r, ϕ e θ can be factored. +38 + +All the previous development allows us to show that the distribution defined +in (66), that we will denote simply by +⟨U, ϕ⟩ = +� +E2 +1, W(1;1)ϕ +� ++ C0ϕ(0), +(73) +is well defined. Thus, we may finally use the eq. (60) to obtain the renormalized +electron self-energy, +E0 = 1 +8π +� +E2 +1, W(1;1)1 +� ++ 1 +8π C0 = C0 +8π . +(74) +Although simple, the eq. (74) bears great physical meaning, concentrating +our results. We have shown that defining the electron self-energy as the applica- +tion of the distribution +1 +8πU ∈ D(R3), which extends (or renormalize) E2, over +the constant function 1, we get rid of the divergence previously obtained. This +divergence appeared when one directly considers E2, which cannot be seen as +an actual distribution.20 Now, E0 becomes an undetermined constant, that we +may control to serve our model. This is the very kernel of a renormalization +method. If, for instance, we assume that the self-energy is, alone, responsible +for the electron mass, we shall take +C0 = 8πmec2, +in a way that E0 = mec2. +To summarize, we have seen how the self-energy problem originates in the +fact that E2 is not a proper element of D(Rn). On the other hand, E2 is, outside +the origin, a smooth function. Thus, we at least have E2 ∈ D′(Rn\{0}), which +means we can use Theorem 3.2 to extend it to a distribution in D′(Rn). +5 +Conclusion +The main objective of this work was to analyze (and renormalize) a sim- +ple but central problem in classical electrodynamics: the electron self-energy. +Although the electrostatics model of a charged point particle implies a linear +divergence on the self-energy, we may skirt this infinity with an extension of the +corresponding distribution. With more details, +1. We have also developed a self-consistent study of the theory of distribu- +tions. The basic aspects, main definitions and examples, operations and key +results were all included. Of course our notes are not supposed to replace the +standard and seminal literature, such as [16,26,27], which is considered utterly +necessary. +However, we provide here the minimum to the interested reader +in maneuvering such a powerful tool for analyzing, for instance, classical and +quantum field theoretical models. +20The statement that E2 /∈ D(R3) is a consequence that the product between distributions is +not, in general, well defined. An alternative method to skirt the electron self-energy divergence +is related to a generalization to the very concept of distributions, working with the generalized +Coulombeau functions. For details, see [24,25] and references therein. +39 + +2. The leading results of extension of distributions, that is, the correspond- +ing renormalization, were all enunciated and demonstrated, following the lines +in [22]. We have focused in a particular subspace of the set of test functions, +namely, the one whose elements vanish in an arbitrary neighborhood of the +origin. We have investigated the behavior of different distributions in the ori- +gin and how one could recover such distributions, in the sense of making them +continuous linear functionals defined over all the space of test functions. Our +thorough demonstrations may serve as an auxiliary/pedagogical pathway on the +subject. +3. We have applied the concepts of distributions and the corresponding ex- +tensions to the classical electron self-energy. At first sight, electrostatics implies +a divergence once we treat the electron as a charged point particle. However, +our construction shows that its self-energy turns out to be an undetermined con- +stant upon renormalization, so that our parameters might be fixed, for example, +appealing to empirical results. +References +[1] A. 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Ryder, Quantum field theory. (Cambridge, New York: Cambridge +University Press, 2nd ed ed., 1996). +[10] N. N. Bogoliubov and D. V. Shirkov, Introduction to the theory of quantized +fields (New York: John Wiley, 2nd american ed., 1980). +40 + +[11] H. Epstein and V. Glaser, “The role of locality in perturbation theory,” +Annales de l’I.H.P. Physique Théorique 19, 211–295 (1973). +[12] J. M. Gracia-Bondía, “Improved Epstein–Glaser renormalization in coor- +dinate space i. euclidean framework,” Mathematical Physics, Analysis and +Geometry 6, 59–88 (2003). +[13] J. +M. +Gracia-Bondia +and +S. +Lazzarini, +“Connes- +Kreimer-Epstein-Glaser +renormalization,” +Available +in +ht tp s: // ar xi v. or g/ ab s/ he p-th /0006106, 2000. +[14] P. Grangé and E. Werner, “UV and IR behaviour for QFT and LCQFT +with fields as operator valued distributions: Epstein and Glaser revisited,” +Nuclear Physics B - Proceedings Supplements 161, 75–80 (2006). +[15] P. +C. +Grange +and +E. +Werner, +“Quantum +fields +as +op- +erator +valued +distributions +and +causality,” +Availabe +in +ht tp s: // ar xi v. or g/ ab s/ ma th -p h/ 0612011v2, 2007. +[16] L. Schwartz, Mathematics for the Physical Sciences (Dover Publications, +2008). +[17] L. Hörmander, The Analysis of Linear Partial Differential Operators I. +(Berlin, Heidelberg: Springer Berlin Heidelberg, 2003). +[18] I. Richards and H. Youn, Theory of distributions: a non-technical introduc- +tion (Cambridge: Cambridge University Press, 1990). +[19] R. G. Bartle, The Elements of Integration and Lebesgue Measure (John +Wiley and Sons, INC, 1995). +[20] G. Arfken, Mathematical Methods for Physicists +(San Diego: Academic +Press, Inc., third ed., 1985). +[21] D. Prange, “Epstein-Glaser renormalization and differential renormaliza- +tion,” Journal of Physics A: Mathematical and General 32, 2225–2238 +(1999). +[22] K. +Fredenhagen, +“Quantum +field +theory,” +Available +in +ht tp s: // ww w. ph ys ik .u ni -h am bu rg .d e/ th 2/ ag -f re de nh ag en /d ok um en te /q ft 09-10. pd f , +2009. +[23] J. J. Duistermaat and J. A. C. Kolk, Distributions: theory and applications +(Cornerstones, Birkhäuser Boston, MA, 2010). +[24] A. Gsponer, “The classical point electron in Colombeau’s theory of non- +linear generalized functions,” Journal of Mathematical Physics 49, 102901 +(2008). +41 + +[25] A. Gsponer, “A concise introduction to Colombeau generalized functions +and their applications in classical electrodynamics,” European Journal of +Physics 30, 109–126 (2009). +[26] L. Schwartz, Théorie des Distributions (Actualités scientifiques et indus- +trielles, Hermann, 1957). +[27] N. A. Lemos, Convite à Física Matemática (São Paulo: Editora Livraria +da Física, 2013). +42 + diff --git a/R9FKT4oBgHgl3EQfjS4w/content/tmp_files/load_file.txt b/R9FKT4oBgHgl3EQfjS4w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..571155783b038de97d0bcd2382cf270f525514d3 --- /dev/null +++ b/R9FKT4oBgHgl3EQfjS4w/content/tmp_files/load_file.txt @@ -0,0 +1,988 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf,len=987 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='11844v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='gen-ph] 23 Dec 2022 A (not so) short comment on the classical electron self-energy H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' de Assis∗1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Rizzuti†2 1Departamento de Matemática, ICE, Universidade Federal de Juiz de Fora, MG, Brazil 2Departamento de Física, ICE, Universidade Federal de Juiz de Fora, MG, Brazil Abstract This paper is devoted to analyze the divergence of the electron self- energy in classical electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' To do so, we develop the basics on the theory of distributions and a method for obtaining corresponding ex- tensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' At first sight, electrostatics implies a divergence once we treat the electron as a charged point particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' However, our construction shows that its self-energy turns out to be an undetermined constant upon renor- malization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Appealing to empirical results we may fix it, demanding, for example, that all its mass comes from an electrostatic origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Keywords: Theory of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Extension of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Elec- tron self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 1 Introduction One of the most compatible matches between theory and experiment in physics is devoted to Quantum Electrodynamics (QED), as the precision on the electron magnetic moment goes far from expected [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Since its early days on QED computations, it became clear that the behavior of fields was more singular than usual functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In turn, this has led the community to stare at fields not as maps, but as distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In fact, for the case of the electric field ⃗E(⃗x, t) originated from a point particle, for instance, one would expect an ultra- violet divergence at origin, while � d3⃗xdt ⃗E(⃗x, t)f(⃗x) = ⃗E(f) is well behaved [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Here, f(·) is a smooth function of compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We are interested, in this manuscript, on the self-energy of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' While it has a fascinating history so depicted in [3], involving an entire war and a new generation of physicists developing regularization and renormalization techniques, the classical counter- part is often subdue, justifying our approach here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' ∗Corresponding author: heitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='ribeiro@estudante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='ufjf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='br †Contact: brunorizzuti@ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='ufjf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='br 1 Simply put, the self-energy of a charged particle, such as the electron, is the measure of the energy which it has when freed from any other interaction, be it with other particles or with given fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' One finds in the study of classical elec- trodynamics that, summed to the kinetic and potential energies given particles might have, a system composed of charged particles has a quantity of energy related to the electromagnetic field it generates [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The electromagnetic system we wish to examine could be seen, at first, as the simplest one: that of an electron, stationary, free from any other interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Seen as a point particle - that is, supposing it has no internal structure and can be solely described by the position in which its whole charge is stored - which is the standard way one encounters at first [4,5], the electric field and potential are given by E(r) = 1 4πǫ0 e r2 ˆr, V (r) = 1 4πǫ0 e r , (1) where e denotes the strength of its charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Meanwhile, the expression for the self-energy for a system with electric field E and magnetic field B is E = ǫ0 2 � R3 � E2 + c2B2� dτ, (2) so that, using (1), we obtain E0 = ǫ0 2 � R3 � 1 4πǫ0 �2 � e r2 �2 dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We denote E0 the self-energy of interest here, as we are allegedly neglecting the magnetic field due to our interest only in the static case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Using spherical coordinates,1 E0 = e2 (4πǫ0)2 �� 2π 0 dφ � �� π 0 sen(θ)dθ � �� +∞ 0 1 r2 dr � = e2 8πǫ0 �1 r �0 +∞ = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (3) The conclusion we arrive then is that there is an infinite amount of energy stored in the field of a simple electron positioned at the origin of our system, if one considers it to be a stationary point particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Needless to say, any sat- isfactory field theory, both classical or quantum, must resolve such type of di- vergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Should we discard the assumption that the electron has no spatial 1Since different materials might use different notations concerning the polar coordinates θ and φ, we make explicit that we are considering here \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 x = r senθ cos φ, y = r senθ senφ, z = r cos θ, where θ ∈ [0, π], φ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 2 extension?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Both theoretical and experimental results seem to point in an oppo- site way [6,7], indicating that we should seek a improvement in the theory and in the conception of self-energy itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Therefore, here we present one of the available methods for the “removal” of such infinite quantities, a process known as renormalization [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The main idea of renormalization is that these infinities can be justified by attributing them to quantities which we cannot directly measure (something that can be seen as a parallel with the acceptance of complex numbers in the formalism of quantum mechanics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Take, for example, our case of the electron and its infi- nite self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In calculating E0, we are, simultaneously, calculating its mass melec arising from the electric field of such particle, since Einstein’s relativity theory affirms that mass and energy are but two manifestations of the same phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In light of this, we conclude that the divergence of E0 implies that the electron possesses infinite inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If, however, we assume there exists another contribution for the effective mass (that is, the one we can actually mea- sure), originated from some unknown effect other than electromagnetism, then we might conceive that this new contribution is negative enough to oppose the infinite appearing from melec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The hypothesis of another source contributing for the effective mass is not something difficultly justified, since we know that neutral bodies are also provided with mass and generate no electric or magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, assuming this new contribution for the mass of the particle, inde- pendent of where it comes from, we can “erase” the infinite we have just found, obtaining the so called mass renormalization of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Such initial method of renormalization gave rise to new and more advanced approaches which came to be used in the renormalization of other infinite quan- tities, specially in the Quantum Field Theory (QFT), which advanced quite a lot in the decades following the emergence of quantum mechanics and pre- sented similar problems with divergent integrals in its equations [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For this new theory, the methods had to be refined in its mathematical formulation and, meanwhile, the development of approaches such as constructive quantum field theory [2] and causal perturbation theory [10, 11] made clear that distribution theory is a crucial subject for the understanding of such new ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It was also through the latter that it was perceived the connection between the appearance of divergent quantities and the product of distributions, whereas distribution theory is strictly linear, as we shall see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This obstacle can be overcome using the idea of extension of distributions, which we will encounter in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' What we shall present here is a direct consequence of the work of Epstein and Glaser [11] and texts which adapted their ideas [12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In view of this, we seek, in the present work, to develop the theory of dis- tributions, elaborating next a method for extension certain distributions and, finally, exemplifying how it can be used in the renormalization of the self-energy of the electron which we have just introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' All this chain was written to be both self-consistent and also, a pedestrian introduction to the theme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 3 2 Distributions Looking through the available literature, we may find different approaches to the theory of distributions, ranging from superficial ones [2,4] to more advanced and detailed studies on the subject [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Here, we shall confine ourselves to a more peripheral point of view, since the reader interested in our work is expected to be, at a certain level, familiar with these ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Moreover, the available literature is rich enough to permit the search for the missing gaps we might leave along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In any case, the manuscript contains the standard definitions/results on the subject and a couple of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' They are intended to make the text as self-consistent as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1 Space of test functions First of all, we must make clear our notation for derivations, since we shall deal quite often with multi-variable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For the euclidean space Rn, we will consider the norm ∥·∥ : Rn → R+ to be ∥x∥ = � x2 1 + x2 2 + · · · + x2n, arising from the scalar product ⟨x, y⟩ = x1y1 + x2y2 + · · · + xnyn, and the topology will be the one derived from the metric our norm provides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We define a multi-index β as an n-tuple (β1, · · · , βn) of natural numbers (also written β ∈ Nn 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Given a multi-index β and a function ϕ : Rn → C, we shall denote the series of partial derivations as follows Dβϕ(x) := ∂|β|ϕ ∂xβ1 1 · · · ∂xβn n (x), when ϕ is such that this new function exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Here, |β| is the order of β, given by |β| = β1 + · · · + βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Furthermore, we define the factorial of β as β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' = β1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' · · · βn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=', notation that will come at hand later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' As elements of the n-dimensional space of natural numbers, β ∈ Nn 0, two multi-index can be added and produce a new one, α+β = (α1+β1, · · · , αn+βn), from which we shall obtain the operation Dα(Dβϕ) = D(α+β)ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For functions which are smooth enough to permit changes in the order of derivations, and this will be always the case here, this operation is guaranteed to be associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Moreover, we are able to define, over Nn 0, a notion of partial order, establishing α ≤ β when αi ≤ βi for all i ∈ {1, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' From two multi-index, α and β, we set min{α, β} = (min{α1, β1}, · · · , min{αn, βn}) and analogously for max{α, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 4 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' As is set by the literature, for the multi-index α = (0, · · · , 0), we appoint Dαϕ = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It is possible to demonstrate the interesting parallel between Leibniz rule applied to multi-variable derivations and the Newton’s binomial expansion Dβ(ϕψ) = � 0≤α≤β β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (β − α)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='Dα(ϕ)Dβ−α(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' As an example of the use of the multi-index notation in dealing with smooth functions, consider two multi-indices β, α and let f ∈ C∞(Rn) be the function f(x) = xβ = xβ1 1 · · · xβn n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In this case, Dαf(x) = � ∂α1 ∂xα1 1 xβ1 1 � � ∂α2 ∂xα2 2 xβ2 2 � · · � ∂αn ∂xαn n xβn n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We can thus see that, if αi > βi for some i ∈ {1, 2, · · · , n}, then the deivation of order αi of the xβi i term will result in the null function, meaning Dβf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If, on the other hand, we have α ≤ β, then Dβf(x) = � β1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (β1 − α1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='xβ1−α1 1 � � β2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (β2 − α2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='xβ2−α2 2 � · · � βn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (βn − αn)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='xβn−αn n � , or, in other words, Dβf(x) = β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (β − α)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='xβ−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In particular, (Dαf)(0) = � α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=', se α = β, 0, se α ̸= β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (4) As we mentioned briefly in the last paragraph, we will always be considering complex functions which are smooth enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' More precisely, we shall work in a subspace of the vector space of infinitely differentiable functions, C∞(Rn), namely the subspace of functions which vanish outside some compact K ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' To make our words more exact and mathematically grounded, we define first the concept of the support of a function ϕ ∈ C∞(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' That is the smallest closed set containing all the points where ϕ does not vanish, in other words, the closure of {x ∈ Rn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' ϕ(x) ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We shall denote the support of ϕ by supp ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We may now define in precise terms the space of functions we deal with when introducing distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We define D(Rn), the space of test functions, as the space containing elements ϕ : Rn → C of C∞(Rn), meaning infinitely diferentiable functions, whose support is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, we can write D(Rn) := {ϕ ∈ C∞(Rn) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' supp ϕ is a compact set}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 5 We encounter no difficulties when trying to prove that D(Rn) is indeed a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The sum of two functions ϕ, ψ ∈ D(Rn), with support K1 = supp ϕ and K2 = supp ψ, will have its support contained in the set K = K1∪K2, which will again be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Besides that, it is trivial the fact that the support of the function zϕ is equal to supp ϕ, for every z ̸= 0, from which we see that the multiplication by a scalar will be closed in D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The reader familiar with the area of functional analysis may recognize D(Rn) by the name of C∞ 0 (Rn) and might question our choice of symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Our notation is justified, however, by the notion of convergence we shall impose over D(Rn), something that is not touched upon when the focus is other than the theory of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Here, a sequence (ϕk)k∈N of functions ϕk ∈ D(Rn) is said to converge to a function ϕ ∈ D(Rn) when There exists a compact set K ⊂ Rn and a natural number k0 such that, for every k ≥ k0, we have supp ϕk ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For every multi-index β, Dβϕk converges uniformly to Dβϕ in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Consider h > 0 and let ψh : (−h, h) −→ R be given by ψh(x) = 1 x2−h2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' First of all, we observe that lim x→h− ψh(x) = lim x→h− x2 (h/x)2 − 1 = +∞ (5) and the same will happen limx→−h+ ψh(x) = +∞, by reasons of symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Moreover, in the interval of definition, ψh is smooth, since it is the composi- tion of smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For that reason, ξh(x) = eψh(x) shall also belong to C∞((−h, h)) and, by (5), limx→h− ξh(x) = limx→−h+ ξh(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Taking the first derivative of ξ, we obtain ξ′ h(x) = ψ′ h(x)eψh(x) = − � 2x (h2 − x2)2 � eψh(x), whose limits in −h and h will both be zero,2 lim x→h− ξ′(x) = − lim x→h− � 2x (h2 − x2)2 � eψh(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Taking one more derivative, we find ξ′′ h(x) = � 4x2 (h2 − x2)4 − 8x2 (h2 − x2)3 − 2 (h2 − x2)2 � eψh(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (6) Once again, taking the limits x → h− e x → −h+, we shall have again zero, since the exponential term will dominate any polynomial term in the denomi- nator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Generally speaking, we can extend this result to any derivative of ξh(x), lim x→−h+ ξ(k) h (x) = (−1)k lim x→h− ξ(k) h (x) = 0, (7) 2From the fact that ψh is an even function, ψ′ h will be odd, so that limx→−h+ ξ′(x) = − limx→h− ξ′(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 6 which permits us to define the following test function ηh ∈ D(R) ηh(x) = � 0, if |x| ≥ h, ξh(x), if |x| ≤ h, (8) whose support is, by definition, supp ηh = [−h, h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' By itself, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2 should be interesting enough to give us the general look of test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Besides that, we can use ηh to construct new test functions which will be even more useful ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Let M, h > 0 be any two positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We define the test function ηM,h ∈ D(R) by simply translating and stretching of the function given in (8), ηM,h(x) = � 0, if x /∈ [M, M + h], 1 Ch ξh(2(x − M − h)), if x ∈ [M, M + h], (9) where Ch is a constant of normalization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=', Ch = � ∞ −∞ ξh(2(t − M − h))dt, which implies � ∞ −∞ ηM,h(t)dt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' As we said, we have translated and stretched the support of our test function, so that now we have supp ηM,h = [M, M + h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' With this, when considering the integral � x −∞ ηM,h(t)dt, we obtain zero if x ≤ M and unity when x ≥ M + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Another fact easily seen is that this function is also infinitely differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We cannot say, however, that ηM,h belongs to D(R), since it does not possesses compact support, the same happening to 1 − ηM,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If, on the other hand, we consider Rn and take the radial coordinate ∥x∥ = � x2 1 + · · · + x2n as input, we can define ζM,h : Rn → R which will be a test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Subsequently, we define ζM,h by ζM,h(x) = 1 − � ∥x∥ −∞ ηM,h(t)dt, (10) obtaining ζM,h infinitely differentiable and with support supp ζM,h = BM+h(0), where BM+h(0) denotes the closed ball centered at the origin and with radius M + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' From that, it follows that ζM,h ∈ D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 7 The importance of this example is better appreciated when we consider the product ζM,hg, where we can, for now, consider g to be only locally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In this case, we obtain a function which is equal to g in BM(0) and zero outside the ball BM+h(0), also satisfying |ζM,h(x)g(x)| ≤ |g(x)| for all x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Since, in our definition of ζM,h, M and h were any positive real numbers, we can make the domain of coincidence of ζM,hg and g as big as we want to and the ring BM+h(0)\\BM(0) as narrow as we wish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Therefore, if we integrate this product and take the limit h → 0, we have, by the Lebesgue Dominated Convergence Theorem (see [19]), lim h→0+ � Rn ζM,h(x)g(x)dnx = � Rn � lim h→0 ζM,h(x)g(x) � dnx = � Rn g(x)χBM (0)(x)dnx, (11) where χBM(0)(x) is the characteristic function of the ball BM(0), defined by χBM(0)(x) = � 1 , if x ∈ BM(0), 0 , if x /∈ BM(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (12) In other words, lim h→0+ � Rn ζM,h(x)g(x)dnx = � BM(0) g(x)dnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This fact will shall be important later ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We have now all the necessary ingredients to define our main objects of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' A distribution is an element of D′(Rn), meaning it is a linear continuous functional, in the sense of convergence defined in D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The action of a given distribution T ∈ D′(Rn) over an element ϕ of D(Rn) may be written in different ways and here we shall denote it by ϕ �−→ ⟨T, ϕ⟩ = T (ϕ), referencing the inner product notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This convention will be justified in more details below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2 Distributions Given a first look at Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2, it may not be clear how one can say that distributions are the generalization of locally integrable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We can, 3As usual, D′(Rn) stands for the dual topological space of D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 8 however, demonstrate that it is so by working on our first example of an element of D′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Given a function f ∈ L1(Rn), let us define the following operation, for each ϕ ∈ D(Rn) , Tf(ϕ) = � Rn f(x)ϕ(x) dnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (13) The condition that f be locally integrable is clearly seen to be necessary for this application to be well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Since ϕ vanishes outside some compact set, the integration is only performed in this set and since it is infinitely differentiable, ϕf will also be locally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We affirm that expression (13) defines a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Indeed, the operations of multiplication by f and integration are well known to be linear, resulting in the linearity of the composition of both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For the continuity, taking a sequence (ϕk)k∈N ⊂ D(Rn) converging to ϕ ∈ D(Rn), it follows from our notion of convergence in D(Rn) that there exists a compact set K containing supp (ϕk−ϕ), for all k sufficiently big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Furthermore, since this convergence is uniform, we have, for every ε > 0 and every x ∈ K, there exists k0 ∈ N such that, for k ≥ k0, |ϕk(x)| ≤ |ϕ(x)| + |ϕk(x) − ϕ(x)| ≤ C + ε, where C = sup {|ϕ(x)| , x ∈ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, |fϕk| is bounded by (C + ε) |f| for all k ≥ k0 and, since this is an uniform bound in K, we can apply the Dominated Convergence Theorem to conclude that lim k→∞⟨Tf, ϕk⟩ = lim k→∞ � Rn f(x)ϕk(x)dnx = � Rn � lim k→∞ f(x)ϕk(x) � dnx = � Rn f(x)ϕ(x)dnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In other words, we have obtained lim k→∞⟨Tf, ϕk⟩ = ⟨Tf, ϕ⟩, proving that Tf is indeed a linear continuous functional over D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Such distributions, characterized by a locally integrable function f, are called regular distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This gives us the idea that we can construct, from a function f ∈ L1(Rn), a distribution Tf ∈ D′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Still, it does not tells yet how we can view such functions as proper elements of D′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Could we, for instance, have different functions representing the same regular distribution?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The answer is yes and is justified by the fact that changing the quantity inside an integral in a discrete set does not alter its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' More generally, any change in a null measure set leaves the integral unaltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This fact implies that two functions which are equal almost everywhere4 define the same regular distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We can prove, however, that this is the only way we get this coincidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Two functions f, g ∈ L1 loc(Rn) define the same distribution, in the sense that Tf = Tg, if and only if f is equal to g almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 4With that we mean that they are equal outside a null measure set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (⇐) Firstly, if A is the null set where f differs from g and B = Rn\\A = Rn ∩ Ac, then for each ϕ ∈ D(Rn), ⟨Tf, ϕ⟩ = � Rn f(x)ϕ(x)dnx = � A f(x)ϕ(x)dnx + � B f(x)ϕ(x)dnx = 0 + � B f(x)ϕ(x)dnx = � B g(x)ϕ(x)dnx = ⟨Tg, ϕ⟩, (14) where we have used f = g in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Since this is valid for every ϕ ∈ D(Rn), we achieve Tf = Tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (⇒) If we suppose now that f and g define the same distribution, then ⟨f, ϕ⟩ = ⟨g, ϕ⟩ for every test function ϕ ∈ D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In particular, we can take ϕ = ζM,h ∈ D(Rn) as constructed in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3, being define by the following: 0 ≤ ζM,h(x) ≤ 1 for all x ∈ Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' ζM,h(x) = 1 for x in the ball centered at x0 and of radius M > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' ζM,h(x) = 0 for x outside the ball centered at x0 and of radius M + h > 0, with h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Therefore, we have ⟨f, ζM,h⟩ = ⟨g, ζM,h⟩ ⇒ � Rn f(x)ζM,h(x)dnx = � Rn g(x)ζM,h(x)dnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' By the conclusion of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3, taking the limit h → 0 we obtain at last � BM(x0) f(x)dnx = � BM(x0) g(x)dnx, which in turn implies that f and g are equal outside a null measure set in BM(x0) (see, for example, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='10 in [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Since x0 is arbitrary and we can cover Rn with a countable number of balls of radius M, we obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In view of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1, if we consider elements of L1 loc(Rn) to be the equiv- alence classes,5 defined by the requirement that two functions are equivalent if, and only if, they are equal almost everywhere, then we shall have an one- to-one correspondence between regular distributions and elements f ∈ L1(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For this reason, it does not clouds our understanding when we refer to a reg- ular distribution the function which characterizes it, justifying also the name generalized functions, sometimes attributed to distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Regular distributions are not, however, the only class of elements in D′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Those which are not defined as in (13) by some f ∈ L1(Rn) are called singular distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The first example of a singular distribution we can give is the famous Dirac delta distribution δ, which can finally put a well defined meaning to symbols such as δ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We define the distribution δ ∈ D′(Rn) by the expression δ(ϕ) = ϕ(0), ∀ ϕ ∈ D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (15) 5Still denoting them by f ∈ L1(Rn), meaning we identify each class with one of its repre- sentatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 10 If ϕ, ψ are functions in D(Rn) and z ∈ C is a complex number, then δ(zϕ + ψ) = (zϕ + ψ)(0) = zϕ(0) + ψ(0) = zδ(ϕ) + δ(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Besides that, if ϕk → ϕ in D(Rn), then the point convergence ϕk(0) → ϕ(0) is promptly assured, so that ⟨δ, ϕk⟩ → ⟨δ, ϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' With this, we prove the continuity of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' To leave things clear, the notation δ(x) or expressions of the form δ(ϕ) = � +∞ −∞ δ(x)ϕ(x)dx (16) become only abuses of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It is not possible to define a legitimate function δ : R −→ R such that δ = Tδ(x), making (16) nothing more than a sometimes useful convention on notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Indeed, if we suppose there exists such function δ(x), then we can show that, in R\\{0}, it must be equal to the null function outside a null measure set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This is given by the fact that, for test functions ϕ such that supp ϕ ⊂ R\\{0}, � A δ(x)ϕ(x)dx = ϕ(0) = 0, where A is a subset of R\\{0} containing supp ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, if N is the mentioned null measure set, N ∪ {0} remains of null measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Therefore, we would have δ(ϕ) = � +∞ −∞ δ(x)ϕ(x)dx = 0, now for ϕ in D(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This is a contradiction with definition (15) of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We shall utilize the symbol δx0, with x0 ∈ Rn to represent the singular distribution δx0(ϕ) = ⟨δx0, ϕ⟩ = ϕ(x0) , ∀ ϕ ∈ D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' With this, the delta distribution as defined in (15) is nothing more than δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We will, however, write the more compact form, for convenience, only explicitly writing the point x0 of evaluation when it is different from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Continuing on the topic of the delta distribution, we may find materials where it is introduced as the limit of a sequence of smooth functions [4, 20], providing but an intuition of the meaning of δ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Now that we have seen the rigorous definition of δ and expressed the space in which it lives, we can in fact solidify this idea of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' As a consequence, we also have, as expected, a generalization of the convergence in the function spaces we have defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Let (Tk)k∈N be a sequence of distributions in D′(Rn) and let T ∈ D′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We say that Tk converges to T if, for every ϕ ∈ D(Rn), we have the convergence (in C) limk→∞⟨Tk, ϕ⟩ = ⟨T, ϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We denote this convergence simply by Tk → T or Tk D′(Rn) −−−−−→ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 11 With this definition, the idea of defining the Dirac delta distribution as a limit of bona fide functions becomes rigorous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This convergence cannot happen point wise, but only if we view the sequence fk as a sequence in D′(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' With this in mind, we next prove a result which allows us to obtain an infinity of sequences of regular distributions converging to δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' These are called Dirac sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Let f : Rn → R be an integrable function such that � Rn f(x)dnx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Then, the sequence fk defined by fk(x) = knf(kx) is such that fk → δ, as a sequence of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If ϕ is an element of D(Rn), then � Rn knf(kx)ϕ(x)dnx y=kx = � Rn f(y)ϕ(y/k)dny = � Rn f(y) (ϕ(y/k) − ϕ(0)) dny + � Rn f(y)ϕ(0)dny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (17) Now, since ϕ is continuous by definition, it follows that lim k→∞ f(y) (ϕ(y/k) − ϕ(0)) = 0, ∀ y ∈ Rn and, since |f(y) (ϕ(y/k) − ϕ(0))| ≤ 2 ∥ϕ∥ |f(y)| , we may apply the Lebesgue Dominated Convergence Theorem, obtaining lim k→∞ � Rn f(y) (ϕ(y/k) − ϕ(0)) dny = � Rn lim k→∞ f(y) (ϕ(y/k) − ϕ(0)) dny = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, by (17), we have lim k→∞ � Rn knf(kx)ϕ(x)dnx = lim k→∞ ϕ(0) � Rn f(y)dny = ϕ(0) = ⟨δ, ϕ⟩, as we wished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Note that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1 claims we can actually construct a Dirac sequence from any integrable function f whose integral is different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' To see this, we need only to take g = Cf, where C ∈ R is the constant �� Rn f(x)dnx �−1, and use then gk(x) = kng(kx) as the sequence contained in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' With Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1 and our last remark, we can now easily obtain some examples of Dirac sequences: fk(x) = k √ 2πe−k2x2 , gk(x) = 1 π k 1 + k2x2 , hk(x) = 1 πk sen2(kx) x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3 New distributions from old ones What we wish to do now is construct some of the operators that take elements of D′(Rn) and return new elements of the same space, much like the operations of sum and product by a scalar z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We are used to such operations acting on function spaces, with derivations, products or convolutions being the main cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Here we are going to translate some of these into the language of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It must be clear that, since we are dealing with a generalization of functions, the operations we want to define should also be generalization of the ones we already know and this, in reality, gives us the insights we need to construct said operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We begin with the concept of the derivative of a distribution T ∈ D′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Given a function f ∈ L1 loc(Rn) which admits a first derivation with respect to some variable xi and whose derivative ∂f ∂xi also belongs to L1 loc(Rn), we have, for ϕ ∈ D(Rn), � ∂f ∂xi , ϕ � = � Rn ∂f ∂xi (x)ϕ(x) dnx = � +∞ −∞ dx1 · · · � +∞ −∞ dxn �� +∞ −∞ ∂f ∂xi (x)ϕ(x)dxi � and, using integration by parts, � +∞ −∞ ∂f ∂xi (x)ϕ(x)dxi = f(x)ϕ(x) ��� xi=∞ xi=−∞ − � +∞ −∞ f(x) ∂ϕ ∂xi (x) dxi = − � +∞ −∞ f(x) ∂ϕ ∂xi (x) dxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (18) Since ∂ϕ ∂xi is again a test function, we can write � ∂f ∂xi , ϕ � = − � f, ∂ϕ ∂xi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' By induction, we can easily extend this result to any multi-index β, since any derivation of a test function will again be a test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Paying attention to the factor of (−1) we must insert at each step, we have at last � Dβf, ϕ � = (−1)|β|� f, Dβϕ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (19) Thus, the logical extension of this result to a general distribution T ∈ D′(Rn) is given by the following Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For any multi-index β, the derivative Dβ of a distribution T ∈ D′(Rn) is a new distribution DβT ∈ D′(Rn) defined by � DβT, ϕ � = (−1)|β|� T, Dβϕ � , ∀ ϕ ∈ D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (20) The fact that equation (20) indeed represents a new distribution is a straight- forward consequence of the facts that the operation Dβ is linear over D(Rn) and that the convergence ϕn → ϕ in D(Rn) implies, by construction, in the conver- gence Dβϕn → Dβϕ, again in D(Rn), whatever β we may have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 13 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Perhaps the most interesting aspect of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='4 - something that can actually be seen as the most interesting aspect of the theory of distri- butions itself - is that, being ϕ ∈ D(Rn) infinitely differentiable, DβT is well defined for every β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In other words, distributions are also infinitely differen- tiable objects, and this is given from the start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We have no need in restraining the space we work with to obtain infinite smoothness, something which is very much desired in almost any theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Consider the uni dimensional case where f is sectionally dif- ferentiable, that is, f ′ exists for every point of R outside a finite set, say {x1, x2, · · · , xm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Suppose, further, that f is such that the limits f(x+ i ) = limx→x+ i f(x) and f(x− i ) = limx→x− i f(x) exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The derivative of the regular distribution Tf is thus given by Tf ′ = Tf ′ + m � i=1 σiδxi, (21) where σi = f(x+ i ) − f(x− i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Indeed, we know that � Tf ′, ϕ � = − � ∞ −∞ f(x)ϕ′(x)dx = − � x1 −∞ f(x)ϕ′(x)dx − m−1 � i=1 � xi+1 xi f(x)ϕ′(x)dx − � ∞ xn f(x)ϕ′(x)dx = − lim ε→0 �� x1−ε −∞ f(x)ϕ′(x)dx + m−1 � i=1 � xi+1−ε xi+ε f(x)ϕ′(x)dx + � ∞ xn+ε f(x)ϕ′(x)dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Applying integration by parts, we obtain � x1−ε −∞ f(x)ϕ′(x)dx = f(x)ϕ(x) ��� x1−ε −∞ − � x1−ε −∞ f ′(x)ϕ(x)dx, � ∞ xn+ε f(x)ϕ′(x)dx = f(x)ϕ(x) ��� ∞ xn+ε − � ∞ xn+ε f ′(x)ϕ(x)dx, � xi+1−ε xi+ε f(x)ϕ′(x)dx = f(x)ϕ(x) ��� xi+1−ε xi+ε − � xi+1−ε xi+ε f(x)ϕ′(x)dx, which implies, passing the limit ε → 0, � Tf ′, ϕ � = � ∞ −∞ f ′(x)ϕ(x)dx + lim ε→0 �m−1 � i=1 f(xi+1 + ε)ϕ(xi+1 + ε) − f(xi − ε)ϕ(xi − ε) � = � ∞ −∞ f ′(x)ϕ(x)dx + m−1 � i=1 σiϕ(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 14 This equation is exactly the equality of distributions we wished to prove, (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' On the other hand, we have no reason to believe that general results such as (21) may be obtained for singular distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The only information we have for DβT in this case is expression (20), defining it as a continuous functional over D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The derivative of order m of the Dirac delta distribution δx0 ∈ D(R), for example, is given by � δ(m), ϕ � = (−1)mϕ(m)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (22) This follows directly from equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Now, the next operation we define over D′(Rn) is the multiplication of dis- tributions by smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We develop these new elements like we have done for DβT , dealing first with regular distributions and then generalizing for the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If T = Tf and h is an infinitely differentiable function, we can write ⟨hf, ϕ⟩ = � Rn h(x)f(x)ϕ(x) dx = ⟨f, hϕ⟩, since hϕ is again a infinitely differentiable function which has support contained in the support of ϕ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=', hϕ ∈ D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If this is the relation which we seek to preserve when working with regular distributions, our definition for the product of a distribution with a function must clearly be the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Given a distribution T and a function h ∈ C∞(Rn), the prod- uct hT ∈ D′(Rn) is defined by ⟨hT, ϕ⟩ = ⟨T, hϕ⟩, ∀ ϕ ∈ D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (23) Alike the case of derivatives of distributions, the linearity of hT follows from the linearity of the operation ϕ �→ hϕ, whereas the continuity is assured by the preservation of the convergence in D(Rn) by this operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This last assertion requires a little more mathematical rigor, which we give now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For any T ∈ D′(Rn) and h ∈ C∞(Rn), expression (23) defines a new distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Indeed, if ϕk → ϕ in D(Rn), ��Dβ {h(ϕk − ϕ)} (x) �� ≤ � α≤β β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (β − α)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' ��Dβ−αh(x) �� |Dα(ϕk − ϕ)(x)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, if K ⊂ Rn is the compact set such that supp ϕk ⊂ K for k sufficiently big, then we have ��Dβ {h(ϕk − ϕ)} (x) �� ≤ � α≤β β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (β − α)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='Cα ∥ϕk − ϕ∥β , 15 where Cα = sup{ ��(Dβ−αh)(x) �� , x ∈ Rn} are constants independent of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' There- fore, lim k→+∞ ∥h(ϕk − ϕ)∥β ≤ � α≤β β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (β − α)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='Cα � lim k→+∞ ∥ϕk − ϕ∥β � = 0, proving that hϕk converges in D(Rn) for hϕ and, consequently, ⟨hT, ϕk⟩ con- verges to ⟨hT, ϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We conslude from this that hT is, indeed, a linear and con- tinuous functional over D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The main example is the one related to the Dirac delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Given any h ∈ C∞(Rn) and a test function ϕ ∈ D(Rn), we have ⟨hδx0, ϕ⟩ = ⟨δx0, hϕ⟩ = h(x0)ϕ(x0) = h(x0)⟨δx0, ϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We see then that, the only value necessary to define hδx0 is h(x0), that means, hδx0 = h(x0)δx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' A particular result, and one of great importance, is when n = 1, x0 = 0 and h(x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In this case, we obtain xδ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 3 Extension of distributions As we have hinted in Section 1, the developments of QFT showed that renor- malization in causal perturbation theory depends heavily on distribution the- ory, more specifically on the procedures for obtaining extensions of distributions whose behavior at the origin (this nomenclature will become clear later) does not allow that we apply these over functions whose support contains the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Our main problem then becomes: Given a distribution T0 which is only well defined when we apply it on test functions ϕ ∈ D(Rn) for which 0 /∈ supp ϕ, how can we construct a new distribu- tion T ∈ D′(Rn) which is the extension of T0, that is, such that ⟨T0, ϕ⟩ = ⟨T, ϕ⟩ whenever 0 /∈ supp ϕ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The method for constructing such extensions is our goal in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The ideas here presented originate mainly from [12, 13, 21] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Some passages in these papers, however, may be too straightforward for an unfamiliar public, perhaps in virtue of the public they are directed to, this being researchers more familiar with the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For that reason, we wish here to perhaps fill some blanks one could find during those reads, providing then a more accessible text to less experienced audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1 Distributions with dependence in one parameter Here, we deviate slightly from the common approach adopted by most of the literature, which takes what we will do now as known, giving more space to certain results crucial in later moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Our main focus now will be distributions with dependence on a real param- eter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' With this, we mean to say that we will study a family of distributions in D′(Rn) of the form {Tµ ∈ D′(Rn) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' µ ∈ R}, and explore the analytic properties of this dependence over µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Firstly, let us see that, for any distinct µ1, µ2 ∈ R, Fµ1,µ2 = Tµ1 − Tµ2 µ1 − µ2 is well defined as an element of D′(Rn), with action given by ⟨Fµ1,µ2, ϕ⟩ = 1 µ1 − µ2 [⟨Tµ1, ϕ⟩ − ⟨Tµ2, ϕ⟩] ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If, moreover, for all test function ϕ and for all µ ∈ R, the limit lim δµ→0⟨Fµ+δµ,µ, ϕ⟩ exists, then we are capable of defining the distribution � d dµTµ, ϕ � := lim δµ→0 1 δµ⟨Tµ+δµ − Tµ, ϕ⟩, ∀ µ ∈ R , ∀ ϕ ∈ D(Rn), which will be the limit, in the sense of distributions, of Fµ+δµ,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Is should be clear that this new distribution appears itself to be equivalent to the derivation of the application µ �→ Tµ and should not be confused with our previous definition of derivative DαT of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Both represent new objects belonging to D′(Rn) and are constructed from T , but whereas the former is given by a differentiation in relation to the parameter µ of the family Tµ, without any mention to test functions, the latter is dependent entirely upon derivations on each ϕ and its variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Let us consider now the following: given the application µ �→ Tµ, we can construct a new, complex, function for each ϕ ∈ D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For this, we take η: R −→ C µ �−→ η(µ) = ⟨Tµ, ϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (24) It follows from this composition that dη dµ(µ) = lim δµ→0 1 δµ (⟨Tµ+δµ, ϕ⟩ − ⟨Tµ, ϕ⟩) = lim δµ→0⟨Fµ+δµ,µ, ϕ⟩, that means, dη dµ(µ) = � d dµTµ, ϕ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (25) 17 This is an important result, justifying then its reiteration in an alternative form: For any ϕ ∈ D(Rn) and µ ∈ R, we have � d dµTµ, ϕ � = d dµ⟨Tµ, ϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (26) Besides that, we can, through η, define a new distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If η is continuous, we know that, for a ∈ R, H(µ) = � µ a η(µ)d µ is a well defined complex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We have thus obtained, from a test function ϕ and a fixed µ ∈ R, a new element I ∈ D′(Rn), characterized by ⟨I, ϕ⟩ = H(µ) = � µ a η(µ)d µ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In reality, we shall utilize another symbol to reference I, defining � µ a d µ Tµ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This nomenclature permits us to write �� µ a d µ Tµ, ϕ � = � µ a d µ ⟨Tµ, ϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In particular, if we take the integration of d dµTµ, we obtain, due to (26), �� µ a d µ � d dµTµ � , ϕ � = � µ a d µ � d dµTµ, ϕ � = � µ a d µ d dµ⟨Tµ, ϕ⟩ = η(µ) − η(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (27) We see then that we have obtained a version of the Fundamental Theorem of Calculus, � µ a d µ d dµTµ = Tµ − Ta, (28) for some mapping from R to D′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This result will be an important piece for our main theorems in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2 Extensions of distributions Let us consider a regular distribution Tf = f ∈ D(Rn) such that f(x) = 0 for every x belonging to a subset U ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It is then evident that, for all ϕ ∈ D(Rn) such that supp ϕ ⊂ U (we write, in this case, ϕ ∈ D(U)), we have 18 ⟨f, ϕ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This can be, therefore, a form of characterizing the support supp f of the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' With the intent of extending this notion to distributions T ∈ D′(Rn), we say that T is zero in a subset U ⊂ Rn when ⟨T, ϕ⟩ = 0 , ∀ ϕ ∈ D(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We are thus able to define the support of a distribution T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For a given T ∈ D′(Rn), the support of T is the subset supp T ⊂ Rn given by supp T := {x ∈ Rn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' x does not contain a neighborhood in which T is zero}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In an equivalent manner, supp T can be seen as the complement of the biggest subset in which T is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This definition, in turn, allows us to define an important subspace of D′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For any subset U ⊂ Rn, the subspace of D′(Rn) given by the distributions such that supp T ⊂ U will be denoted by D′(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This notation comes from the clear idea that we can associate this subset with the space of distributions whose arguments are test functions in D(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Given the distribution δx0 ∈ D′(Rn), we know that, for any ϕ ∈ D(Rn) such that ϕ(x0) = 0, that is, such that x0 /∈ supp ϕ, we have ⟨δx0, ϕ⟩ = ϕ(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It follows from this that supp δx0 = {x0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We can obtain a reciprocate from this last result with the following lemma (for the proof, see for example [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If T ∈ D′(Rn) is such that supp T = {x0}, then there exists m ∈ N and constants cν, |ν| ≤ m, such that T = � |ν|≤m cνDνδx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (29) In what follows, we define a quantity which probes the behavior of a dis- tribution T in the origin, in terms of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Since we have drawn a clear distinction between bona fide functions (regular distributions) and general (singular) distributions, we must develop for the later the idea of studying the behavior of T at some point in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It is in this ground that we introduce the concept of pull-back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This definition is characterized by a transformation Φ : Rn → Rn, which we consider here to be invertible for simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For a function f ∈ L1 loc(Rn), the pull-back Φ∗f : Rn → C of f over Φ is a new complex function given by Φ∗f(x) = f(Φ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 19 Therefore, if viewing f as a regular distribution, we will have, for every ϕ ∈ D(Rn),6 ⟨Φ∗f, ϕ⟩ = � Rn f(Φ(x))ϕ(x)dnx = � Rn f(y)ϕ(Φ−1(y)) ��DΦ−1(y) �� dny and, finally, ⟨Φ∗f, ϕ⟩ = ⟨f, |DΘ(y)| Θ∗ϕ⟩ , Θ = Φ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' With this in mind, we can finally define the pull-back of a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The pull-back of T ∈ D′(Rn) over a invertible transformation Φ : Rn → Rn is a new distribution Φ∗T , defined by ⟨Φ∗T, ϕ⟩ = ⟨T, |DΘ(y)| Θ∗ϕ⟩ , ∀ ϕ ∈ D(Rn), (30) where Θ = Φ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It is common the notation T (Φ(x)) (which we shall employ from here on, for conformity) for the distribution Φ∗T , making reference to the definition of the pull-back of a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We reiterate that the use of Φ invertible is a particular case of a definition that can be made more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For the generalization, it is defined Φ∗T as the limit, in the distribution sense, of the regular distributions Φ∗fn if fn is a sequence converging to T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The reader can utilize [17] for a source of deeper reading in the matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Let us take here some more lines to achieve a better understand- ing of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Despite the indication given by the notation T (Φ(x)), the pull-back of a distribution should not be read as an ordinary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If T is singular, then Φ∗T is only well defined as a new distribution, which will again be singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, we should not interpret T (Φ(x)) as an object which varies with x ∈ Rn, but actually as a distribution dependent on the transformation Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Nonetheless, when regarding simple cases, such as Φ(x) = λx (λ > 0), it is easier and perhaps more didactic to express the transformation directly as the argument of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For regular functions, Tf(Φ(x)) will be indeed a bona fide function, just like Tf itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Moreover, let us see that, if ϕ ∈ D(Rn) is a test function such that supp ϕ ⊂ BM(0), then supp ϕ(λ−1x) ⊂ BλM(0), so that ⟨f(λx), ϕ⟩ = � Rn f(λx)ϕ(x)dnx = � Rn f(x)ϕ(λ−1x)λ−ndnx = λ−n � BλM(0) f(x)ϕ(λ−1x)dnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (31) As we take the limit λ → 0+, the integration is performed over a ball with ever decreasing radius, that means, we evaluate the behavior of f in smaller and smaller neighborhoods of the origin, as indicated in our discussion preceding Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 6Here the symbol |DF (y)| represents the Jacobian of F : Rn −→ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 20 After such remarks, we present the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Let T ∈ D′(Rn) be a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The scaling degree of T is the real number (or ±∞), denoted here by σ(T ), such that σ(T ) = inf{s ∈ R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' λsT (λx) λ→0+ −−−−→ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='7 The singular order of T , denoted by ω(T ), is the value ω(T ) = [σ(T )] − n, where [m] denotes the biggest integer smaller (or equal) to m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Let us see some examples of distributions and their respective scaling degrees, making our definitions clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The examples will be useful to our further discussions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If T = δ ∈ D′(R), then σ(δ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This is given by the result δ(λx) = λ−1δ(x), which follows directly from (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' More generally, dealing with the n-dimensional case, δ ∈ D′(Rn), we have ��DΦ−1(y) �� = λ−n, if Φ(x) = λx, implying that ⟨δ(λx), ϕ⟩ = λ−n⟨δ, ϕ⟩, that means, σ(δ) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If T = P 1 x, then λs⟨T (λx), ϕ⟩ = λs−1P � +∞ −∞ 1 xϕ(λ−1x)dx = λs−1P � +∞ −∞ ϕ(y)λdy λy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Therefore, λsT (λx) λ→0+ −−−−→ 0 if, and only if, s > 1, that is, σ(T ) = 1, again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If f is a continuous function, of one variable, homogeneous with degree m, meaning f(λx) = λmf(x), then ⟨λsf(λx), ϕ⟩ = λs � +∞ −∞ λmf(x)ϕ(x)dx = λs+m⟨f, ϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It follows immediately from this that σ(f) = −m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, the scaling degree of a homogeneous function is the (additive) inverse of its homogeneity degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 7Here the convergence is in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 21 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We have seen that σ(δ) = n and, therefore, ω(δ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Let us see now what occurs for a derivation T = Dαδ of the delta distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We know that, for any ϕ ∈ D(Rn), ⟨Dαδ, ϕ⟩ = (−1)|α|⟨δ, Dαϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (32) With this, we can verify that, for λ > 0, ⟨T (λx), ϕ⟩ = (1/λn+|α|)⟨δ, ϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Indeed, from (32) and given Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2, λs⟨Dαδ(λx), ϕ⟩ = (−1)|α|λs−n� δ, Dαϕ(λ−1x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' From this, and from the equality Dαϕ(λ−1x) = λ−|α|Dαϕ(x), we conclude that λs⟨Dαδ(λx), ϕ⟩ = λs−(n+|α|)⟨Dαδ, ϕ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This implies that σ(T ) = n + |α| and, thus, ω(T ) = |α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In other words, the derivative Dα of the delta distribution increases its initial scaling degree by a factor ∥α∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Now that we have gone through some examples to fixate this new concepts and definitions, we shall cite a lemma which gather their main properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' De- spite its importance, we feel it is not necessary that we give here the complete demonstration of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For the idea of its demonstration, we refer the reader to [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Consider T, S ∈ D′(Rn), c ∈ C and β a multi-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Then, we have the following 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' σ(xβT ) = σ(T ) − |β|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' σ(DβT ) = σ(T ) + |β|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' σ(cT ) = σ(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' σ(ϕ) ≤ 0 and σ(ϕT ) ≤ σ(T ), for every ϕ ∈ D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' σ(T + S) ≤ max{σ(T ), σ(S)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=') of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2 comes from the evident fact that, if s ∈ R is such that λsT (λx) and λsS(λx) both converge to zero, the we cannot have anything other than λs(T +S)(λx) → 0, in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The reciprocal, however, is not necessarily true, which gives rise to the inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' However, for the particular case when T and S are both derivatives of the Dirac delta, say T = Dα1δ and S = Dα2δ, then we obtain the equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 22 Indeed, let us suppose, without loss of generality, that |α1| > |α2|, from which max{σ(T ), σ(S)} = σ(T ) = n + |α1|, by Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Now, taking s < n + |α1|, we have, for ϕ ∈ D(Rn), λs⟨(T + S)(λx), ϕ⟩ = λs [⟨Dα1δ(λx), ϕ⟩ + ⟨Dα2δ(λx), ϕ⟩] = λs � (−1)|α1|λ−n−|α1|⟨δ, Dα1ϕ⟩ + (−1)|α2|λ−n−|α2|⟨δ, Dα2ϕ⟩ � = (−1)|α1|λs−n−|α1|Dαϕ(0) + (−1)|α2|λs−n−|α2|Dαϕ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (33) Hence, if ϕ is such that Dα1ϕ(0) ̸= 0, then lim λ→0+ λs−n−|α1|Dαϕ(0) = ±∞, that means, λs⟨(T + S)(λx), ϕ⟩ diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This proves that n + |α1| must be a lower bound to {s ∈ R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' λs(T + S)(λx) λ→0+ −−−−→ 0}, which implies max{σ(T ), σ(S)} ≤ σ(T + S) and, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2, max{σ(T ), σ(S)} = σ(T + S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (34) We now posses the appropriate tools for proving the results that concern the proper extension of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' As we shall see, we must separate our problem into two cases, differentiated by a condition over the singular order of T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The biggest difference between the two cases, which are characterized by ω(T ) < 0 and ω(T ) ≥ 0, is in the uniqueness of our extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Let T0 ∈ D′(Rn\\{0}) such that σ(T0) = s < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Then there exists a unique distribution T ∈ D′(Rn) such that σ(T ) = s and ⟨T, ϕ⟩ = ⟨T0, ϕ⟩ , ∀ ϕ ∈ D(Rn\\{0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We first prove uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Indeed, if T1, T2 both are extensions of T0 in D′(Rn), then ⟨T1 − T2, ϕ⟩ = 0 for any function ϕ ∈ D(Rn\\{0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' On one hand, supp (T1 − T2) = {0} and, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1, we have T1 − T2 = � |ν|≤m cνDνδ, for some set of complex constants {cν ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' |ν| ≤ m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Supposing that some of these constants are not zero, we can use Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='4 to conclude that σ(T1 − T2) ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' On the other hand, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2 affirms that σ(T1 − T2) ≤ max{σ(T1), σ(T1)} < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This is a contradiction and from that we take that T1 − T2 ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For the existence, we first take χ ∈ D(Rn) a test function such that χ(x) = 1 in a neighborhood of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='8 For every µ > 0 and ϕ ∈ D(Rn), (1 − 8A clear possibility would be to take χ as in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 23 χ(µx))ϕ(x) = 0 is also a neighborhood of the origin, from which (1−χ(µx))ϕ(x) ∈ D(Rn\\{0}) and, therefore, the distribution Tµ = (1 − χ(µx))T0 (35) is well defined as an element of D′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Moreover, for every ϕ ∈ D(Rn\\{0}), taking µ0 > 0 big enough, we have χ(µx)ϕ(x) = 0 , ∀ x ∈ Rn and for µ ≥ µ0, thus the limit T = (Tµ)µ→∞ is defined for every argument in D′(Rn\\{0}) and lim µ→∞⟨Tµ, ϕ⟩ = ⟨T0, ϕ⟩ , ∀ ϕ ∈ D′(Rn\\{0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For the proof that T is defined over the whole space of test functions, take any ϕ ∈ D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' By the result (28) of subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1, we can rewrite Tµ as Tµ = T1 + � µ 1 d µ d dµTµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' On the other side, we know from (26) that � d dµTµ, ϕ � = d dµ⟨Tµ, ϕ⟩ = d dµ (⟨T0, (1 − χ(µx))ϕ⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (36) The dependence of this term on µ is now entirely within the argument and, since T0 is a continuous functional, we are able to transfer the derivation to the inside of the brackets, giving d dµ (⟨T0, (1 − χ(µx))ϕ⟩) = − � T0, d dµ(χ(µx)ϕ(x)) � = − n � i=1 ⟨T0, xi(∂iχ)(µx)ϕ(x)⟩, (37) where it was used that d dµ(χ(µx)) = n � i=1 d dµ(µxi) ∂χ ∂xi (µx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Now, for each term of the sum (37), we see that ⟨T0, xi(∂iχ)(µx)ϕ(x)⟩ = µ−1⟨ϕT0, (µxi)(∂iχ)(µx)⟩ = µ−(n+1)� (ϕT0)(µ−1x), xi(∂iχ)(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (38) Thus, from (36), (37) and (38), we obtain � d dµTµ, ϕ � = −µ−(n+1) n � i=1 � (ϕT0)(µ−1x), xi(∂iχ)(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (39) 24 Let us then take ε such that σ(T0) < ε < n, λ = µ−1 and ψi(x) = xi(∂iχ)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2, we know that σ(ϕT0) ≤ σ(T0) and, by the definition of scaling degree of T0, lim µ→∞ µ−ε� (ϕT0)(µ−1x), xi(∂iχ)(x) � = lim λ→0+ λε⟨(ϕT0)(λx), ψi⟩ = 0, (40) for all i ∈ {1, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Therefore, for λ small enough, that is, for µ big enough, it follows from (39) and (40) that µn+1−ε ���� � d dµTµ, ϕ ����� ≤ 1, that means, ���� � µ 1 d µ � d dµTµ, ϕ ����� ≤ � µ 1 (µ)ε−n−1d µ = 1 ε − n(µε−n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We have just proven that T = (Tµ)µ→∞ exists as an element of D′(Rn) and, by definition of Tµ in (35), T will also have scaling degree equal to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Furthermore, we have already seen that T will be the only extension of T0 with such scaling degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1 will also be used to help us prove our next result, which has the same objective as our last, but now to distributions such that σ(T ) ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Before that, however, we shall need to define some more concepts which will be an important part of our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We will denote by Dω(Rn) the subspace of D(Rn) composed by functions such that their derivatives up to order ω vanish in the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, for some natural number ω > 0, Dω(Rn) = {ϕ ∈ D(Rn) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Dαϕ(0) = 0 , |α| ≤ ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Now, for every function ϕ ∈ D(Rn), its Taylor expansion will be given by ϕ(x) = � ν xν ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Dνϕ(0), (41) from which, separating the terms whose multi-index have norm |ν| > ω, we have ϕ(x) = � |ν|≤ω xν ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Dνϕ(0) + � |ν|>ω xν ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Dνϕ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (42) Therefore, the inclusion ϕ ∈ Dω(Rn) is equivalent to saying that the first summation in (42) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Meanwhile, the second summation term will then belong to Dω(Rn), being actually equal to ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Furthermore, using Lagrange Remainder formula, we can affirm the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 25 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Any function ϕ ∈ Dω(Rn) can be written as ϕ(x) = � |α|=ω+1 xαgα(x), where gα ∈ D(Rn) for all multi-index in this summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' To prove this, apply, for every ϕ ∈ Dω(Rn), the Taylor Theorem with Remainder in the multi variable case,9 so that we have ϕ(x) = � |β|≤ω Dβϕ(0) β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' xβ + � |α|=ω+1 xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' fα(x), where fα(x) = (ω + 1) � 1 0 (1 − t)ωDαϕ(tx)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Since ϕ ∈ Dω(Rn), the first summation term vanishes for any x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Furthermore, being ϕ a infinitely differentiable function with compact support, each fα(x) must also be infinitely differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' They may, however, not be of compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Nonetheless, we can take a function ψ(x) ∈ D(Rn) such that ψ(x) = 1 for x ∈ supp ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We shall have ψ(x)ϕ(x) = ϕ(x) and the product gα = ψfα ∈ Dω(Rn) will satisfy ϕ(x) = � |α|=ω+1 xαgα(x), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Not only the subspace Dω(Rn) plays a crucial role to us, but also the projec- tion of D(Rn) on Dω(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Let W = Dω(Rn)⊥ be the orthogonal complement of Dω(Rn) so that D(Rn) = Dω(Rn) ⊕ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='10 The direct sum of both subspaces guarantees that functionals in D′(Rn) may be written as T = Tω ⊕ l , Tω ∈ D′ ω(Rn) , l ∈ W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We ask, then, what are the functionals in W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Well, they are characterized as elements l ∈ D′(Rn) such that ⟨l, ϕ⟩ = 0 for any ϕ ∈ Dω(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Hence, we have, in particular, ⟨l, ϕ⟩ = 0 for every ϕ ∈ D(Rn\\{0}) ⊂ Dω(Rn) and, therefore, supp l ⊂ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' According to the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1, it also follows that l = � |α|≤m cαDαδ , m ∈ N, cα ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 9For the reader interested in the proof of this version of the Taylor Theorem, see, for example, [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 10Dω(Rn) is evidently closed due to the continuity of any Dαϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 26 We state that m ≤ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In fact, if we suppose that m > ω, then we may take ϕ ∈ Dω(Rn) such that Dβϕ(0) ̸= 0, with |β| = m, which implies ⟨l, ϕ⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This contradicts l ∈ W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Reciprocally, if l = � |α|≤ω cαDαδ, then ⟨l, ϕ⟩ will only possess derivations of ϕ in x = 0 with order lesser or equal to ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, if ϕ ∈ Dω(Rn), ⟨l, ϕ⟩ = 0, that is, l ∈ W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thereby we have just proved the following Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For an arbitrary real number ω > 0, we have W′ = {l ∈ D′(Rn) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' l = � |α|≤ω cαDαδ , cα ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Once W′ is also a linear space, it is a consequence of the last Lemma that B = {Dαδ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' |α| ≤ ω} is a basis to the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Actually, we won’t use this set of distributions to represent the orthogonal projection of D′(Rn) on D′ ω(Rn), but another one that depends upon a particular test function w ∈ D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We will show that, for any function w ∈ D(Rn) such that w(0) ̸= 0, the set11 C = {Dαδ(w−1·) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' |α| ≤ ω} is a basis to W′ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The number of elements of C equals the number of elements in B, thus, it remains to show that the elements of the latter may be generated by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We use the Leibniz rule so pointed out in Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2, which implies that, for any multi-index α and any ψ ∈ D(Rn) that does not vanish in x ∈ Rn, Dαϕ(x) = 1 ψ(x)Dα(ψϕ)(x) − 1 ψ(x) � 0≤β≤α β̸=α α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (α − β)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (Dβϕ)(x)(Dα−βψ)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Hence, if ⟨Dγδ, ϕ⟩ = (−1)|γ|Dγϕ(0) may be written as a linear combination of terms such as � Dβδ(w−1·), ϕ � = (−1)|β|Dβ(w−1ϕ)(0) for arbitray12 γ < α, then the same holds for Dαϕ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We conclude our proof if we note that the result is valid for α = (0, · · · , 0), ⟨Dαδ, ϕ⟩ = ϕ(0) = w(0) ϕ(0) w(0) = w(0) � Dαδ(w−1·), ϕ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Just as any element in B may be written as a linear combination of elements in C, the same is true for W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In addition, the set E = { (−1)|α| α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' w(x)xα ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' |α| ≤ ω} generates W and will be a basis to the dual of C, once13 � Dαδ(w−1·), (−1)|β| β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' w(x)xβ � = (−1)|α|+|β| β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (Dαxβ)(0) = δα,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 11This means the action of an element of C over a ϕ ∈ D(Rn) is � Dαδ(w−1·), ϕ � = � Dαδ, w−1ϕ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 12This inequality means that γ ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' However γ ̸= α, that is, at least one of its coordinates γi is strictly lesser than αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 13We have used the generalization of the Kroenecker delta for multiple variables, which is 1 whenever α = β, and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 27 Then, we can write the projection operator of D(Rn) on Dω(Rn) in the form W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w) : D(Rn) −→ Dω(Rn) ϕ(x) �−→ ϕ(x) − w(x) � |α|≤ω xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' � Dα ϕ w � (0), (43) for any w ∈ D(Rn) such that w(0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In fact, if ϕ ∈ D(Rn) is arbitrary, then fixing γ a multi-index such that |γ| ≤ ω, we have (DγW(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)ϕ)(0) = Dγϕ(0) − � |α|≤ω � Dγwxα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' � (0) � Dα ϕ w � (0) and we know, due to Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2, that � Dγw(x)xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' � (0) = � β≤γ γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (γ − β)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (Dγ−βw)(0) � Dβ xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' � (0) = � β≤γ γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (γ − β)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (Dγ−βw)(0)δβ,α = γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (γ − α)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (Dγ−αw)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (44) That way, we obtain (DγW(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)ϕ)(0) = Dγϕ(0) − � |α|≤ω γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (γ − α)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (Dγ−αw)(0) � Dα ϕ w � (0) = Dγϕ(0) − Dγ(w ϕ w )(0) = Dγϕ(0) − Dγϕ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (45) Once again we have utilized, in the second equality, what we obtained in the Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, we have just proved that (W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)ϕ)(x) is indeed a test function whose derivatives up to order ω vanish at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We may check that W is indeed a projection by showing that it is idempotent, W 2 = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For that, we observe that in Dω(Rn), W is the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In fact, for a function ϕ ∈ Dω(Rn), we have � Dα ϕ w � (0) = 0 for any |α| ≤ ω, in a way that (W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)ϕ)(x) = ϕ(x) − � |α|≤ω xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' � Dα ϕ w � (0) = ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We chose w−1 instead of the function w itself because the former allows us to write an interesting and useful property, to be used ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Namely, the operator W satisfies W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)(wϕ) = wW(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1)(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (46) 28 We point out that there is no problem at all when using W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='ψ), with ψ being the constant function equals the unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Although W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1)ϕ is not a test function, once its support is not compact, it is infinitely differentiable and, hence, its product with w will be, in fact, in D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In accordance with (46), we will have, if |α| ≤ ω, W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)(wxα) = wW(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1)xα, that is, W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)(wxα)(x) = w(x) \uf8ee \uf8f0xα − � |β|≤ω xβ β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' � Dβxα� (0) \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' By the Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1, the sum reduces to xα, W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)(wxα) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (47) Our next result is the following Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Let T0 ∈ D′(Rn\\{0}) such that σ(T0) = s ≥ n and ω = ω(T0) = s − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Moreover, given w ∈ D(Rn), with w(0) ̸= 0, and constants Cα ∈ C for all multi-index α, with |α| ≤ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' There exists one, and only one, distribution T ∈ D′(Rn) such that σ(T ) = s and satisfying 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' ⟨T, ϕ⟩ = ⟨T0, ϕ⟩ , ∀ ϕ ∈ D(Rn\\{0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' ⟨T, wxα⟩ = Cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Specifically, T is given by ⟨T, ϕ⟩ = � Tω, W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)ϕ � + � |α|≤ω Cα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' � Dα ϕ w � (0), (48) where Tω is the only extension guaranteed by the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1 and W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w) is the operator W, defined in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We shall begin, once again, with the uniqueness first, supposing the existence prior to its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If T1 and T2 are both extensions of T0 in D(Rn), then, for ϕ ∈ D(Rn\\{0}), ⟨T1 − T2, ϕ⟩ = ⟨T1, ϕ⟩ − ⟨T2, ϕ⟩ = 0, which implies supp (T1 − T2) ⊂ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Again, according to the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1, T1 − T2 = � |ν|≤m cνDνδ, so that σ(T1 − T2) = n + |ν| and we must have, by hypothesis, m ≤ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, we have, for any α such that |α| ≤ ω, ⟨T1 − T2, wxα⟩ = ⟨T1, wxα⟩ − ⟨T2, wxα⟩ = Cα − Cα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' On the other hand, if we first take |α| = m, then ⟨T1 − T2, wxα⟩ = � |ν|≤m cν(Dνwxα)(0) 29 and, with the Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1, we have ⟨T1 − T2, wxα⟩ = � |ν|≤m cν \uf8eb \uf8ed� β≤ν ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (ν − β)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (Dβxα)(0)(Dν−βw)(0) \uf8f6 \uf8f8 = � |ν|=m cν \uf8eb \uf8ed� β≤ν ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (ν − β)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (Dβxα)(0)(Dν−βw)(0) \uf8f6 \uf8f8 , (49) where we have used that all derivatives (Dβxα)(0) will be zero whenever |ν| < |α| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Furthermore, the only non-vanishing term appears when β = ν = α, so that the we are left with only ⟨T1 − T2, wxα⟩ = cα(Dαxα)(0)w(0) = α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' cα w(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We have obtained that cν = 0 once |ν| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Analogously, if we consider |α| = m − i, 1 ≤ i ≤ m, then we will find the same result, canceling all the constants cα, finally getting T1 = T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For the existence, we first restrict T0 to the subspace D′ ω(Rn\\{0}) of D′(Rn\\{0}) and, denoting this restriction by ˜T0, we have, by the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3, � ˜T0, ϕ � = � |α|=ω+1 ⟨xαT0, gα⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Using now (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=') and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=') from the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2, we obtain σ( ˜T0) ≤ σ(T0) − ω − 1 < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Therefore, the restriction of T0 in D′ ω(Rn\\{0}) has, due to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1, an extension14 in D′ ω(Rn), which we denote by Tω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' However, we seek an extension over the whole space D′(Rn), so that we still need to extend Tω to general elements of D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If we obtain such T ∈ D′(Rn), it is evident that it will be an extension of T0, since D(Rn\\{0}) ⊂ Dω(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Now, since we are dealing with a closed orthogonal subspace of D′(Rn), extensions of Tω will be simply characterized by T = Tω ⊕ l , l ∈ W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It is through the operator W, which projects functions from D(Rn) onto the subspace Dω(Rn), that we are capable of applying Tω over any ϕ ∈ D(Rn), placing between then the action of W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Since the projection operator is unique (once we have chosen w), each extension T = Tω ◦ W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w) ⊕ l 14The more careful reader may ponder if this extension will in fact belong to D′ ω(Rn), since Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1 affirms only that the extension will be an element of D′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For that, we note that our construction made no reference to the behavior of ϕ ∈ D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' From that, we see that the restrictions over the application of ˜ T0 ∈ D′ ω(Rn\\{0}) will be inherited by Tω ∈ D′ ω(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 30 will be unique except by the change of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Since C = {Dαδ(w−1·) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' |α| ≤ ω} is a basis to W′, the constants Cα must define such functional l, which implies that T = Tω ◦ W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w) ⊕ � |α|≤ω Cα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' � Dα ϕ w � (0) is the only extension of T0 satisfying ⟨T, wxα⟩ = Cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We can simplify even further our calculations of the extension of T0 if we restrict the class of functions permitted to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' More specifically, if we take w such that (Dαw)(0) = δα,0, which is equivalent to taking w equal to 1 in a neighborhood of the origin, we have (Dα ϕ w )(0) = � 0≤β≤α β̸=α α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (α − β)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (Dβϕ)(0)(Dα−βw−1)(0) = Dαϕ(0), (50) since any derivation of w−1 will also carry some derivative of w itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' From that, it follows that W reduces to (W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)ϕ)(x) = ϕ(x) − w(x) � |α|≤ω xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Dαϕ(0) and the extension T given by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2 can, therefore, be written as ⟨T, ϕ⟩ = � Tω, W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)ϕ � + � |α|≤ω Cα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (Dαϕ) (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (51) Such functions, satisfying (Dαw)(0) = δα,0, are called Epstein-Glaser func- tions (see, for example, [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3 Dependence of the extension on the test function w(x) We have seen that, for the case when σ(T0) ≥ n, it does not seem possible to get rid of the dependence of the extension T ∈ D′(Rn) on the test function w ∈ D(Rn) chosen to construct the projection of D(Rn) over Dω(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We can, nonetheless, observe the behavior of this dependence, mainly by studying the term � Tω, W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)ϕ � , which we denote, following [21], by the integral kernel of the extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Here, we will stick with the supposition that w is an Epstein-Glaser function, in the sense defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 31 Thus, choosing two Epstein-Glaser test functions w1, w2 ∈ D(Rn), we have (W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w1)ϕ)(x) = ϕ(x) − w1(x) � |α|≤ω xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Dαϕ(0) = ϕ(x) − w2(x) � |α|≤ω xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Dαϕ(0) + (w2(x) − w1(x)) � |α|≤ω xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Dαϕ(0) = (W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w2)ϕ)(x) + (w2(x) − w1(x)) � |α|≤ω xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Dαϕ(0) (52) and, applying Tω over this expression, we obtain (bearing in mind that Tω is unique, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1) � Tω, W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w1)ϕ � = � Tω, W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w2)ϕ � + � |α|≤ω � Tω, (w2(x) − w1(x))xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' � Dαϕ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (53) We therefore conclude that the application of Tω over different projections differ only by a linear combination of terms of the form ⟨Dαδ, ϕ⟩, that is, by application, over the test function ϕ, of operators belonging to W′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Our goal ahead will be, however, to get rid of the restriction that w be a test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' As we have already mentioned, this seems to be a crucial condition for defining W as a projection operator, since it is responsible for the fact that the term w(x) � |α|≤ω xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Dαϕ(0) has compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Nonetheless, very often (as will be the case ahead) a distribution which is not well behaved at the origin will behave nicely at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' To be more precise, we mean that such distributions will be well defined when applied to functions ϕ ∈ C∞(Rn) whose support may not be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In that sense, taking a sequence (wk) ⊂ D(Rn) whose pointwise limit w is a function15 in C∞(Rn) and T0 is such that lim k→∞ � Tω, W(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='wk)ϕ � ∈ C, ∀ ϕ ∈ D(Rn), then there is no motive to not consider w in our renormalization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We shall see how this is an important part for our application of this method of extension of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 4 Application to the electron self-energy In this section, we finally attack the electron self-energy problem, seen as a point particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We posses now sufficient machinery to see it as a pathology to be faced with extension of distributions defined over D(Rn\\{0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' To clar- ify our aims, we will translate the issues of electrostatics to the language of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 15Some works, such as [21], go even further as to only ask that w ∈ D′(Rn) be a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We will not need this generality, so that we have preferred to omit this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 32 The first and default example is the charge distribution of an electron, seen as a charged point particle, which is represented by the Dirac delta δ(·), cen- tered where we suppose the whole electric charge of the particle should be concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Actually, this particular case is not the only one where we con- sider the charge distribution ρ as a generalized function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' After all, it would be an incredible coincidence in nomenclature if the charge distributions presented in the electrodynamics realm were not presented by distributions ρ ∈ D′(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This is one of the main contributions of the theory of generalized functions to electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It not only incorporates more general charge distributions ρ (which could not be defined just as real functions in R3), but also eases its manipulations, employing the properties so described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' To illustrate our last paragraph, let us analyze the representa- tion of an electric dipole as a distribution in D(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The pure dipole, considered an idealization such as point charges, is constructed as follows: start by setting two charges, q and −q, separated by a fixed distance ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Suppose that they lie in the x coordinate, with −q at the origin and q at x = ε, accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' That way, the distribution ρ ∈ D(R) will be ρ = −qδ0 + qδε = q(δε − δ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If we set q = 1/ε, then, in the limit ε → 0+ we obtain a distribution whose total charge vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This result is not correct nonetheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' A null charge distribution would lead to a zero electric field, which is not the case for a dipole (see [4] for the expression of ⃗E in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This simple argument indicates that the distribution formalism is more than necessary to an accurate description of even the basics of electrostatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Now, we apply ρε = ε−1(δε − δ0) over any ϕ ∈ D(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We have ⟨ρε, ϕ⟩ = 1 ε⟨δε − δ0, ϕ⟩ = ϕ(ε) − ϕ(0) ε and, therefore, we shall obtain, in the limit ε → 0+, ⟨ρε, ϕ⟩ → ϕ′(0) = −⟨δ′, ϕ⟩, that is, ρdip = −δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The generalization to the three-dimensional case is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In this case, the dipole moment p := q⃗ε, where ⃗ε is the displacement vector that con- nects the negative to the positive charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Once again we take the limit |⃗ε| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The way we defined q above, we guaranteed that the dipole moment was kept constant, even when the charges are close enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' With this premise, the charge distribution will be given by ⟨ρε, ϕ⟩ → − ∂ ∂pϕ(0) , ∀ ϕ ∈ D(R3), that we denote by ρdip = −p · ∇δ ∈ D(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 33 We return to the main problem we intend to attack, namely, the divergence of the electron self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In this context, the fact that the charge is fully concentrated in the origin is seen by the application of qδ on different test functions ϕ ∈ D(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For any test function whose support does not contain the origin, we have � supp ϕ ρ(x)ϕ(x)d3x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (54) At the same time, the distribution possess a finite charge once the integration of ρ over R3 is equivalent to apply qδ on a test function ϕ ∈ D(R3) whose support contains the origin such that ϕ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' � R3 ρ(x)ϕ(x) d3x = � supp ϕ ρ(x)ϕ(x) d3x = ⟨qδ, ϕ⟩ = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (55) This charge distribution produces both a potential V and an electric field E, which are also new distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We haven’t considered vector fields as distributions so far, just like E, but this generalization is quite natural and E acts on an element of D(R3) according to ⟨E, ϕ⟩ = (⟨Ex, ϕ⟩, ⟨Ey, ϕ⟩, ⟨Ez, ϕ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The application of vector fields as distributions were mentioned here just for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It will be no longer necessary hence forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We can see that, in fact, V represents an element of D′(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The explicit formula for the potential is given by V (r) = e r, where e is the electron charge and we are using unities in which 4πǫ0 = 1, with no further implications to the final results whatsoever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' As usual, r represents the radial coordinate of a spherical coordinate system centered on the charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' V (·) is a smooth function for any r ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Hence, we just have to be concerned to the convergence of ⟨V, ϕ⟩, for an arbitrary ϕ ∈ D(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In effect, if R > 0 is such that K = BR(0) ⊃ supp ϕ and M = maxx∈R3 ϕ(x), then ���� � R3 V (x)ϕ(x)d3x ���� ≤ M � K V (x)d3x = M �� 2π 0 dφ � π 0 senθdθ � � R 0 e r r2dr = (2πMe)R2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (56) An analogous consideration may be done for the electric field, since E ∼ 1 r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In view of that, the radial integral will converge as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We can thus turn our attention to the stored self-energy of a charged system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' As we have already seen, for the particular case of an electron there is a divergence at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We are considering here only the static case, so we don’t have to worry about magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We can consider, however, other cases where such divergence does not appear and we are thus able to calculate the system self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If we consider, 34 for example, the electron as a uniformly charged spherical shell of radius a, then E(r) = � 0 , se r ≤ a, (e/r2) ˆr , se r > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (57) Therefore, the self-energy, see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (2), will be given by W = 1 8π � R3 E2dτ = 1 8π (4π) � ∞ a e2 r2 dr = 1 2 e2 a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (58) In distribution parlance, the last equation is but the application of the reg- ular distribution 1 8πE2 over the function ϕ(x) ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We point out that, even with ϕ /∈ D(R3), � E2, ϕ � does exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It is a consequence of the behavior of E2 at infinity, which is good enough that we do not need to restrict the range of integration to a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For general distributions, with unknown behavior at infinity, this restriction is considered by supposing that ϕ is a test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We have already mentioned, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3, that it is often advantageous (or even necessary) to work with functions that are not compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This is allowed once our distribution possesses the necessary conditions so that its application over this larger class of functions is well behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For instance, the Dirac delta may be applied on any function ϕ continuous at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In our specific case, we will see how the electron self-energy (∼ E2) behavior far from the origin permits such loosening of the conditions over ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' More precisely, E2 = e2 r4 , r > 0 (59) is not well defined as a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In effect, for any test function obeying ϕ(x) = 1 for x in a neighborhood V of the origin, say, a ball, we have � E2, ϕ � = � V e2 r4 d3x + � R3\\V e2 r4 ϕ(x)d3x = +∞, once the first term is linear divergent due to the fourth-order homogeneity of E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='16 Outside the origin, however, E2 is a smooth function, and as such, locally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Hence, we have, at least, E2 ∈ D′(R3\\{0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For this reason, we may extend (renormalize) the distribution E2 to a new distribution U ∈ D′(Rn) with the methods exposed previously in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In that way, we expect that the electron self-energy E0 will be well defined as the application of 1 8πU over the function ϕ ≡ 1, E0 = 1 8π ⟨U, 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (60) According to what we have made so far, let us first determine the scaling degree and the singular order of E2, which are key to Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 16The classification as a linear divergence may be justified in polar coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Taking R = 1/r, we find � +∞ 0 dr r2 = � +∞ 0 dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' See [3] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 35 We observe that it is a homogeneous function of order −4, so that, as expressed through the Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2, λsE2(λx) = λs e2 (λr)4 = λs−4E2, which implies, λsE2(λx) λ→0+ −−−−→ 0 ⇐⇒ s > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' That is, σ(E2) = 4, and also ω(E2) = σ(E2) − n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Therefore, for a test function w ∈ D(R3) and constants C0, C(1,0,0) ≡ C1, C(0,1,0) ≡ C2, C(0,0,1) ≡ C3, we obtain, according to the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2, a distri- bution U ∈ D′(R3) defined by ⟨U, ϕ⟩ = � E2 1, W(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)ϕ � + � |α|≤1 Cα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' � Dα ϕ w � (0) = � E2 1, W(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)ϕ � + C0 ϕ(0) w(0) + 3 � i=1 Ci∂xi � ϕ w � (0) (61) satisfying ⟨U, ϕ⟩ = � E2, ϕ � , ∀ ϕ ∈ D(R3\\{0}), (62) ⟨U, wxα⟩ = Cα , |α| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (63) Moreover, if w is chosen as an Epstein-Glaser function, we have ⟨U, ϕ⟩ = � E2 1, W(1,w)ϕ � + C0ϕ(0) + � i=3 Ci(∂xiϕ)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (64) Before moving on, there is a comment in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The charge distribution of a (charged) point particle possesses spherical symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' It is not only due to the concentration of charge in a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In fact, a electric dipole has also a distribution whose support is contained in the origin, although the allegedly spherical symmetry is broken once the moment p defines a privileged direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The additional fact that our point particle model admits no internal structure imposes the constraint of having no special direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Since we would like to preserve such symmetry when extending E2, we must choose Ci = 0, for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This is justified because the last term in (64) does not behave like a scalar under rotations of our coordinate system, unless the three constants vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We can promptly see this by writing � i=3 Ci(∂xiϕ)(0) = C · (∇ϕ)(0) , C = (C1, C2, C3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Now, ∇ϕ does behave as a vector, however we cannot say the same for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For this reason, the only way to keep this sum inert under rotations is to set C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 36 From this, we may rewrite U as ⟨U, ϕ⟩ = � E2 1, W(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='w)ϕ � + C0ϕ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (65) Then, we seek, pretty much like what has been done in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3, to relax the conditions under w employed in the renormalization of E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Our path will be to take a sequence of test functions wM that converges pointwise to w(x) = 1 ∈ C∞(R3), obtaining a well defined distribution given by ⟨U, ϕ⟩ = lim M→∞ � E2 1, W(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='wM)ϕ � + C0ϕ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (66) The last equation suggests that we will take all wM as Epstein-Glaser func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Specifically, each wM shall be taken as in the Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3, wM(x) = ζM,1(x) = 1 − � r −∞ ηM,h(t)dt, where r = ∥x∥ is a radial coordinate and ηM,h(t) is given in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We can write this sequence of functions in a convenient manner that will be useful ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The comments in the Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3 suggest that wM equals 1 within the ball BM(0) and 0 outside the ball BM+1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In the ring BM+1(0)\\BM(0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=', for M ≤ r ≤ M +1, we may write wM as a radial smooth function17 χ(r −M) such that |χ(s)| ≤ 1, s ∈ [0, 1], χ(0) = 1 and χ(1) = 0, wM(x) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1, if x ∈ BM(0), χ(r − M), if x ∈ BM+1(0)\\BM(0), 0, if x /∈ BM+1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (67) Therefore, we have (DαwM)(0) = δα,0 and wM(x) → 1 for any x ∈ R3 in the limit M → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Moreover, due to the result (53), for any two naturals M1, M2 ∈ N (say, M1 < M2), we have � E2 1, W(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='wM2 )ϕ � = � E2 1, W(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='wM1 )ϕ � + � |α|≤1 � E2 1, (wM1(x) − wM2(x))xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' � Dαϕ(0), wherein wM1(x) − wM2(x) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 0, if x ∈ BM1(0), χ(r − M1) − 1, if x ∈ BM1+1(0)\\BM1(0), −1, if x ∈ BM2(0)\\BM1+1(0), −χ(r − M2), if x ∈ BM2+1(0)\\BM2(0), 0, if x /∈ BM2+1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (68) 17which shall not be confused with the characteristic function χA on a set A ⊂ R3, also present in the Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 37 That way, if we denote (aM) the real sequence whose elements are � E2 1, W(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='wM)ϕ � , then we will show that it converges, proving that (aM) is a Cauchy sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' In fact, we have just seen that18 aM1 − aM2 = � E2 1, W(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='wM2 )ϕ � − � E2 1, W(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='wM1 )ϕ � = � |α|≤1 � E2, (wM1(x) − wM2(x))xα α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' � Dαϕ(0), (69) so that, if we limit this sum, then we will also limit the difference aM1 − aM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Now, given ε > 0, we take M ∈ N such that 1 M < ε 8πe2 and M1, M2 ∈ N, with M ≤ M1 < M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Since E2 acts on test functions whose support does not contain the origin, we can employ the formula (59), obtaining � E2, (wM1(x) − wM2(x)) � = � R3 e2 r4 (wM1(x) − wM2(x))d3x, � E2, (wM1(x) − wM2(x))xi � = � R3 e2 r4 (wM1(x) − wM2(x))xid3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Now, once i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' wM1 − wM2 is radial, ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' has support in BM2+1(0)\\BM1(0) and iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' assumes values only in [0, 1], we have ��� E2, (wM1(x) − wM2(x)) ��� ≤ (4πe2) � M2+1 M1 1 r2 dr = (4πe2) �1 r �M1 M2+1 ≤ (8πe2) 1 M1 , that is, ��� E2, (wM1(x) − wM2(x)) ��� ≤ 8πe2 M < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (70) Meanwhile, � E2, (wM1(x) − wM2(x))xi � = � R3 e2 r4 (wM1(r) − wM2(r))xid3x = 0, (71) because for any i = 1, 2, 3, corresponding to the three Euclidean axis x, y, z, respectively, the integration in ϕ or in θ will vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='19 In fact, for i = 1 and i = 2, the integrals in ϕ vanish, � 2π 0 dφ cos φ = � 2π 0 dφ senφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Now, for i = 3, we find � π 0 dθ cos θ senθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Hence, with the help of (70) and (71), we may conclude that |aM1 − aM2| ≤ εϕ(0), (72) which, in turn, implies that (aM) will be a Cauchy sequence, that is, a convergent one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 18Since wM1(x) − wM2(x) ∈ D(Rn\\{0}), the action of E2 1 may be replaced by E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 19This is only possible because wM is a sequence of radial functions and as such, the integration in r, ϕ e θ can be factored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 38 All the previous development allows us to show that the distribution defined in (66), that we will denote simply by ⟨U, ϕ⟩ = � E2 1, W(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1)ϕ � + C0ϕ(0), (73) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, we may finally use the eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (60) to obtain the renormalized electron self-energy, E0 = 1 8π � E2 1, W(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='1)1 � + 1 8π C0 = C0 8π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (74) Although simple, the eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' (74) bears great physical meaning, concentrating our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We have shown that defining the electron self-energy as the applica- tion of the distribution 1 8πU ∈ D(R3), which extends (or renormalize) E2, over the constant function 1, we get rid of the divergence previously obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This divergence appeared when one directly considers E2, which cannot be seen as an actual distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='20 Now, E0 becomes an undetermined constant, that we may control to serve our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' This is the very kernel of a renormalization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' If, for instance, we assume that the self-energy is, alone, responsible for the electron mass, we shall take C0 = 8πmec2, in a way that E0 = mec2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' To summarize, we have seen how the self-energy problem originates in the fact that E2 is not a proper element of D(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' On the other hand, E2 is, outside the origin, a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Thus, we at least have E2 ∈ D′(Rn\\{0}), which means we can use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content='2 to extend it to a distribution in D′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 5 Conclusion The main objective of this work was to analyze (and renormalize) a sim- ple but central problem in classical electrodynamics: the electron self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Although the electrostatics model of a charged point particle implies a linear divergence on the self-energy, we may skirt this infinity with an extension of the corresponding distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' With more details, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We have also developed a self-consistent study of the theory of distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The basic aspects, main definitions and examples, operations and key results were all included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Of course our notes are not supposed to replace the standard and seminal literature, such as [16,26,27], which is considered utterly necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' However, we provide here the minimum to the interested reader in maneuvering such a powerful tool for analyzing, for instance, classical and quantum field theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 20The statement that E2 /∈ D(R3) is a consequence that the product between distributions is not, in general, well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' An alternative method to skirt the electron self-energy divergence is related to a generalization to the very concept of distributions, working with the generalized Coulombeau functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' For details, see [24,25] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' The leading results of extension of distributions, that is, the correspond- ing renormalization, were all enunciated and demonstrated, following the lines in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We have focused in a particular subspace of the set of test functions, namely, the one whose elements vanish in an arbitrary neighborhood of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We have investigated the behavior of different distributions in the ori- gin and how one could recover such distributions, in the sense of making them continuous linear functionals defined over all the space of test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' Our thorough demonstrations may serve as an auxiliary/pedagogical pathway on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' We have applied the concepts of distributions and the corresponding ex- tensions to the classical electron self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' At first sight, electrostatics implies a divergence once we treat the electron as a charged point particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' However, our construction shows that its self-energy turns out to be an undetermined con- stant upon renormalization, so that our parameters might be fixed, for example, appealing to empirical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FKT4oBgHgl3EQfjS4w/content/2301.11844v1.pdf'} +page_content=' References [1] A.' 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b/VtE5T4oBgHgl3EQfBg7N/content/tmp_files/2301.05388v1.pdf.txt @@ -0,0 +1,787 @@ +arXiv:2301.05388v1 [astro-ph.GA] 13 Jan 2023 +Astronomy & Astrophysics manuscript no. 45021corr-modified-copy-copy +©ESO 2023 +January 16, 2023 +Letter to the Editor +The mixing of dust and gas in the high latitude translucent cloud +MBM 40 +Marco Monaci1, Loris Magnani2, and Steven N.Shore1 +1 Dipartimento di Fisica, Università di Pisa, Largo Bruno Pontecorvo 3, Pisa +e-mail: monaci93@gmail.com +e-mail: steven.neil.shore@unipi.it +2 Department of Physics and Astronomy, University of Georgia, Athens, GA 30602-2451 +e-mail: loris@uga.edu +Received 20 September 2022 ; accepted 25 October 2022 +ABSTRACT +Context. High latitude molecular clouds (hereafter HLMCs) permit the study of interstellar gas dynamics and astrochemistry with +good accuracy due to their proximity, generally clear lines of sight, and lack of internal star-forming activity which can heavily modify +the physical context. MBM 40, one of the nearest HLMCs, has been extensively studied, making it a superb target to infer and study +the dust-to-gas mixing ratio (DGMR). +Aims. The mixing of dust and gas in the interstellar medium remains a fundamental issue to keep track of astrochemistry evolution +and molecular abundances. Accounting for both molecular and atomic gas is difficult because H2 is not directly observable and H i +spectra always show different dynamical profiles blended together which are not directly correlated with the cloud. We used two +independent strategies to infer the molecular and atomic gas column densities and compute the dust-to-gas mixing ratio. +Methods. We combined H i 21 cm and 12CO line observations with the IRAS 100 µm image to infer the dust-to-gas mixing ratio +within the cloud. The cloud 21 cm profile was extracted using a hybrid Gaussian decomposition where 12CO was used to deduce the +total molecular hydrogen column density. Infrared images were used to calculate the dust emission. +Results. The dust-to-gas mixing ratio is nearly uniform within the cloud as outlined by the hairpin structure. The total hydrogen +column density and 100 µm emissivity are linearly correlated over a range in N(Htot) of one order of magnitude. +Key words. ISM: clouds – dust, extinction – molecules – structure +1. Introduction +The admixture of dust and gas is an essential, but poorly de- +termined, property of the interstellar medium. It affects the +modeling of radiative balance, moderates astrochemical pro- +cesses, and affects star formation within molecular clouds (for +the theoretical aspect see, e.g., Lee et al. (2017), Tricco et al. +(2017), Marchand et al. (2021); for the observational picture, +e.g., Reach et al. (2017)). The widely adopted value for the mass +ratio in our Galaxy is ∼ 100 − 150 (Hildebrand (1983)), but it +varies in other galaxies (Young et al. (1986)) and even in clouds +in our Galaxy (Reach et al. (2015)).1 +The usual procedure to obtain the dust-to-gas ratio (here- +after DGR) infers the neutral hydrogen column densities from +the dust extinction which is then used to obtain the dust mass. +Alternatively, atomic resonance transitions have been used to in- +fer the atomic column densities, but these are altered by density- +dependent depletion in the gas phase and from the spotty sam- +pling of any intervening clouds because of the sparse coverage +by stars and extragalactic sources. The sparseness of background +sources also complicates the distance determination of clouds +using standard methods such as star counting, Wolf diagrams, +or multiband extinction measurements (e.g., Sun et al. (2021), +1 We must emphasize at the beginning of this Letter that our aim is +to study the mixing of the components and not the mass ratio, as con- +ventionally discussed, because we are trying to avoid the uncertainties +induced by adopting specific dust models. +Lv et al. (2018), Liljestrom & Mattila (1988)). These techniques +are hampered by the proximity of the translucent and high lat- +itude molecular clouds (see, e.g., Magnani & Shore (2017)). +Moreover, different lines of sight through these clouds may sam- +ple very different structures (e.g., Lombardi et al. (2014)) . If the +DGRs differ, this complicates extinction corrections and infer- +ences about the cloud masses based on infrared imaging. It is +now possible, however, to obtain the atomic column densities di- +rectly following the completion of all-sky H i surveys that com- +plement the infrared maps obtained for Galactic and cosmologi- +cal surveys. +The object of our study, MBM 40, is a small molecular +structure embedded in a larger H i flow (Shore et al. (2003)). +The molecular gas distribution has been discussed extensively +(Magnani et al. (1985), MBM; Lee et al. (2002); Shore et al. +(2003), SMLM; Shore et al. (2006), SLCM; Chastain et al. +(2009), CCM10). The CO(1-0) molecular gas distribution +is complex with the denser gas in a hairpin structure sur- +rounded by a diffuse envelope. The distance to the cloud is 93 +pc (Zucker et al. (2019)) and previous studies (Magnani et al. +(1996b); SMLM; CCM10) have derived a molecular mass of 20 - +40 M⊙. There is no evidence of star formation despite a rigorous +search (Magnani et al. (1996a)). This cloud provides an exem- +plar of a non-star-forming medium where no internal processing +has affected the dust properties. We should, therefore, see a more +Article number, page 1 of 7 + +A&A proofs: manuscript no. 45021corr-modified-copy-copy +nearly pristine presentation of the DGR and its uniformity than +would be obtained from a more active source. +2. Data +We have used a range of archival data for this study. We briefly +describe here the individual data sets. +2.1. GALFA +The Galactic Arecibo L-Band Feed Array HI (GALFA-HI, +Peek et al. (2011),) is an extended survey between −1◦ ≲ δ ≲ +38◦ with an angular resolution of 4′ and a 0.184 km s−1 spec- +tral resolution using the William E. Gordon 305-m telescope +at the Arecibo Observatory (Peek et al. (2018)).2. We used the +narrow bandwidth data repository (see Peek (2017)) centered at +0 km s−1; each spectrum has 2 048 channels spanning |vLSR| ≲ +188 km s−1. We used ancillary data furnished with the GALFA +datacube to correct for stray radiation and cropped the PPV dat- +acube in position to match the extension and velocity (|vLSR| ≲ +15 km s−1) of MBM 40, which is detected between ∼ 2 and +∼ 4 km s−1 in molecular gas. The top panel of Fig. 1 shows an +example of a GALFA narrowband spectrum (within the range +|vLSR| ≤ 20 km s−1) with a signal-to-noise ratio (S/N) ∼ 60; each +spectrum within the cloud has a similar S/N. +0 +5 +10 +15 +20 +25 +30 +-15 +-10 +-5 +0 +5 +10 +15 +0 +1 +2 +3 +Fig. 1. Sample spectra of MBM 40 used in this study. (Top panel.) +Single GALFA spectrum at α = 242.7583◦, δ = +21.8084◦ (J2000.0), in +the lower part of the western ridge of MBM 40. (Bottom panel.) Sample +FCRAO 12CO spectrum near the same position as the H i, obtained by +averaging 64 spectra in an 8×8 pattern to enhance the S/N. The vertical +dashed line shared by two plots indicates the bulk velocity of the cloud +(3.35 km s−1). The H i spectrum shows the brightness temperature (TB), +and the 12CO spectrum is in terms of the radiation temperature (TR). +2.2. FCRAO +The 12CO observations were obtained using the Five College Ra- +dio Astronomy Observatory (FCRAO3) in early 2000. The full +datacube is composed of 24 576 frequency-switchedspectra with +a velocity resolution of about 0.05 km s−1. We used only the 56 +central channels centered at 3 km s−1. The average rms noise +2 During these observations the Arecibo Observatory was operated +by SRI International under a cooperative agreement with the National +Science Foundation, and in alliance with Ana G. Méndez-Universidad +Metropolitana, and the Universities Space Research Association. +3 The FCRAO was supported in part by the National Science Founda- +tion and was operated with the permission of the Metropolitan District +Commission, Commonwealth of Massachusetts. +value is 0.7 K (Shore et al. (2003)). The 12CO radiation temper- +ature (TR) was calculated taking the antenna temperature and +then dividing by ηfssηc, where ηfss is the forward-scattering and +spillover efficiency (≃ 0.7) and ηc is the source filling factor, +which we assumed to be unity (see SMLM for a complete dis- +cussion of FCRAO observations). The bottom panel of Fig. 1 +shows the equivalent FCRAO 12CO profile near the same posi- +tion of the H i GALFA spectrum. The H i profile shows different +components, one of which is compatible with the MBM 40 bulk +velocity of ∼ 3 km s−1. +242 +242.2 +242.4 +242.6 +242.8 +243 +243.2 +21 +21.2 +21.4 +21.6 +21.8 +22 +22.2 +22.4 +22.6 +A +B +10 +15 +20 +25 +30 +35 +Fig. 2. H i brightness temperature at 3.4 km s−1 with integrated line +12CO intensity (black contours at levels 2, 3, and 4 K km s−1). White +circles denote the positions to which we refer in the text. In this velocity +slice (very near the molecular cloud bulk velocity), the H i emission and +12CO emission (position A) are anticorrelated. +The HI and 12CO temperature contours for a restricted ve- +locity range are shown in Fig. 2. Here we identify two positions +to which we refer in the following sections. +2.3. IRAS 100 µm dust map +For the dust distribution, we used the IRAS +100 µm +IRIS images that have a spatial resolution of about 2′ +(Miville-Deschênes & Lagache (2005)). These were published +after our first study of MBM 40 and have reduced striping, im- +proved zodiacal light subtraction, and a zero level compatible +with DIRBE. The image is an interpolated 50 × 50 pixel ma- +trix with the same H i and 12CO spatial resolution centered at +RA = 16h10m57s.19, DEC = +21◦52′29′′.28. We did not use +Planck data for the principal study because the spatial resolu- +tion is slightly lower and further reduced by a required interpo- +lation to a common coordinate grid (from galactic to equatorial). +We show in the Appendix, however, that our conclusions are un- +changed based on Planck. +Article number, page 2 of 7 + +Marco Monaci et al.: The mixing of dust and gas in the high latitude translucent cloud MBM 40 +3. Data analysis +3.1. Velocity slices +The comparison between 12CO and H i profiles in different ve- +locity slices is shown in Fig. 3. Because the spectral resolu- +tion of H i is coarser than 12CO, a linear interpolation was per- +formed to match the velocity resolution of the FCRAO data +(see the next subsection for further details). The H i is more +extended in velocity than the molecular gas traced by 12CO. +It appears as a "cocoon" (Shore et al. (2003),Verschuur (1974)) +around cloud molecular gas and shows some internal structures. +We note that H i is present at both lower and higher velocities +(i.e., v < 2.25 km s−1 or v > 4 km s−1) where no 12CO is present, +tracing foreground and background gas. +The H i spatial resolution is sufficient to compare the atomic +and molecular gas structures mapped by 12CO. The velocity +slices show an anticorrelation between atomic and molecular +gas, especially at position A, marked by a white circle in the +3 km s−1 slice in Fig. 3. The 12CO enhancement is associated +with a lower atomic gas column density, indicative of a phase +transition. This condensation is visible in all velocity slices. +3.2. Data rebinning and interpolation +The 21 cm and 12CO maps differ in spectral and spatial reso- +lutions. To compare these data, spectral and spatial linear inter- +polations were performed. Each H i spectrum was linearly inter- +polated using the 12CO FCRAO velocities (∆v = 0.05 km s−1) +as query points. The final spectral H i resolution is 3.5 times +higher than the original GALFA resolution. Each H i spectrum +has been checked for artifacts and, although linear interpolation +is sensitive to S/N, in our case S/N is over 60 for each spectrum. +We then performed a 2D linear interpolation for both H i and +FCRAO data to a common 50 × 50 pixel grid, covering about +0◦.8 in RA and 1◦ in DEC. Consequently, using this procedure +at each position yields linked GALFA and FCRAO spectra with +the same velocity resolution. +3.3. H i profile decomposition +Each H i profile is a blend of multiple components. Because +the 21 cm transition is almost always optically thin for lines +of sight toward high-latitude clouds, the full line of sight con- +tributes to the observed profile, not only to gas associated with +MBM 40, and significant effort was made to decompose each HI +spectrum into simpler Gaussian components (see Murray et al. +(2021), Lindner et al. (2015), Pingel et al. (2013)). In this work +we effected a two-step decomposition of each profile guided by +velocity information from 12CO as explained below. We used +Gaussian profiles to separate the foreground and background as +well as cloud neutral hydrogen. The 12CO observations constrain +a velocity window in which the cloud is embedded, so it is possi- +ble to use this information to trim the H i spectra. We emphasize +that the choice of Gaussian fitting was used only to select that +gas that is connected to the cloud. This is different from recent +Gaussian decomposition studies aimed at characterizing the fine +structure of the cold neutral medium (e.g., Murray et al. (2021)). +We started the H i decomposition by subtracting a very +broad, diffuse component (see, Verschuur & Magnani (1994)) +from each GALFA profile, fitting a Gaussian only to the wings +of the profile outside the interval −7.55 ≤ vLSR ≤ 7.35 km s−1, +and then we fit a second Gaussian to the residual emission. Fig. +4 (left panel) shows an example of the procedure. +We excluded positions where there is no clear double profile +(i.e., where only diffuse gas is present in the neighborhood of +MBM 40), and we fit only the redward wing of each H i resid- +ual emission, avoiding points below 2 km s−1 (see Fig. 4, right +panel). This procedure cannot map all the gas because we do +not know the parent distribution; however, the narrow Gaussian +component is at least proportional to the total gas connected with +MBM 40. +3.4. Molecular and atomic gas column density +If H i is optically thin, as in MBM 40, and if we assume that +one temperature dominates each velocity channel, then the total +atomic hydrogen column density is given by the following (see +for example Draine (2010)): +N(H i) = 1.813 · 1018 +� +∞ +−∞ +TB,Hi(v) dv +cm−2, +(1) +where TB,Hi(v) is the brightness temperature in K and v is the +velocity in km s−1. +The molecular hydrogen column density, N(H2), was ob- +tained using the integrated 12CO line in TR multiplied by a con- +version factor (XCO in units of cm−2 K−1 km−1 s): +N(H2) = XCOW(COJ=1→0) +cm−2, +(2) +where W(CO) is the velocity-integrated 12CO radiation temper- +ature. +The value of XCO is not constant in the Galaxy or even in- +side the same cloud (Bolatto et al. (2013)), and it is a source of +systematic uncertainty. Cotten & Magnani (2013) found that for +MBM 40, the XCO factor spans from (0.6−3.3)·1020 with an av- +erage of 1.3 · 1020: we adopted this value as well as a systematic +uncertainty of about a factor of two. For each FCRAO position, +we evaluated N(H2) as follows: +N(H2) = 1.3 · 1020 K−1 km−1 s +� +∞ +−∞ +TR,CO(v) dv +cm−2, +(3) +where TR,CO(v) is the radiation temperature for +12CO (cf. +SMLM). +Fig. 5 shows the derived column densities for atomic +and molecular hydrogen. Using equations 1 and 3, we obtain +N(Htot) = 2N(H2) + N(H i) within the cloud boundary. +Fig. 6 shows N(Htot). The enhancement is quite steep in- +side the cloud, where the total column density reaches N(Htot) ≈ +16 · 1020 cm−2. We assumed 93 pc as the distance of the cloud +(Zucker et al. (2019)). Thus, the spatial separation between dif- +fuse gas and the maximum hydrogen column density near po- +sition A is ≈ 0.18 pc, similar to that in 12CO, where the 12CO +fades out more or less ten beams away. The cloud shows a broad +atomic gas environment within the N(Htot) = 2 · 1020 cm−2 con- +tour (i.e., the outermost contour in Fig. 6) where the 12CO is +too faint to be detected with reasonable integration times. The +distribution of atomic gas is consistent with Shore et al. (2003), +where it was modeled with a "cocoon" shape, but using higher +fidelity from GALFA revealed some internal structure. Position +B (the red dot on the top of Fig. 6) shows 12CO weak lines with +virtually no 13CO. +Article number, page 3 of 7 + +A&A proofs: manuscript no. 45021corr-modified-copy-copy +242.4 +242.7 +243 +21.4 +21.6 +21.8 +22 +22.2 +VEL = 2 km/s +242.4 +242.7 +243 +21.4 +21.6 +21.8 +22 +22.2 +VEL = 2.25 km/s +242.4 +242.7 +243 +21.4 +21.6 +21.8 +22 +22.2 +VEL = 2.5 km/s +242.4 +242.7 +243 +21.4 +21.6 +21.8 +22 +22.2 +VEL = 2.75 km/s +242.4 +242.7 +243 +21.4 +21.6 +21.8 +22 +22.2 +VEL = 3 km/s +242.4 +242.7 +243 +21.4 +21.6 +21.8 +22 +22.2 +VEL = 3.25 km/s +242.4 +242.7 +243 +21.4 +21.6 +21.8 +22 +22.2 +VEL = 3.5 km/s +242.4 +242.7 +243 +21.4 +21.6 +21.8 +22 +22.2 +VEL = 3.75 km/s +242.4 +242.7 +243 +21.4 +21.6 +21.8 +22 +22.2 +VEL = 4 km/s +242.4 +242.7 +243 +21.4 +21.6 +21.8 +22 +22.2 +VEL = 4.25 km/s +10 +15 +20 +25 +30 +35 +Fig. 3. H i velocity slices (color scale) compared to the same 12CO velocity slices (black contours). Contours span from 1 K to 5 K in steps of 1 K. +The white circle in the 3 km s−1 panel marks position A in the western ridge. +-15 +-10 +-5 +0 +5 +10 +15 +-5 +0 +5 +10 +15 +20 +25 +30 +-10 +-5 +0 +5 +10 +-5 +0 +5 +10 +15 +20 +25 +30 +Fig. 4. (Left panel.) Subtraction of extremely diffuse gas from H i sam- +ple profile. Bold gray lines are wings fitted by a single Gaussian (black +dashed line) and the gray dashed line is the section we excluded from +the fit; the solid black line is the result of the subtraction, where the dif- +fuse component is severely reduced. (Right panel.) Narrow component +extraction: crosses indicate points below 2 km s−1 which we suppose are +not directly linked with gas traced by 12CO and which were excluded +for the narrow Gaussian fit (black solid line). Circles denote points we +used for the fit. +3.5. Dust-to-gas mixing ratio (DGMR) +The DGMR was obtained using the IRIS 100 µm image. We lin- +early interpolated the image to obtain a 50 × 50 matrix, in which +each pixel has a relative H i and 12CO spectra, so we were able +to directly perform a simple division to obtain the DGMR. The +mean 100 µm of the MBM 40 complex is a few MJy sr−1 and the +hydrogen column density is about 1020 cm−2, so we scaled the +latter by 10−20 to obtain comparable values with dust emission +242.4 +242.6 +242.8 +243 +21.4 +21.6 +21.8 +22 +22.2 +0 +0.5 +1 +1.5 +2 +1020 +242.4 +242.6 +242.8 +243 +0 +1 +2 +3 +4 +5 +6 +7 +81020 +Fig. 5. Total column density of H i (left panel) and H2 (right panel). Out- +side the jagged boarders visible via N(H i), the column densities were +not calculated due to a lack of 12CO emission. We note that near posi- +tion A, there is a deficit in atomic hydrogen where an enhancement in +12CO is present. +and then used the following ratio: +DGMR = +E100 +N(Htot) 1020 � +MJy sr−1 cm2� +, +(4) +where E100 is the total emission of the IRIS 100 µm image and +N(Htot) is the total hydrogen column density. +The derived ratio (Fig. 7) is nearly constant within the com- +plex, indicating that the dust and gas are well mixed. We tested +the robustness of our procedure by calculating the DGMR using +the complete H i spectra, that is without Gaussian decomposi- +tion and including the atomic gas over all velocities. The result +is nearly the same because the majority of gas is in molecular +form (mapped by 12CO). Considering the full H i velocity range, +Article number, page 4 of 7 + +Marco Monaci et al.: The mixing of dust and gas in the high latitude translucent cloud MBM 40 +242.4 +242.5 +242.6 +242.7 +242.8 +242.9 +243 +21.5 +21.6 +21.7 +21.8 +21.9 +22 +22.1 +22.2 +22.3 +2 +4 +6 +8 +10 +12 +14 +16 +Fig. 6. Total hydrogen column density. The colorbar values must be +multiplied by a factor of 1020. Red dots indicate the positions discussed +in Fig. 2. +the N(H i) is nearly doubled, but it accounts for about 30% of the +total gas. However, without any Gaussian decomposition, the re- +sult is much more sensitive to the XCO factor: if we use a lower +value, that is to say 0.6 (instead of 1.3), the DGMR changes +abruptly and the MBM 40 cloud is barely visible. Conversely, +if we decompose the H i profiles, the DGMR remains constant +and also the cloud structure is clearly visible. Therefore, we can +conclude that not all H i gas mapped by GALFA is linked with +MBM 40, but only the narrow component that we extracted. The +qualitative appearance from the map in Fig. 7 is quantified by the +linearity of the relation of the scatterplot of the total gas column +density versus 100 µm emissivity (see Fig. 8). +242.4 +242.5 +242.6 +242.7 +242.8 +242.9 +243 +21.5 +21.6 +21.7 +21.8 +21.9 +22 +22.1 +22.2 +22.3 +0 +0.5 +1 +1.5 +2 +2.5 +Fig. 7. Map of DGMR. The colorbar values are expressed in +1020 MJy sr−1 cm2. Red dots indicate the same positions as reported +in Fig. 2. +2 +4 +6 +8 +10 +12 +14 +16 +18 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Fig. 8. Scatter plot of the total column density versus 100 µm emissiv- +ity showing a linear relation. The red dashed line is the best fit with +a slope of 0.316 ± 0.005 MJy sr−2 cm2 and a correlation coefficient +ρ = 0.757+0.031 +−0.035. Errors were evaluated by the standard deviation of the +background where no signal is present in either N(Htot) or 100 µm emis- +sivity images. +4. Discussion and conclusions +We have shown that the dust is well mixed with gas within +the translucent cloud MBM 40, both in densest and rarefied +regions, indicating that HLMCs similar to MBM 40 are ideal +comparisons for modelling how dust affects gas condensation. +Liseau et al. (2015) used a different molecule, N2H+ instead of +CO which they argued would be more condensed, to study the +dust-to-gas mass ratio in ρ Oph, a dense star-forming region. In +contrast, we are only concerned with the degree to which the +atomic and molecular gas, and the dust, are mixed within this +region. Consequently, we do not need to make any assumptions +about the intrinsic dust properties, only that the dust tempera- +ture is approximately constant across the cloud. This is valid for +MBM 40, as confirmed by the Planck dust temperature map. +Murray et al. (2021) provide a detailed picture of the neutral +hydrogen column densities and optical depths of diffuse gas at +high Galactic latitudes. Their mean value for N(H i), around 2 × +1020cm−2, is similar to the highest values we have for the cocoon +of MBM 40. In contrast, within the cloud boundaries, the neutral +hydrogen is depleted relative to inferred H2 and there we find +a total hydrogen column density an order of magnitude greater +(Fig. 6), similar to the much coarser result presented in SMLM. +The maximum in N(Htot) corresponds to the minimum in N(H i). +As Murray et al. found for their sample, the gas in MBM 40 +is not associated with any large structure, such as a supernova +remnant or bubble, but it appears to be a transition to molecular +gas within a more extended neutral hydrogen filamentary shear +flow. The cloud-associated atomic gas is connected in space and +radial velocity to more extended regions at distances up to 10 pc +in which there is no evidence for excess IRIS 100 µm emission, +even in those locations where N(H i) is about the same as for +MBM 40. +The comparatively small scale of the molecular structures in +MBM 40 provides a testbed for understanding the phase transi- +Article number, page 5 of 7 + +A&A proofs: manuscript no. 45021corr-modified-copy-copy +tion of the diffuse gas in isolated environments. The gas and dust +are well mixed regardless of the total gas density. This result +should inform turbulent mixing simulations and studies of grain +chemistry. Our next paper, which is currently in preparation, will +present the dynamical and astrochemical tracers. +Acknowledgements. We thank the Arecibo William E. Gordon Observatory staff +and GALFA team for HI 21cm datacube. The 12CO FCRAO data are obtained +from SMLM 2003 study. IRIS infrared images are obtained using the SkyView +Virtual Observatory and NASA/IPAC Infrared Science Archive. We also thank +the referee for valuable suggestions that extended the discussion. +References +Bolatto, A. D., Wolfire, M., & Leroy, A. K. 2013, ARA & A, 51, 207 +Chastain, R. J., Cotten, D., & Magnani, L. 2009, AJ, 139, 267 +Cotten, D. L. & Magnani, L. 2013, MNRAS, 436, 1152 +Draine, B. T. 2010, Physics of the interstellar and intergalactic medium (Prince- +ton University Press) +Hildebrand, R. H. 1983, QJRAS, 24, 267 +Lee, H., Hopkins, P. F., & Squire, J. 2017, MNRAS, 469, 3532 +Lee, Y., Chung, H. S., & Kim, H. 2002, JKAS, 35, 97 +Liljestrom, T. & Mattila, K. 1988, A & A, 196, 243 +Lindner, R. R., Vera-Ciro, C., Murray, C. E., et al. 2015, AJ, 149, 138 +Liseau, R., Larsson, B., Lunttila, T., et al. 2015, A&A, 578, A131 +Lombardi, M., Bouy, H., Alves, J., & Lada, C. 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The re- +sult is shown in Fig. 9. +There is also a DGMR correlation. The greater dispersion +results from the reduced resolution caused by coordinate inter- +polation. But, in addition, the derivation of a single temperature +for each pixel affects the correlation isolated to within the cloud. +4 Based +on +observations +obtained +with +Planck +(http://www.esa.int/Planck), an ESA science mission with instru- +ments and contributions directly funded by ESA Member States, +NASA, and Canada. +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +4 +5 +6 +7 +8 +9 +10 +Fig. 9. DGMR using Planck data. +Each pixel was integrated over the line of sight and the spec- +tral energy distribution was modeled for each of these to give a +unique temperature, but we also know from the velocity analysis +that there is extended foreground and background cold dust that +is not associated with MBM 40. We therefore used only the dust +emission from the 100 µm IRAS image, without any assump- +tion as to temperature modeling, and we discuss only the mixing +ratio. +Dust temperature using Planck +We selected a PLA region including MBM 40 to check whether +the dust temperature is also constant through the whole cloud. +The result, shown in Fig. 10, is that the temperature is nearly +constant with a mean value of ∼ 18 K. It is notable that the in- +ternal cloud structure is rendered uniform, confirming that the +visible substructures are regions differing in density, not temper- +ature. +Article number, page 6 of 7 + +Marco Monaci et al.: The mixing of dust and gas in the high latitude translucent cloud MBM 40 +36.6 +36.8 +37 +37.2 +37.4 +37.6 +37.8 +38 +38.2 +38.4 +43.8 +44 +44.2 +44.4 +44.6 +44.8 +45 +45.2 +45.4 +45.6 +15 +16 +17 +18 +19 +20 +21 +22 +Fig. 10. Dust temperature from Planck data. Black solid con- +tours denote MBM40 (from Planck 545 GHz channel). The two +bright sources are galaxies, SDSS J161101.90+215839.6 and SDSS +J160808.48+213111.7. +Article number, page 7 of 7 + diff --git a/VtE5T4oBgHgl3EQfBg7N/content/tmp_files/load_file.txt b/VtE5T4oBgHgl3EQfBg7N/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3b4bedea80427b35f7f029f5fe5413330232f76c --- /dev/null +++ b/VtE5T4oBgHgl3EQfBg7N/content/tmp_files/load_file.txt @@ -0,0 +1,646 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf,len=645 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='05388v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='GA] 13 Jan 2023 Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 45021corr-modified-copy-copy ©ESO 2023 January 16, 2023 Letter to the Editor The mixing of dust and gas in the high latitude translucent cloud MBM 40 Marco Monaci1, Loris Magnani2, and Steven N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='Shore1 1 Dipartimento di Fisica, Università di Pisa, Largo Bruno Pontecorvo 3, Pisa e-mail: monaci93@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='com e-mail: steven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='neil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='shore@unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='it 2 Department of Physics and Astronomy, University of Georgia, Athens, GA 30602-2451 e-mail: loris@uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='edu Received 20 September 2022 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' accepted 25 October 2022 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' High latitude molecular clouds (hereafter HLMCs) permit the study of interstellar gas dynamics and astrochemistry with good accuracy due to their proximity, generally clear lines of sight, and lack of internal star-forming activity which can heavily modify the physical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' MBM 40, one of the nearest HLMCs, has been extensively studied, making it a superb target to infer and study the dust-to-gas mixing ratio (DGMR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The mixing of dust and gas in the interstellar medium remains a fundamental issue to keep track of astrochemistry evolution and molecular abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Accounting for both molecular and atomic gas is difficult because H2 is not directly observable and H i spectra always show different dynamical profiles blended together which are not directly correlated with the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We used two independent strategies to infer the molecular and atomic gas column densities and compute the dust-to-gas mixing ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We combined H i 21 cm and 12CO line observations with the IRAS 100 µm image to infer the dust-to-gas mixing ratio within the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The cloud 21 cm profile was extracted using a hybrid Gaussian decomposition where 12CO was used to deduce the total molecular hydrogen column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Infrared images were used to calculate the dust emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The dust-to-gas mixing ratio is nearly uniform within the cloud as outlined by the hairpin structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The total hydrogen column density and 100 µm emissivity are linearly correlated over a range in N(Htot) of one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' ISM: clouds – dust, extinction – molecules – structure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Introduction The admixture of dust and gas is an essential, but poorly de- termined, property of the interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' It affects the modeling of radiative balance, moderates astrochemical pro- cesses, and affects star formation within molecular clouds (for the theoretical aspect see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=', Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2017), Tricco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2017), Marchand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' for the observational picture, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=', Reach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The widely adopted value for the mass ratio in our Galaxy is ∼ 100 − 150 (Hildebrand (1983)), but it varies in other galaxies (Young et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (1986)) and even in clouds in our Galaxy (Reach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='1 The usual procedure to obtain the dust-to-gas ratio (here- after DGR) infers the neutral hydrogen column densities from the dust extinction which is then used to obtain the dust mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Alternatively, atomic resonance transitions have been used to in- fer the atomic column densities, but these are altered by density- dependent depletion in the gas phase and from the spotty sam- pling of any intervening clouds because of the sparse coverage by stars and extragalactic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The sparseness of background sources also complicates the distance determination of clouds using standard methods such as star counting, Wolf diagrams, or multiband extinction measurements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=', Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2021), 1 We must emphasize at the beginning of this Letter that our aim is to study the mixing of the components and not the mass ratio, as con- ventionally discussed, because we are trying to avoid the uncertainties induced by adopting specific dust models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Lv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2018), Liljestrom & Mattila (1988)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' These techniques are hampered by the proximity of the translucent and high lat- itude molecular clouds (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=', Magnani & Shore (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Moreover, different lines of sight through these clouds may sam- ple very different structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=', Lombardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2014)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' If the DGRs differ, this complicates extinction corrections and infer- ences about the cloud masses based on infrared imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' It is now possible, however, to obtain the atomic column densities di- rectly following the completion of all-sky H i surveys that com- plement the infrared maps obtained for Galactic and cosmologi- cal surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The object of our study, MBM 40, is a small molecular structure embedded in a larger H i flow (Shore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2003)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The molecular gas distribution has been discussed extensively (Magnani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (1985), MBM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Shore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2003), SMLM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Shore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2006), SLCM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Chastain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2009), CCM10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The CO(1-0) molecular gas distribution is complex with the denser gas in a hairpin structure sur- rounded by a diffuse envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The distance to the cloud is 93 pc (Zucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2019)) and previous studies (Magnani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (1996b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' SMLM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' CCM10) have derived a molecular mass of 20 - 40 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' There is no evidence of star formation despite a rigorous search (Magnani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (1996a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' This cloud provides an exem- plar of a non-star-forming medium where no internal processing has affected the dust properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We should, therefore, see a more Article number, page 1 of 7 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 45021corr-modified-copy-copy nearly pristine presentation of the DGR and its uniformity than would be obtained from a more active source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Data We have used a range of archival data for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We briefly describe here the individual data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' GALFA The Galactic Arecibo L-Band Feed Array HI (GALFA-HI, Peek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2011),) is an extended survey between −1◦ ≲ δ ≲ 38◦ with an angular resolution of 4′ and a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='184 km s−1 spec- tral resolution using the William E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Gordon 305-m telescope at the Arecibo Observatory (Peek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We used the narrow bandwidth data repository (see Peek (2017)) centered at 0 km s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' each spectrum has 2 048 channels spanning |vLSR| ≲ 188 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We used ancillary data furnished with the GALFA datacube to correct for stray radiation and cropped the PPV dat- acube in position to match the extension and velocity (|vLSR| ≲ 15 km s−1) of MBM 40, which is detected between ∼ 2 and ∼ 4 km s−1 in molecular gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 1 shows an example of a GALFA narrowband spectrum (within the range |vLSR| ≤ 20 km s−1) with a signal-to-noise ratio (S/N) ∼ 60;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' each spectrum within the cloud has a similar S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 0 5 10 15 20 25 30 15 10 5 0 5 10 15 0 1 2 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Sample spectra of MBM 40 used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (Top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=') Single GALFA spectrum at α = 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7583◦, δ = +21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8084◦ (J2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='0), in the lower part of the western ridge of MBM 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (Bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=') Sample FCRAO 12CO spectrum near the same position as the H i, obtained by averaging 64 spectra in an 8×8 pattern to enhance the S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The vertical dashed line shared by two plots indicates the bulk velocity of the cloud (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='35 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The H i spectrum shows the brightness temperature (TB), and the 12CO spectrum is in terms of the radiation temperature (TR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' FCRAO The 12CO observations were obtained using the Five College Ra- dio Astronomy Observatory (FCRAO3) in early 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The full datacube is composed of 24 576 frequency-switchedspectra with a velocity resolution of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='05 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We used only the 56 central channels centered at 3 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The average rms noise 2 During these observations the Arecibo Observatory was operated by SRI International under a cooperative agreement with the National Science Foundation, and in alliance with Ana G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Méndez-Universidad Metropolitana, and the Universities Space Research Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 3 The FCRAO was supported in part by the National Science Founda- tion and was operated with the permission of the Metropolitan District Commission, Commonwealth of Massachusetts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 K (Shore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2003)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The 12CO radiation temper- ature (TR) was calculated taking the antenna temperature and then dividing by ηfssηc, where ηfss is the forward-scattering and spillover efficiency (≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7) and ηc is the source filling factor, which we assumed to be unity (see SMLM for a complete dis- cussion of FCRAO observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 1 shows the equivalent FCRAO 12CO profile near the same posi- tion of the H i GALFA spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The H i profile shows different components, one of which is compatible with the MBM 40 bulk velocity of ∼ 3 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 242 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 243 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 21 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 A B 10 15 20 25 30 35 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' H i brightness temperature at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 km s−1 with integrated line 12CO intensity (black contours at levels 2, 3, and 4 K km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' White circles denote the positions to which we refer in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' In this velocity slice (very near the molecular cloud bulk velocity), the H i emission and 12CO emission (position A) are anticorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The HI and 12CO temperature contours for a restricted ve- locity range are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Here we identify two positions to which we refer in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' IRAS 100 µm dust map For the dust distribution, we used the IRAS 100 µm IRIS images that have a spatial resolution of about 2′ (Miville-Deschênes & Lagache (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' These were published after our first study of MBM 40 and have reduced striping, im- proved zodiacal light subtraction, and a zero level compatible with DIRBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The image is an interpolated 50 × 50 pixel ma- trix with the same H i and 12CO spatial resolution centered at RA = 16h10m57s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='19, DEC = +21◦52′29′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We did not use Planck data for the principal study because the spatial resolu- tion is slightly lower and further reduced by a required interpo- lation to a common coordinate grid (from galactic to equatorial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We show in the Appendix, however, that our conclusions are un- changed based on Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Article number, page 2 of 7 Marco Monaci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' : The mixing of dust and gas in the high latitude translucent cloud MBM 40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Data analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Velocity slices The comparison between 12CO and H i profiles in different ve- locity slices is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Because the spectral resolu- tion of H i is coarser than 12CO, a linear interpolation was per- formed to match the velocity resolution of the FCRAO data (see the next subsection for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The H i is more extended in velocity than the molecular gas traced by 12CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' It appears as a "cocoon" (Shore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2003),Verschuur (1974)) around cloud molecular gas and shows some internal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We note that H i is present at both lower and higher velocities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=', v < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='25 km s−1 or v > 4 km s−1) where no 12CO is present, tracing foreground and background gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The H i spatial resolution is sufficient to compare the atomic and molecular gas structures mapped by 12CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The velocity slices show an anticorrelation between atomic and molecular gas, especially at position A, marked by a white circle in the 3 km s−1 slice in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The 12CO enhancement is associated with a lower atomic gas column density, indicative of a phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' This condensation is visible in all velocity slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Data rebinning and interpolation The 21 cm and 12CO maps differ in spectral and spatial reso- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' To compare these data, spectral and spatial linear inter- polations were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Each H i spectrum was linearly inter- polated using the 12CO FCRAO velocities (∆v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='05 km s−1) as query points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The final spectral H i resolution is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='5 times higher than the original GALFA resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Each H i spectrum has been checked for artifacts and, although linear interpolation is sensitive to S/N, in our case S/N is over 60 for each spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We then performed a 2D linear interpolation for both H i and FCRAO data to a common 50 × 50 pixel grid, covering about 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 in RA and 1◦ in DEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Consequently, using this procedure at each position yields linked GALFA and FCRAO spectra with the same velocity resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' H i profile decomposition Each H i profile is a blend of multiple components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Because the 21 cm transition is almost always optically thin for lines of sight toward high-latitude clouds, the full line of sight con- tributes to the observed profile, not only to gas associated with MBM 40, and significant effort was made to decompose each HI spectrum into simpler Gaussian components (see Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2021), Lindner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2015), Pingel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' In this work we effected a two-step decomposition of each profile guided by velocity information from 12CO as explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We used Gaussian profiles to separate the foreground and background as well as cloud neutral hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The 12CO observations constrain a velocity window in which the cloud is embedded, so it is possi- ble to use this information to trim the H i spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We emphasize that the choice of Gaussian fitting was used only to select that gas that is connected to the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' This is different from recent Gaussian decomposition studies aimed at characterizing the fine structure of the cold neutral medium (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=', Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We started the H i decomposition by subtracting a very broad, diffuse component (see, Verschuur & Magnani (1994)) from each GALFA profile, fitting a Gaussian only to the wings of the profile outside the interval −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='55 ≤ vLSR ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='35 km s−1, and then we fit a second Gaussian to the residual emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 4 (left panel) shows an example of the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We excluded positions where there is no clear double profile (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=', where only diffuse gas is present in the neighborhood of MBM 40), and we fit only the redward wing of each H i resid- ual emission, avoiding points below 2 km s−1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 4, right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' This procedure cannot map all the gas because we do not know the parent distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' however, the narrow Gaussian component is at least proportional to the total gas connected with MBM 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Molecular and atomic gas column density If H i is optically thin, as in MBM 40, and if we assume that one temperature dominates each velocity channel, then the total atomic hydrogen column density is given by the following (see for example Draine (2010)): N(H i) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='813 · 1018 � +∞ −∞ TB,Hi(v) dv cm−2, (1) where TB,Hi(v) is the brightness temperature in K and v is the velocity in km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The molecular hydrogen column density, N(H2), was ob- tained using the integrated 12CO line in TR multiplied by a con- version factor (XCO in units of cm−2 K−1 km−1 s): N(H2) = XCOW(COJ=1→0) cm−2, (2) where W(CO) is the velocity-integrated 12CO radiation temper- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The value of XCO is not constant in the Galaxy or even in- side the same cloud (Bolatto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2013)), and it is a source of systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Cotten & Magnani (2013) found that for MBM 40, the XCO factor spans from (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='3)·1020 with an av- erage of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='3 · 1020: we adopted this value as well as a systematic uncertainty of about a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' For each FCRAO position, we evaluated N(H2) as follows: N(H2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='3 · 1020 K−1 km−1 s � +∞ −∞ TR,CO(v) dv cm−2, (3) where TR,CO(v) is the radiation temperature for 12CO (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' SMLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 5 shows the derived column densities for atomic and molecular hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Using equations 1 and 3, we obtain N(Htot) = 2N(H2) + N(H i) within the cloud boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 6 shows N(Htot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The enhancement is quite steep in- side the cloud, where the total column density reaches N(Htot) ≈ 16 · 1020 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We assumed 93 pc as the distance of the cloud (Zucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Thus, the spatial separation between dif- fuse gas and the maximum hydrogen column density near po- sition A is ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='18 pc, similar to that in 12CO, where the 12CO fades out more or less ten beams away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The cloud shows a broad atomic gas environment within the N(Htot) = 2 · 1020 cm−2 con- tour (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=', the outermost contour in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 6) where the 12CO is too faint to be detected with reasonable integration times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The distribution of atomic gas is consistent with Shore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2003), where it was modeled with a "cocoon" shape, but using higher fidelity from GALFA revealed some internal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Position B (the red dot on the top of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 6) shows 12CO weak lines with virtually no 13CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Article number, page 3 of 7 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 45021corr-modified-copy-copy 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 VEL = 2 km/s 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 VEL = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='25 km/s 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 VEL = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='5 km/s 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 VEL = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='75 km/s 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 VEL = 3 km/s 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 VEL = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='25 km/s 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 VEL = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='5 km/s 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 VEL = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='75 km/s 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 VEL = 4 km/s 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 VEL = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='25 km/s 10 15 20 25 30 35 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' H i velocity slices (color scale) compared to the same 12CO velocity slices (black contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Contours span from 1 K to 5 K in steps of 1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The white circle in the 3 km s−1 panel marks position A in the western ridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 15 10 5 0 5 10 15 5 0 5 10 15 20 25 30 10 5 0 5 10 5 0 5 10 15 20 25 30 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (Left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=') Subtraction of extremely diffuse gas from H i sam- ple profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Bold gray lines are wings fitted by a single Gaussian (black dashed line) and the gray dashed line is the section we excluded from the fit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' the solid black line is the result of the subtraction, where the dif- fuse component is severely reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (Right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=') Narrow component extraction: crosses indicate points below 2 km s−1 which we suppose are not directly linked with gas traced by 12CO and which were excluded for the narrow Gaussian fit (black solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Circles denote points we used for the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Dust-to-gas mixing ratio (DGMR) The DGMR was obtained using the IRIS 100 µm image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We lin- early interpolated the image to obtain a 50 × 50 matrix, in which each pixel has a relative H i and 12CO spectra, so we were able to directly perform a simple division to obtain the DGMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The mean 100 µm of the MBM 40 complex is a few MJy sr−1 and the hydrogen column density is about 1020 cm−2, so we scaled the latter by 10−20 to obtain comparable values with dust emission 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='5 2 1020 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 243 0 1 2 3 4 5 6 7 81020 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Total column density of H i (left panel) and H2 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Out- side the jagged boarders visible via N(H i), the column densities were not calculated due to a lack of 12CO emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We note that near posi- tion A, there is a deficit in atomic hydrogen where an enhancement in 12CO is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' and then used the following ratio: DGMR = E100 N(Htot) 1020 � MJy sr−1 cm2� , (4) where E100 is the total emission of the IRIS 100 µm image and N(Htot) is the total hydrogen column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The derived ratio (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 7) is nearly constant within the com- plex, indicating that the dust and gas are well mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We tested the robustness of our procedure by calculating the DGMR using the complete H i spectra, that is without Gaussian decomposi- tion and including the atomic gas over all velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The result is nearly the same because the majority of gas is in molecular form (mapped by 12CO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Considering the full H i velocity range, Article number, page 4 of 7 Marco Monaci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' : The mixing of dust and gas in the high latitude translucent cloud MBM 40 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='5 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='9 243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='9 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='3 2 4 6 8 10 12 14 16 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Total hydrogen column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The colorbar values must be multiplied by a factor of 1020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Red dots indicate the positions discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' the N(H i) is nearly doubled, but it accounts for about 30% of the total gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' However, without any Gaussian decomposition, the re- sult is much more sensitive to the XCO factor: if we use a lower value, that is to say 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 (instead of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='3), the DGMR changes abruptly and the MBM 40 cloud is barely visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Conversely, if we decompose the H i profiles, the DGMR remains constant and also the cloud structure is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Therefore, we can conclude that not all H i gas mapped by GALFA is linked with MBM 40, but only the narrow component that we extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The qualitative appearance from the map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 7 is quantified by the linearity of the relation of the scatterplot of the total gas column density versus 100 µm emissivity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='5 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='9 243 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='9 22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Map of DGMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The colorbar values are expressed in 1020 MJy sr−1 cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Red dots indicate the same positions as reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 2 4 6 8 10 12 14 16 18 4 5 6 7 8 9 10 11 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Scatter plot of the total column density versus 100 µm emissiv- ity showing a linear relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The red dashed line is the best fit with a slope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='316 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='005 MJy sr−2 cm2 and a correlation coefficient ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='757+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='031 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Errors were evaluated by the standard deviation of the background where no signal is present in either N(Htot) or 100 µm emis- sivity images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Discussion and conclusions We have shown that the dust is well mixed with gas within the translucent cloud MBM 40, both in densest and rarefied regions, indicating that HLMCs similar to MBM 40 are ideal comparisons for modelling how dust affects gas condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Liseau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2015) used a different molecule, N2H+ instead of CO which they argued would be more condensed, to study the dust-to-gas mass ratio in ρ Oph, a dense star-forming region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' In contrast, we are only concerned with the degree to which the atomic and molecular gas, and the dust, are mixed within this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Consequently, we do not need to make any assumptions about the intrinsic dust properties, only that the dust tempera- ture is approximately constant across the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' This is valid for MBM 40, as confirmed by the Planck dust temperature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2021) provide a detailed picture of the neutral hydrogen column densities and optical depths of diffuse gas at high Galactic latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Their mean value for N(H i), around 2 × 1020cm−2, is similar to the highest values we have for the cocoon of MBM 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' In contrast, within the cloud boundaries, the neutral hydrogen is depleted relative to inferred H2 and there we find a total hydrogen column density an order of magnitude greater (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 6), similar to the much coarser result presented in SMLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The maximum in N(Htot) corresponds to the minimum in N(H i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' As Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' found for their sample, the gas in MBM 40 is not associated with any large structure, such as a supernova remnant or bubble, but it appears to be a transition to molecular gas within a more extended neutral hydrogen filamentary shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The cloud-associated atomic gas is connected in space and radial velocity to more extended regions at distances up to 10 pc in which there is no evidence for excess IRIS 100 µm emission, even in those locations where N(H i) is about the same as for MBM 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The comparatively small scale of the molecular structures in MBM 40 provides a testbed for understanding the phase transi- Article number, page 5 of 7 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 45021corr-modified-copy-copy tion of the diffuse gas in isolated environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The gas and dust are well mixed regardless of the total gas density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' This result should inform turbulent mixing simulations and studies of grain chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Our next paper, which is currently in preparation, will present the dynamical and astrochemical tracers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We thank the Arecibo William E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Gordon Observatory staff and GALFA team for HI 21cm datacube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The 12CO FCRAO data are obtained from SMLM 2003 study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' IRIS infrared images are obtained using the SkyView Virtual Observatory and NASA/IPAC Infrared Science Archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We also thank the referee for valuable suggestions that extended the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' References Bolatto, A.' 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al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 2019, ApJ, 879, 125 Appendix: Analysis based the Planck dust maps Dust-to-gas mass ratio from Planck The Planck Legacy Archive (PLA) 4 provides high-level maps, including the dust mass column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We extracted a dust map for MBM 40 in M⊙ pc−2 and converted our total gas column density to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 mass units to obtain the DGMR (Planck Collaboration IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Planck 2018 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The re- sult is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' There is also a DGMR correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The greater dispersion results from the reduced resolution caused by coordinate inter- polation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' But, in addition, the derivation of a single temperature for each pixel affects the correlation isolated to within the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 4 Based on observations obtained with Planck (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='int/Planck), an ESA science mission with instru- ments and contributions directly funded by ESA Member States, NASA, and Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='45 4 5 6 7 8 9 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' DGMR using Planck data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Each pixel was integrated over the line of sight and the spec- tral energy distribution was modeled for each of these to give a unique temperature, but we also know from the velocity analysis that there is extended foreground and background cold dust that is not associated with MBM 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' We therefore used only the dust emission from the 100 µm IRAS image, without any assump- tion as to temperature modeling, and we discuss only the mixing ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Dust temperature using Planck We selected a PLA region including MBM 40 to check whether the dust temperature is also constant through the whole cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The result, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 10, is that the temperature is nearly constant with a mean value of ∼ 18 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' It is notable that the in- ternal cloud structure is rendered uniform, confirming that the visible substructures are regions differing in density, not temper- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Article number, page 6 of 7 Marco Monaci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' : The mixing of dust and gas in the high latitude translucent cloud MBM 40 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 37 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 38 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 44 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='8 45 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 15 16 17 18 19 20 21 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Dust temperature from Planck data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Black solid con- tours denote MBM40 (from Planck 545 GHz channel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' The two bright sources are galaxies, SDSS J161101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='90+215839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='6 and SDSS J160808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='48+213111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} +page_content=' Article number, page 7 of 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE5T4oBgHgl3EQfBg7N/content/2301.05388v1.pdf'} diff --git a/W9FRT4oBgHgl3EQf9zjY/content/tmp_files/2301.13689v1.pdf.txt b/W9FRT4oBgHgl3EQf9zjY/content/tmp_files/2301.13689v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b09f6fd126c96fdafc240584248f8f47cc78149 --- /dev/null +++ b/W9FRT4oBgHgl3EQf9zjY/content/tmp_files/2301.13689v1.pdf.txt @@ -0,0 +1,1633 @@ +MNRAS 000, 1–11 (2015) +Preprint 1 February 2023 +Compiled using MNRAS LATEX style file v3.0 +Diffusive shock acceleration at EeV and associated multimessenger flux +from ultra-fast outflows driven by Active Galactic Nuclei +Enrico Peretti1★, Alessandra Lamastra2, Francesco Gabriele Saturni2,3, +Markus Ahlers1, Pasquale Blasi4,5, Giovanni Morlino6 and Pierre Cristofari7 +1 Niels Bohr International Academy, Niels Bohr Institute,University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark +2 INAF – Osservatorio Astronomico di Roma, Via Frascati 33, I-00078 Monte Porzio Catone (RM), Italy +3 ASI – Space Science Data Center, Via del Politecnico snc, I-00133 Roma, Italy +4 Gran Sasso Science Institute, Via F. Crispi 7, 67100, L’Aquila, Italy +5 INFN/Laboratori Nazionali del Gran Sasso, Via G. Acitelli 22, Assergi (AQ), Italy +6 INAF, Osservatorio Astrofisico di Arcetri, L.go E. Fermi 5, I-50125 Firenze, Italy +7 Observatoire de Paris, PSL Research University, LUTH, 5 Place J. Janssen, 92195 Meudon, France +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Active galactic nuclei (AGNi) can launch and sustain powerful winds featuring mildly relativistic velocity and wide opening +angle. Such winds, known as ultra-fast outflows (UFOs), can develop a bubble structure characterized by a forward shock +expanding in the host galaxy and a wind termination shock separating the fast cool wind from the hot shocked wind. In this +work we explore whether diffusive shock acceleration can take place efficiently at the wind termination shock of UFOs. We +calculate the spectrum of accelerated particles and find that protons can be energized up to the EeV range promoting UFOs to +promising candidates for accelerating ultra-high energy cosmic rays (UHECRs). We also compute the associated gamma-ray +and neutrino fluxes and compare them with available data in the literature. We observe that high-energy (HE) neutrinos are +efficiently produced up to hundreds of PeV while the associated gamma rays are efficiently absorbed beyond a few tens of GeV. +By assuming a typical source density of non-jetted AGNi we expect that UFO could play a dominant role as diffuse sources of +UHECRs and HE neutrinos. We finally apply our model to the recently observed NGC1068 and we find out that an obscured +UFO could provide a sizeable contribution to the observed gamma-ray flux while only contributing up to ∼ 10% to the associated +neutrino flux. +Key words: cosmic rays – Active Galactic Nuclei – particle acceleration – gamma rays – neutrinos +1 INTRODUCTION +Fast wide angle winds are one of the most intriguing feedback mech- +anisms of Active Galactic Nuclei (AGNi) (Silk & Rees 1998). Their +impact on the host galaxies has long been considered to affect the +dynamical evolution of the interstellar medium (ISM) and act as a +regulator of star formation (Crenshaw et al. 2003). The discovery +of blue-shifted Fe K absorption lines in X-ray spectra of AGNi (see +e.g. Chartas et al. 2002) brought compelling evidence of mildly rel- +ativistic velocities typically ranging from 0.1 𝑐 to 0.3 𝑐 (where 𝑐 is +the speed of light) in such winds, which thereafter have been of- +ten referred to as ultra-fast outflows (UFOs). Recently UFOs have +been systematically detected in both radio-quiet and radio-loud AGNi +(see e.g. Markowitz et al. 2006; Braito et al. 2007; Cappi et al. 2009; +Tombesi et al. 2010a,b, 2015). The number of observations keeps +increasing with time, despite of the observational challenges, also +thanks to high resolution grating spectra in the soft X-ray (see e.g. +Pounds et al. 2003) and the discovery of ultraviolet (UV) lines in +addition to those already known in the X-ray (Kriss et al. 2018; +★ E-mail: peretti@nbi.ku.dk +Mehdipour et al. 2022). The search for a launching mechanism of the +UFOs has not found a definitive answer yet, although there is a gen- +eral agreement to ascribe this phenomenon to the accretion activity +of the super massive black hole (SMBH) (see Laha et al. 2021, for +an updated review). +The dynamics of the wind and the associated feedback on the +host galaxy depend on whether the outflow conserves energy or +momentum (King & Pounds 2015). In particular, if the wind plasma +does not cool efficiently when it shocks with the surrounding ISM, +the system is energy-conserving and the momentum flux is boosted +while the wind sweeps up the ISM matter. In the opposite scenario, +namely if the wind radiates most of its thermal energy, it evolves +conserving its momentum. In spite of the strong radiation fields +that could strongly affect the cooling of electrons, Faucher-Giguère +& Quataert (2012) showed that two-temperature plasma effects are +likely to slow down radiative losses for protons thereby favoring an +energy-conserving dynamics. +A recent Fermi-LAT analysis (Ajello et al. 2021) showed that +UFOs are a new class of gamma-ray emitters. In this analysis the +average gamma-ray emission from a sample of 11 nearby (z<0.1) +radio-quiet AGNi with an UFO is derived by adopting a stacking +© 2015 The Authors +arXiv:2301.13689v1 [astro-ph.HE] 31 Jan 2023 + +2 +E. Peretti et al. +analysis. The average best-fit gamma-ray spectral slope is measured +to be 2.1 ± 0.3, and the gamma-ray luminosity is found to scale with +the AGN bolometric luminosity. +AGN-driven outflows, similar to stellar winds (see e.g. Weaver +et al. 1977; Koo & McKee 1992a,b), are expected to develop a struc- +ture characterized by an inner wind termination shock (hereafter +wind shock), a contact discontinuity and an outer forward shock. +The forward shock has been proposed as a plausible site for particle +acceleration (see e.g. Lamastra et al. 2016; Wang & Loeb 2016a; +Lamastra et al. 2019; Ajello et al. 2021) where ideal conditions for +efficient production of gamma rays and high-energy (HE) neutri- +nos are expected (see also McDaniel et al. 2023, for a recent study +on molecular outflows). Wang & Loeb (2017) highlighted also the +possibility that in somewhat extreme conditions a fast AGN-driven +wind could have the energy budget to accelerate CRs up to the ultra- +high-energy (UHE) range. The associated cumulative contribution +of AGN-driven winds to the diffuse gamma-ray and neutrino flux +has been also explored (see e.g. Wang & Loeb 2016a,b; Lamastra +et al. 2017; Liu et al. 2018). In addition, the amplitude of the recently +observed spectrum of the diffuse neutrino flux (Abbasi et al. 2020), +in light of the constraints imposed by the diffuse gamma-ray flux +observed by Fermi-LAT (Ackermann et al. 2015), suggests that there +could be a class of sources at least partially opaque to gamma rays +(see e.g. Murase et al. 2016). +Indeed, a search for time-integrated point-like neutrino sources +(Aartsen et al. 2020) highlighted an excess in the direction of the +Seyfert galaxy NGC1068. The most intriguing aspect of the emission +of such galaxy is the lack of gamma rays in the TeV band (Acciari et al. +2019), where the neutrino flux is observed. The natural implication of +an highly opaque cosmic particle accelerator triggered several studies +on the multimessenger implications of HE particles populating the +innermost region of AGN such as disks and accretion flows (see e.g. +Kimura et al. 2019; Gutiérrez et al. 2021) and AGN corona (see +e.g. Murase et al. 2020; Inoue et al. 2020; Kheirandish et al. 2021; +Inoue et al. 2022; Eichmann et al. 2022; Murase 2022). Interestingly, +we recently witnessed a growth in the statistical significance of the +signal from NGC1068 (Abbasi et al. 2022a). +Even though there is an increasing evidence for HE particles pop- +ulating the innermost regions of active galaxies, our understanding +of the acceleration mechanisms at play in such environments is still +incomplete. The mechanisms capable of energizing HE particles in +the vicinity of SMBH is indeed a partially unexplored field we aim to +assess in this work together with its multimessenger consequences. +Therefore, we develop a model for particle acceleration explor- +ing the diffusive shock acceleration (DSA) mechanism at the wind +shock of UFOs. At this shock, unique conditions for acceleration of +protons at energies as high as ∼ 1018 eV can be found. In addition, +the medium property could make such sources bright in HE neutri- +nos while being partially opaque to gamma rays beyond 10 − 102 +GeV. We discuss the role of UFOs as UHE cosmic ray (UHECR) +sources in light of the spectral behavior of the particle flux escaping +the AGN wind bubble. In particular, we show that at the highest +energies spectral features harder than 𝐸−2 could appear in the spec- +trum of escaping particles due to the interplay of diffusion-advection +and energy losses. This result is of great interest in light of the phe- +nomenological results aiming at modeling the spectrum and mass +composition observed by the Pierre Auger Observatory which sug- +gest UHECRs to be characterized by hard spectra (see e.g., Unger +et al. 2015, and references therein). Moreover, if UFOs were com- +mon in galaxies this would produce important implications for their +diffuse multimessenger emission. We provide here an order of mag- +nitude estimate of the potential role of UFOs for a diffuse flux of HE +Figure 1. Structure of the wind bubble. The SMBH (BH) responsible for +the wind launching is located on the left of the sketch. The blue (red) arrow +corresponds to the cool (shocked) wind of the upstream (downstream) region. +The wind shock (𝑅sh) separates these two regions. The SAM is located +between the contact discontinuity (𝑅cd) and the forward shock (𝑅fs) which +bounds the system (credit: I. Peretti). +neutrinos and CR at the ankle. Finally we explore whether an UFO +could play a role in the neutrino emission observed in NGC1068 +discussing possible realizations of such a system. +The manuscript is organized as follows: in § 2 we describe the +model for the wind bubble and the associated particle acceleration +and transport formalism; in § 3 we discuss our results in terms of +spectra of accelerated and escaping particles, we perform a parameter +space scan focusing on the maximum energy. In § 4 we discuss the +multimessenger implications in terms of HE photons and neutrinos +and in § 5, we specialize our model by applying it to the case of +NGC1068 and discuss possible model improvements and alternative +scenarios. We draw our conclusions in § 6. +2 MODEL FOR PARTICLE ACCELERATION AND +MULTIMESSENGER EMISSION IN UFO +The fast wind launched and sustained by an AGN expands with large +opening angle and mildly relativistic velocity. At such speed the +outflow is supersonic, therefore it drives a forward shock expanding +in the external medium, while a contact discontinuity separates the +shocked wind (SW) material from the shocked ambient medium +(SAM). The impact of the wind on the external medium creates a +shock inside the wind material, the wind shock, which is trailing +behind the contact discontinuity and it is oriented towards the central +engine. The outflow, characterized by these three discontinuities, +features a bubble structure as sketched in Fig. 1. +In a uniform medium of density 𝑛0 the wind expands with constant +velocity up to the radius at which the swept-up mass roughly balances +the whole mass of the outflow. After the swept-up mass becomes dy- +namically relevant, the outflow starts decelerating. During the decel- +eration phase the forward shock 𝑅fs and the wind shock 𝑅sh evolve +self-similarly according to different scaling laws: 𝑅fs ∼ 𝑡3/5 and +𝑅sh ∼ 𝑡2/5 (see also Weaver et al. 1977; Koo & McKee 1992a,b, for +detailed discussions and Appendix A for additional details). Since the +wind shock decelerates faster than the forward shock, the hot bubble, +namely the spherical shell between the wind shock and the contact +discontinuity, grows with time while remaining approximately adia- +MNRAS 000, 1–11 (2015) + +upstream +downstream +BH +accretion +R +R +R +disk +sh +cd +fsParticle acceleration and multimessenger radiation from UFOs +3 +batic. In this context, the wind bubble evolution can be considered as +energy-conserving (see e.g., Faucher-Giguère & Quataert 2012). On +the other hand, losses in the SAM can be efficient (see e.g. Nims et al. +2015), so that the whole swept-up mass is eventually compressed into +a relatively thin layer between the contact discontinuity, 𝑅cd, and 𝑅fs. +While the wind bubble slows down its expansion, the relative veloc- +ity between the plasma and the wind shock remains high. We refer +to the innermost region of free expanding wind and to the shocked +wind respectively as upstream and downstream. Different from the +wind shock, the Mach number of the forward shock strongly depends +on the temperature and conditions of the surrounding ISM medium. +Therefore, it is not guaranteed that the forward shock is strong for a +sufficient amount of time. +We assume a spherically symmetric geometry for the outflow and, +since the wind launching region has a negligible size compared to +the whole bubble, we also assume a constant upstream velocity 𝑢1. +The shocked wind is adiabatic, therefore the velocity profile reads: +𝑢2(𝑟) = 𝑢2(𝑅sh/𝑟)2, where 𝑢2 = 𝑢1/4. In agreement with observa- +tions of UFOs, we limit our investigation to a plasma velocity lower +than ∼ 0.3𝑐. We thus neglect relativistic effects due to the mildly rel- +ativistic motion of the plasma. The target density in the upstream re- +gion scales as 𝑛1(𝑟) = �𝑀/[4𝜋𝑟2𝑢1𝑚 𝑝], while in the downstream re- +gion it is constant and equal to 𝑛2 = 4𝑛1(𝑅sh). The target density be- +tween 𝑅cd and 𝑅fs depends on the amount of matter accumulated dur- +ing the outflow evolution, 𝑛SAM = 𝑛0𝑅3 +fs/(𝑅3 +fs − 𝑅3 +cd), where we as- +sume 𝑅cd ≃ 0.9 𝑅fs. We postulate a turbulent nature for the magnetic +field and we estimate its amplitude in the upstream region under the +assumption that a fraction 𝜖𝐵 ≲ 10% of the ram pressure is converted +into magnetic field energy density, namely 𝑈𝐵(𝑟) = 𝜖𝐵𝑚 𝑝𝑛1(𝑟)𝑢2 +1. +At the wind shock we assume that the magnetic field gets com- +pressed by a factor +√ +11, typical of strong shocks, and remains con- +stant throughout the whole downstream region. We adopt the quasi- +linear theory of diffusion, 𝐷(𝑟, 𝑝) = 𝑣(𝑝)𝑟2−𝛿 +𝐿 +(𝑟, 𝑝)𝑙 𝛿−1 +𝑐 +/3, where +𝑣 is the particle velocity, 𝑟𝐿 is the Larmor radius, 𝛿 is the slope +of the turbulence power spectrum and 𝑙𝑐 is the coherence length of +the magnetic field that we assume to be comparable in size with the +launching radius of the wind. In addition, we account for the small +angle scattering regime of diffusion, namely 𝐷 ∝ 𝑟2 +𝐿, which takes +place when 𝑟𝐿 > 𝑙𝑐 (see e.g. Subedi et al. 2017; Dundovic et al. +2020). +The average lifetime of AGNi is inferred to be ≲107 yr (Yu & +Tremaine 2002). During this time, the AGN is expected to show +multiple episodes of activity with duty cycles of ≲105 yr duration +(Schawinski et al. 2015). This suggests that UFOs could have a +lifetime ranging from hundreds to several thousands of years. In this +context, the dynamical evolution of the system becomes soon slower +than all relevant timescales involving HE particles. Hence the process +of particle acceleration and transport can be treated as stationary (see +Fig. 2 in Sec. § 3 where the typical timescales for HE particles in a +prototype UFO are discussed). We assume a spherically symmetric +transport where particles are injected via DSA at the wind shock +whereas, once they reach the forward shock location, they freely +escape the wind bubble. The transport equation reads: +𝑟2𝑢𝜕𝑟 𝑓 = 𝜕𝑟 [𝑟2𝐷𝜕𝑟 𝑓 ] + 𝑝 +3 𝜕𝑝 𝑓 𝜕𝑟 [𝑟2𝑢] + 𝑟2[𝑄 − 𝜆 𝑓 ] , +(1) +where 𝑢 = 𝑢(𝑟) is the wind velocity profile, 𝐷 = 𝐷(𝑟, 𝑝) is the diffu- +sion coefficient, 𝑄 = 𝑄(𝑟, 𝑝) is the injection term and 𝜆 = 𝜆(𝑟, 𝑝) is +the loss rate accounting for pp and p𝛾 interactions (see Appendix B +for additional details). As boundary conditions, consistently with the +spherical symmetry, we assume a null net flux at the center of the +system, 𝑢 𝑓 − 𝐷𝜕𝑟 𝑓 |𝑟=0 = 0, while, as mentioned earlier, we regard +the forward shock as a free escape boundary, 𝑓 (𝑅fs, 𝑝) = 0. The +Table 1. Parameters of the benchmark UFO and of the three alternative +scenario considered for the multimessenger emission. +Parameter +benchmark +A +B +C +𝑢1/𝑐 +0.2 +- +- +- +�𝑀 [M⊙ yr−1] +10−1 +- +- +- +𝜉CR +0.05 +0.087 +0.1 +0.12 +𝜖B +0.05 +- +- +- +𝑙𝑐 [pc] +10−2 +- +- +- +𝛿 +3/2 +- +- +- +𝐿𝑋 [erg s−1] +1044 +- +- +- +𝑛ISM [cm−3] +104 +2 · 103 +5 · 102 +2 · 102 +𝑡age [yr] +103 +3 · 103 +104 +2 · 104 +injection term reads: +𝑄(𝑟, 𝑝) = 𝑄0(𝑝)𝛿[𝑟 − 𝑅sh] = 𝜂CR𝑢1𝑛1 +4𝜋𝑝2 +𝛿[𝑝 − 𝑝inj]𝛿[𝑟 − 𝑅sh], (2) +where 𝑝inj = 1 GeV/c is the injection momentum of particles (picked +up from the plasma) that enter the DSA process and 𝜂CR is the +efficiency factor, normalized such that the CR pressure at the shock +is a small fraction (≲ 10%) of the plasma ram pressure. +We solve Eq. (1) following the same procedure developed in Mor- +lino et al. (2021) and Peretti et al. (2022) with the modification that, +in the present work, energy losses in the downstream region are also +accounted for due to their possible dynamical relevance. In fact the +relative small distance from an active SMBH makes the environment +potentially hostile for HE particles where energy losses could affect +the acceleration and limit the escape. The details of the calculation +are reported in Appendix B while the general form of the solution at +the wind shock reads: +𝑓sh(𝑝) = 𝐶𝑝−𝑠e−Γcut( 𝑝), +(3) +where 𝐶 is a constant (see Appendix B), 𝑠 is the spectral index (𝑠 = 4 +in strong shocks) and Γcut is a HE cut-off function (Γcut = Γ𝑙 + Γ𝑒 +as detailed in Appendix B) which increases with momentum. +As the background photon field, we use the spectral energy dis- +tribution (SED) model shown in the top panel of Fig. A1 provided +in Appendix A. Such a SED is characterized by the big blue bump +and an X-ray power-law component as described in Marconi et al. +(2004). Such field is assumed to decrease with the second power of +distance from the central engine. Consistently with the AGN SED +we also account for the far infrared (FIR) component produced by a +dusty torus (Mullaney et al. 2011). +Gamma rays and neutrinos from pp and p𝛾 interactions are com- +puted following Kelner et al. (2006) and Kelner & Aharonian (2008), +respectively. The energy losses due to Bethe-Heitler pair-production +are taken into account as energy loss mechanism taking place at a +rate 𝑡−1 +BH (see Appendix B). Gamma-gamma absorption on the AGN +SED including the torus is also accounted for by adopting the cross +section appropriate for the case of an isotropic photon field (see +Aharonian 2004). Finally, the associated flux-density of HE protons +escaping the system is computed self-consistently from the solution +to the transport equation as 𝑗esc(𝑝) = −𝐷2𝜕𝑟 𝑓 |𝑟=𝑅 𝑓 𝑠. +The calculations illustrated here have been carried out in the con- +text of the thin shell approximation, in which the SW is perfectly +separated from the SAM. This implies that the cold gas acting as a +target for pp interactions (see below) is all located close to the 𝑅fs. It +is worth pointing out that the spatial distribution of the gas in the SW +might be affected by instabilities and mixing that may lead to a more +MNRAS 000, 1–11 (2015) + +4 +E. Peretti et al. +106 +107 +108 +109 +Energy [GeV] +10 +1 +100 +101 +102 +103 +104 +105 +Time [yr] +age +acc +adv +diff +p +BH +Figure 2. Typical timescales regulating the transport of HE particles com- +pared with the age of the system (thick black line). The blue dashed line +represents the acceleration timescale, while energy losses via photomeson +and Bethe-Heitler pair-production are represented respectively by red and +magenta dot-dashed lines. Advective and diffusive escape are represented by +orange and green dotted lines. +pervasive, though clumpy structure of the gas, which in turn might +affect the spatial and spectral properties of the secondary emission. +These effects will be investigated in a future dedicated work. +3 RESULTS +In order to present our model and discuss its physical implications +we assume a set of typical parameters, hereafter referred to as our +benchmark scenario. The parameters defining our benchmark sce- +nario, summarized in Tab. 1, have been chosen according to the +following criteria: we assume 𝑢1 = 0.2 𝑐 as the average value for the +terminal wind speed of UFOs; �𝑀 = 10−1 M⊙ yr−1 has been chosen +such that the total kinetic power �𝐸 = �𝑀𝑢2 +1/2 matches about ∼ 3% +of the total AGN bolometric luminosity (Fiore et al. 2017) charac- +terized by an X-ray luminosity 𝐿𝑋 = 1044 erg s−1; 𝑙𝑐 = 10−2 pc is +compatible with the launching radius of the wind as predicted by +accretion disk wind models (see e.g. Murray et al. 1995); the age of +the system 𝑡age = 103 yr has been chosen in order to assure stationary +conditions, which are not guaranteed for much younger systems (the +resulting shock radii at such an age are 𝑅sh ≈ 0.8 pc and 𝑅fs ≈ 3 pc); +𝜖𝐵 = 0.05 guarantees a minor dynamical impact of the turbulent +magnetic field; 𝛿 = 3/2 is motivated by an MHD-like (Kraichnan) +turbulence cascade; finally, we assume 𝑛ISM = 104 cm−3 as a typical +value for the external ambient medium that can be found in the core +of luminous infrared galaxies (see e.g. Downes & Solomon 1998; +Faucher-Giguère & Quataert 2012). We assume such a density also +as the representative value for the medium of broad line regions of +active galaxies which is characterized by dense clouds (of density up +to 𝑛𝑐 ≲ 1010 cm−3) embedded in a hot and pressurized gas of lower +density. +Even though we study particle acceleration and transport by solv- +ing the stationary transport equation, Eq. (1), a more direct under- +standing of the physics property of the solution can be obtained +by analyzing the typical timescales of the different competing pro- +cesses. Fig. 2 illustrates the typical timescales for HE particles as +computed at 𝑅sh for the benchmark scenario. Here, the age of +the system (𝜏age, thick black line) is compared with the following +timescales: acceleration (𝜏acc ≈ 𝑠𝐷1/𝑢2 +1, blue dashed line), diffusion +(𝜏diff = (𝑅esc − 𝑅sh)2/𝐷2, green dotted line the diffusion), advection +(𝜏adv = (𝑅esc − 𝑅sh)/< 𝑢2 >, orange dotted), the 𝑝𝛾 photomeson +(𝜏𝑝𝛾, red dot-dashed line) and Bethe-Heitler pair-production (𝜏BH, +magenta dot-dashed). Inelastic pp collisions were also taken into ac- +count, however at the wind shock the target density is of the order +of 20 cm−3, so that the associated timescale exceed 105 yr, the upper +limit of the plot. Therefore, pp interactions are dynamically irrele- +vant for the acceleration. On the other hand, this does not exclude +them as relevant loss mechanism in the SAM where the density is +orders of magnitude higher. From the interplay between the differ- +ent timescales it is possible to to observe: 1) 𝜏acc ≪ 𝜏age and the +minimum between losses and escape is also smaller than the age +supporting the stationary assumption; 2) 𝜏acc as well as 𝜏diff feature +a break between 102 PeV and 1 EeV due to the transition in the diffu- +sion coefficient from the QLT behavior (𝑟L < 𝑙c) to the small angle +scattering regime (𝑟L > 𝑙c); 3) energy losses via 𝑝𝛾 photomeson pro- +duction play a dominant role at the highest energies and are expected +to set the maximum energy; 4) 𝜏𝑝𝛾 and 𝜏BH increase with the second +power of the distance moving outward from 𝑅sh to 𝑅fs, therefore +the transport in the downstream region is characterized by a close +competition between energy losses and escape; 5) the Bethe-Heitler +pair production does not play a dominant role since at low energy +the transport is advection-dominated while at the highest energies is +regulated by the photomeson production. +The spatial transport of particles is regulated by advection at low +energy and by diffusion at the highest energies while energy losses, +both adiabatic and inelastic collisions, can affect the normalization +and/or introduce spectral features. The top panel of Fig. 3 shows +the spatial distribution for three different CR energies. The upstream +region (𝑅/𝑅sh < 1) is characterized by the competition between dif- +fusion, which tries to spatially homogenized particles, and advection +which prevents low energy particles to diffuse upwind. In particu- +lar, one can see that the higher the energy the stronger the impact +of diffusion. The red dotted line illustrates the spatial distribution +at low energies while the blue and black lines show results for an +intermediate energy and near the exponential cut off, respectively. In +the downstream region (𝑅/𝑅sh > 1) one can see that, different from +the upstream one, advection-dominated transport leads to a spatially +homogenized solution whereas diffusion-dominated transport leads +to a number suppression while approaching the free escape bound- +ary. This behavior is a natural result of the spherical geometry of the +system. +The bottom panel of Fig. 3 illustrates the spectrum of accelerated +particles at the shock (thick green line), at different radii in the down- +stream region (dotted curves where the red one is the closest to 𝑅sh +while the blue one approaches 𝑅fs) and the associated spectrum of +the escaping flux (purple dashed line). The spectrum of accelerated +particles, as suggested by Eq. (3) and as naturally predicted by DSA +in a finite system, is a power-law of index 𝑠 with maximum energy +𝐸max ≃ 1 EeV and does not show any relevant additional spectral +feature. On the other hand, the particle spectrum gradually steepens +in the downstream region moving from the wind shock to the forward +shock as a result of escape and 𝑝𝛾 energy losses. In the downstream +region, energy losses play a crucial role in shaping the spectrum of +particles escaping the system. In particular, the photomeson interac- +tions on the big blue bump occur faster than escape at ∼ 1017 eV as +one can also deduce from Fig. 2. This results in a dip in the spectrum +at such energy, whereas at higher energies the escape is more efficient +so that the spectrum hardens at the highest energies. +A comment on the maximum energy is in order: the exponential +function regulating the cutoff, Γcut, accounts for the geometry of the +MNRAS 000, 1–11 (2015) + +Particle acceleration and multimessenger radiation from UFOs +5 +100 +2 × 100 +3 × 100 +4 × 100 +R / Rsh +10 +4 +10 +3 +10 +2 +10 +1 +100 +F (r, p) [arbitrary units] +F (r, 10 PeV) +F (r, 102 PeV) +F (r, 103 PeV) +10 +3 +10 +2 +10 +1 +100 +101 +E [1018 eV] +106 +107 +108 +ps F(r, p) [arbitrary units] +Fsh +Jesc +F2 +Figure 3. Top panel: Spatial distribution of the CR phase space density. Low +energy particles behave in the system as illustrated by the red dotted line, high +energy particle behavior is represented by the blue dot-dashed curve while the +black curve shows the behavior of particles at the maximum energy. Bottom +panel: Spectrum of particles at the shock (thick green line) compared to the +spectral shape of the escaping flux (dashed magenta line). The dotted curves +represent the particle spectra in the downstream region from the wind shock +(red) to the escape radius (blue). +system and loss mechanisms, so that it cannot be simplified as a ratio +𝐸/𝐸max. Therefore, here we define 𝐸max as the energy where 𝑝𝑠 𝑓sh +is suppressed by one 𝑒-fold. In what follows we describe in detail +the impact of different realizations of the system to the maximum +energy. +3.1 Impact of parameters on the maximum energy +A qualitative estimate of the maximum energy set by the geometry +of the system can be obtained by comparing the upstream diffusion +length, 𝐷1/𝑢1, with the size of such region, 𝑅sh (see also Morlino +et al. 2021; Peretti et al. 2022, for additional discussion). Since at +the highest energies 𝑟L is already larger than 𝑙𝑐 one can write the +maximum energy as follows: +𝐸max = 𝑞𝐵 +√︂ +6 +𝑐 +� 𝜖𝐵 �𝑀𝑙𝑐 +𝑅sh +�1/2 +𝑢1 +≃ 1.4 EeV +� 𝜖𝐵 +0.05 +�𝑀 +10−1M⊙yr−1 +𝑙𝑐 +10−2 pc +1 pc +𝑅sh +�1/2 +𝑢1 +0.2 𝑐 . +(4) +As one can see from Eq. (4), the maximum energy for DSA at the +wind shock of UFOs turns out to be of the order of EeV for standard +values of parameters. +Tab. 2 highlights the impact of different parametric assumptions +on the maximum energy. In particular, we see that, according to +Table 2. Impact on the maximum energy of a parameter variations. All +parameters are set to the benchmark UFO values shown in Table 1 except +for those indicated in the first two columns. The last row shows the result for +benchmark values for comparison. +Parameter(s) +Variation(s) +𝐸max [EeV] +𝑢1 [𝑐] +0.03 / 0.1 / 0.3 +0.03 / 0.31 / 1.86 +�𝑀 [𝑀⊙ yr−1] +10−2 / 1 +0.29 / 2.82 +𝜖B +0.01 / 0.1 +0.53 / 1.41 +𝑙𝑐 [pc] +3 · 10−3 / 10−1 +0.81 / 0.24 +𝑡age [yr] +102 / 104 / 105 +0.58 / 1.12 / 0.88 +𝛿 +5/3 (Kolmogorov) +1.02 +𝑛ISM [cm−3] +103 +1.11 +𝑈rad +none / double +2.04 / 0.77 +( �𝑀, 𝑢1) +pessimistic / optimistic +0.01 / 4.53 +no variations (benchmark) +1.06 +Eq. (4), 𝐸max scales roughly linearly with 𝑢1 and with the square +root of �𝑀 and 𝜖𝐵. The impact of 𝑙𝑐 on 𝐸max can be understood as +follows: when 𝑙𝑐 ≫ 10−2 pc, the diffusion coefficient is much larger +than the benchmark scenario so that the diffusion length reaches the +size of the system at lower energies; when 𝑙𝑐 ≪ 10−2 pc the energy +at which the diffusion coefficient changes regime (from the stan- +dard quasi-linear theory ∼ 𝐸2−𝛿 to the small pitch-angle scattering +regime ∼ 𝐸2) shifts to lower energies thereby resulting in a larger +value of 𝐷 at the highest energies. Therefore, since at the highest +energies, diffusion dominates, a local maximum in 𝐸max appears for +𝑙𝑐 ≃ 10−2 pc. The age of the system does not have a strong impact +on 𝐸max which is affected by less than a factor 2 for the wide range +of alternatives considered. Similarly, different assumptions on the +slope of the turbulence cascade (Kolmogorov-like) and the external +medium density have a negligible impact on 𝐸max. In fact, while +the former is irrelevant because at the highest energies 𝑟L > 𝑙𝑐 and +diffusion has changed regime, the latter impacts mostly the dynamics +of the bubble. Interestingly, as also highlighted in Fig. 2, different +assumptions in the photon field highlight a trend which suggests that +the 𝑝𝛾 interactions on the infrared field of the torus regulate the max- +imum energy. In particular, 𝐸max increases by a factor 2 when the +photon field is removed, while it decreases when a stronger photon +field is considered. This suggests that the infrared field of the torus +could play a crucial role in regulating the maximum energy achiev- +able in UFOs. We finally explore the combined effect of maximum +(minimum) values of 𝑢1 and �𝑀 corresponding to a plausible opti- +mistic (pessimistic) scenario. In this context one can see that UFOs +can be responsible for particle acceleration with 𝐸max ranging from +10 PeV up to 5 EeV. In particular, the objects in the high luminosity +end of a hypothetical luminosity function of UFOs are candidate +acceleration sites of UHECRs where protons could reach a few EeV. +Heavier nuclei could be accelerated to higher total energies provided +they survive photodisintegration. The latter possibility depends on +the photon background present at the acceleration site and on the +relative distance between 𝑅sh and 𝑅fs. +4 GAMMA-RAYS AND HE NEUTRINOS FROM UFOS AND +CONSTRAINTS TO THEIR LOCAL DENSITY +The gas swept-up from the dense environment of the SMBH as well +as the strong radiation field of the AGN can make hadronic interac- +tions dynamically relevant in UFOs. Since interactions are copiously +MNRAS 000, 1–11 (2015) + +6 +E. Peretti et al. +100 +102 +104 +106 +108 +E (GeV) +10 +14 +10 +13 +E2dN/dE (erg cm +2 s +1) +UFO SED +SAM (pp) +SAM + (pp) +SW+SAM + (p ) +Figure 4. Gamma-ray (thick blue line) and HE neutrino flux produced in +the benchmark UFO (𝑅sh ≃ 0.8 pc) via pp (dotted red line) and p𝛾 (orange +dot-dashed line) interactions. The acronym SAM (SW) refers to the shocked +ambient medium (shocked wind). The thin blue-to-cyan lines represents the +gamma-ray flux computed respectively for scenario A, B and C (see Tab. 1) in +order to illustrate the dependence of the gamma-ray absorption on the bubble +expansion (where 𝑅sh ≃ 2, 5 and 8 pc respectively). The UFO is assumed to +be located at z=0.013 in order to be directly compared with the best-fit UFO +SED provided in Ajello et al. (2021), where the gray band represent the 1 𝜎 +uncertainty band of such a best-fit UFO SED. +taking place in the UFO wind bubble, a high level of gamma-ray +and HE neutrino emission can be expected. Figure 4 illustrates the +gamma-ray (thick blue line) and the single-flavor neutrino flux (dot- +ted red for pp and dot-dashed orange for p𝛾) flux expected from +the benchmark UFO placed at a redshift z=0.013. In particular, the +gamma-ray flux is compared with the typical UFO spectral energy +distribution (SED) as found in Ajello et al. (2021). +Despite the fact that the benchmark scenario represents an average +UFO in terms of power and maximum energy, the gamma-ray flux of +the benchmark UFO (𝑅sh ≃ 0.8 pc and 𝑅fs ≃ 3 pc) cannot in fact be +representative of the whole class due to the strong radial-dependence +of the 𝛾𝛾 absorption. Therefore, in Fig. 4 we also compare it with the +expected gamma-ray fluxes as predicted from scenario A (𝑅sh ≃ 2 pc +and 𝑅fs ≃ 8 pc), B (𝑅sh ≃ 5 pc and 𝑅fs ≃ 22 pc) and C (𝑅sh ≃ 8 pc +and 𝑅fs ≃ 40 pc) as described in Tab. 1. Scenarios A, B and C do +not differ from the benchmark scenario in terms of total power but +illustrate alternative realizations of it, having a larger size resulting +from a longer evolution in a less dense environment. One can observe +that these scenarios enhance the gamma-ray emission above ∼ 10 +GeV due to a weaker gamma-gamma absorption and allow a better +agreement with the UFO sample observed by Ajello et al. (2021). +Regardless of the age of the system, gamma-rays of energy greater +than a few TeV are completely absorbed by the infrared radiation +field of the torus. Therefore, pp neutrinos in the 10 TeV - 1 PeV band +as well as p𝛾 neutrinos in the energy band 102 TeV - 102 PeV would +be produced without their gamma-ray counterpart. UFOs are thus +expected to be bright neutrino sources featuring spectra as hard as +∼ 𝐸−2, while being highly opaque to TeV (and possibly 10 − 102 +GeV) gamma-rays. +UFOs could be common in nearby luminous infrared galaxies +(LIRGs) such as active starburst galaxies and Seyfert galaxies. How- +ever, the abundance and distribution of these objects throughout +the Universe as well as their luminosity function are poorly known. +Therefore, in what follows we estimate the order of magnitude of their +diffuse multimessenger emission in terms of EeV cosmic rays and +associated HE neutrinos and gamma rays. Since the horizon for the +Bethe-Heitler pair-production suffered by UHECRs on the cosmic +microwave background (CMB) is placed beyond 𝑧 > 2 we neglect +such a loss mechanism in our calculations. +We first assume as a prototype UFO the EeV-atron described by the +benchmark scenario presented in Tab. 1. As discussed in Appendix B, +the escaping flux is regulated by the interplay between diffusion, +advection and energy losses. However, despite its complex analytic +expression, assuming an ∼ 𝐸−2 spectrum the power contained by the +escaping particles can be approximated as follows: +𝐿CR = +∫ +𝑑𝑝 4𝜋𝑝2[𝑝𝑐]4𝜋𝑅2 +esc 𝑗esc ≃ 3 +4 𝜉CR 𝜂loss �𝑀 𝑢2 +1 +≃ 2 · 1043 𝜂loss +𝜉CR +0.05 +�𝑀 +0.1 M⊙/yr +� 𝑢1 +0.2 𝑐 +�2 erg +s , +(5) +where 𝑗esc is the escaping flux of protons as defined in Appendix B, +𝜂loss ≤ 1 is an age-dependent parameter which accounts for the +relative reduction in the escaping flux due to energy losses, while the +other parameters are normalized to the values shown in Tab. 1. The +CR luminosity is related to the CR spectral injection rate QCR (units +of GeV−1s−1) as 𝐿CR = +∫ +d𝐸 𝐸QCR(𝐸). For simplicity, we assume +in the following that the CR emission follows 𝐸−2 from GeV to EeV, +such that 𝐸2𝑝QCR ≃ 𝜒𝐿CR with 𝜒 ≡ 1/ln(EeV/GeV) ≃ 0.05. +In general, the locally observed CR spectrum 𝜙CR (units of +GeV−1cm−2s−1sr−1) is related to the spectral emission rate of ex- +tragalactic sources via a set of transport equations. For CR protons +in the EeV energy range we can assume that the transport is domi- +nated by continuous energy loss due to the expansion of the Universe +while we neglect the effect of intergalactic magnetic fields. Following +the notation of Ahlers & Halzen (2018), we can estimate the local +contribution of UFO EeV-atrons as: +[𝐸2 +𝑝𝜙CR]EeV ≃ 𝜉𝑧 +4𝜋 +𝑐 +𝐻0 +𝜌0[𝐸2 +𝑝QCR]EeV . +(6) +The factor 𝜉𝑧 is of order unity accounting for the integral in redshift +of the source distribution. In particular, 𝜉𝑧 ≃ 0.5(2.6) for a flat (star- +formation rate) distribution. The parameter 𝜌0 represents the local +comoving density of sources for which we assume 𝜌0 = 10−5 Mpc−3 +as a reference. Such a value is often quoted as a typical density of +AGNi with X-ray luminosity of the order of 𝐿𝑋 ≃ 1044 erg s−1 (see +e.g. Fiore et al. 2017; Ueda et al. 2014; Murase & Waxman 2016). +It is worth mentioning that a source density ∼ 10−4 − 10−5 Mpc−3 +matches also the number density inferred for powerful starbursts such +as luminous and ultra-luminous infrared galaxies (see Gruppioni et al. +2013; Peretti et al. 2020; Condorelli et al. 2022) which are currently +also considered to be plausible hosts for UHECR accelerators (Aab +et al. 2018). Expressing the contribution of EeV CR protons to the +spectral emission as [𝐸2𝑝QCR]EeV = 𝜒𝐿CR with 𝜒 ≃ 0.05 we arrive +at: +[𝐸2 +𝑝𝜙CR]EeV ≃ 3 · 10−7 +𝜌0 +10−5 Mpc−3 +𝜉𝑧 +2.6 +𝐿CR +1043erg s−1 +GeV +cm2s sr . +(7) +In +comparison, +the +observed +CR +spectrum +has +value +of +[𝐸2𝜙CR]EeV ≃ 2·10−7GeVcm−2s−1sr−1, very close to our estimate. +The associated all-flavor neutrino spectral injection rate Q𝜈 result- +ing from pp interactions can be related to QCR as: +𝐸2 +𝜈Qall 𝜈 ≃ 1 +2 𝜅 𝜎pp 𝑐 𝑡age 𝑛ISM 𝜂−1 +loss 𝐸2 +𝑝QCR +≃ 0.07 𝑛ISM,b +104cm−3 +𝑡age +103yr 𝜂−1 +loss 𝐸2 +𝑝QCR , +(8) +where 𝜅 ≃ 0.5 is the inelasticity of pp interactions with cross section +MNRAS 000, 1–11 (2015) + +Particle acceleration and multimessenger radiation from UFOs +7 +𝜎pp ≃ 3 · 10−26cm2 and the CR proton energy is related to neutrino +energy as 𝐸𝜈 ≃ 0.05𝐸p. Notice that in this order of magnitude +estimate we consider only neutrinos resulting from pp interaction +since they dominate up to at least 102 TeV. Neutrinos from p𝛾 as +previously shown, do not feature a different order of magnitude in +the HE SED therefore the pp neutrino flux can be assumed as a good +approximation also for the order of magnitude flux reached by the +p𝛾 component. +Again, following Ahlers & Halzen (2018), we can now estimate +the isotropic neutrino flux as: +𝐸2 +𝜈𝜙all𝜈 = 𝜉𝑧 +4𝜋 +𝑐 +𝐻0 +𝜌0𝐸2 +𝜈Qall𝜈 . +(9) +Adopting in Eq. (9) the expressions for the CR and neutrino luminos- +ity as described in Eq. (5) and Eq. (8) together with 𝐸2𝑝QCR ≃ 𝜒𝐿CR +one obtains an estimate of the total flux of EeV CRs and single flavor +neutrino flux at Earth: +𝐸2 +𝜈𝜙all𝜈 ≃ 2 · 10−8 GeV +cm2s sr +× +𝜌0 +10−5Mpc−3 +𝜉𝑧 +2.6 +𝑛ISM,b +104cm−3 +𝑡age +103yr +𝐿CR +1043erg s−1 . +(10) +This flux is comparable to the level of the diffuse neutrino flux +observed by IceCube (see Abbasi et al. 2022b) for energies larger +than 100 TeV. +These estimates indicate that if UFOs were as abundant as typi- +cal non-jetted AGNi and powerful starbursts they could potentially +be the dominant sources of cosmic rays at EeV while also strongly +contributing to the diffuse neutrino flux observed by IceCube above +100 TeV. In this context, the associated gamma-ray flux would not +be expected to contribute substantially to the diffuse gamma-ray flux +observed by Fermi-LAT (Ackermann et al. 2015) due to the flat spec- +tral shape and the strong absorption taking place inside the source +environment above ∼ 102 GeV that would limit the energy budget of +the electromagnetic cascade in the propagation of the gamma rays to +the Earth. In a scenario of minimal absorption (scenario C) one can +expect at most a gamma-ray flux at ≲ 1 TeV at the same level of the +neutrino one, namely 𝐸2𝛾𝜙𝛾(𝐸𝛾) ∼ 𝐸2𝜈𝜇 𝜙𝜈𝜇 (𝐸𝜈𝜇). +Directly observed UFOs as well as X-ray AGNi and luminous in- +frared galaxies where a UFO can be obscured, represent a promising +source class where UHECRs could be accelerated. The detection of +a flat (∼ 𝐸−2) neutrino spectrum extending in the PeV range from the +innermost core of these galaxies could serve as a hint to discriminate +whether DSA is taking place as we propose in this work. +5 APPLICATION TO NGC1068 +In what follows we specialize our calculations to the nearby Seyfert +galaxy NGC1068 located at a distance 𝐷𝐿 = 14 Mpc. Mizu- +moto et al. (2019) reported some indications of an UFO hosted +in the core of NGC1068. We compute the gamma-ray and associ- +ated neutrino emission through pp and p𝛾 interactions by adopting +�𝑀 = 2 · 10−1 M⊙ yr−1, 𝑢1 = 0.1 𝑐 and 𝑙𝑐 = 10 pc while assuming +all other parameters as in the benchmark scenario. In particular, the +coherence length assumed here to be as large as the system size +(2 𝑅fs ≃ 𝑙𝑐 ≃ 10 pc) result in a maximum energy of approximately +𝐸max = 5 PeV. This is found in agreement with the trend presented +in the previous section (§ 3.1). Notice that the value of 𝑙𝑐 required for +this calculation is considerably larger than in our benchmark scenario +discussed in Sec. 3. In fact, assuming smaller values of 𝑙𝑐 leads to a +neutrino spectrum extending up to several PeV with an approximately +∼ 𝐸−2 spectrum, in contradiction with the IceCube sensitivity. +10 +2 +100 +102 +104 +106 +108 +E (GeV) +10 +15 +10 +14 +10 +13 +10 +12 +10 +11 +10 +10 +E2dN/dE (erg cm +2 s +1) +Fermi +MAGIC +IceCube +SAM (pp) +SAM + (pp) +SW (pp) +SW + (pp) +SW+SAM + (p ) +Figure 5. Multimessenger emission for the case of NGC1068. The thick blue +line represents the gamma-ray emission dominated by pp interactions in the +shocked ambient medium (SAM), while the red dotted line is the associated +neutrino flux. The emission from the shocked wind (SW) (gamma-ray in cyan +dashed and neutrino in magenta dot-dot-dashed) as well as the photomeson +(orange dot-dashed) emission are subdominant. The model prediction is com- +pared with Fermi-LAT (Abdollahi et al. 2020), MAGIC (Acciari et al. 2019) +and IceCube (Abbasi et al. 2022a) data. +Figure 5 illustrates the multimessenger flux produced by the ac- +celerated particles in the system and under the assumption that the +pressure of accelerated particles is ∼ 5% of the ram pressure at the +wind termination shock. One can see that the gamma-ray emission +(thick lines) is dominated by the pp contribution in the shocked am- +bient medium (blue line), whereas the emission from the shocked +wind (purple line) is more than two order of magnitude dimmer. As +discussed in Sec. § 4, the strong photon field associated with the ac- +cretion disk and torus makes the source opaque to gamma rays above +a few tens GeV, so that all TeV photons are completely absorbed. The +neutrino flux is dominated by the pp channel in the range from GeV +up to ∼ 102 TeV where it features a reduction due to the maximum +energy. Photomeson interactions take also place in the UFO environ- +ment. However, they produce a negligible impact on the spectrum. +Overall the neutrino flux shows a remarkable flat spectrum over more +than five orders of magnitude where the associated gamma-ray coun- +terpart gets absorbed beyond ∼ 10 GeV. It is possible to notice that 1) +a UFO could dominate the gamma-ray flux observed by Fermi-LAT +and 2) in the TeV range the UFO could contribute from a few up +to ∼ 10% of the flux measured by IceCube (Abbasi et al. 2022a) +leaving room for other possible sources such as the AGN corona, the +molecular outflow or the starburst ring. A standard UFO seems there- +fore disfavoured for explaining the level of neutrino flux observed by +IceCube in light of the stringent upper limits imposed by MAGIC as +well as the detected Fermi-LAT flux at lower energies. +5.1 Forward shock scenario for NGC1068 +As an alternative scenario to the acceleration at the wind termination +shock one could explore the same UFO at an early stage during which +the forward shock is expected to play the most relevant role in terms +of particle acceleration. +The forward shock expanding in the unperturbed ISM of density +𝑛ISM ∼ 104 cm−3 can indeed foster particle acceleration through +DSA. Let us consider a young UFO which has not entered the de- +celeration phase yet, namely it would be expanding with constant +velocity ∼ 𝑢1. In this context, the forward shock is expected to +be strong (M ≫ 1) thereby efficient in accelerating particles. The +MNRAS 000, 1–11 (2015) + +8 +E. Peretti et al. +spectrum of accelerated particles at such shock can be written as +𝑓 (𝑝) = 𝐴(𝑝/𝑝0)−𝛼 exp(−𝑝/𝑝max), where the normalization 𝐴 is +estimated assuming that a fraction 𝜉fs ∼ 0.1 of the ram pressure +𝑚 𝑝𝑛ISM𝑢2 +1 is converted into energized particles. In the test-particle +limit, 𝛼 = 4, while an upper limit on the maximum momentum can +be estimated with the Hillas criterion, leading to: +𝑝max,fs ≃ 𝜉fs +0.1 +𝑅fs +pc +𝑢1 +108cm/s +𝐵 +𝜇G103 GeV c−1 . +(11) +An acceleration site with 𝑅fs ∼ 10−2 pc and 𝐵 ∼ 102 − 103 𝜇G +allows for 𝑝max ≳ 104 − 105 GeV c−1, in principle sufficient for +the pp and p𝛾 processes to produce gamma rays and neutrinos in an +energy range accessible to current instruments. +Assuming that the accelerated protons fill at most a volume 𝑉 ∼ +4 · 10−6(𝑅fs/10−2pc)3 pc3, in which the interactions with the gas +and the AGN photon field are taking place, the UFO is expected to +produce a gamma-ray luminosity (before any absorption effects are +taken into account) of the order of 𝐿𝛾(103 − 104GeV) ≃ 10−15 − +10−14 erg cm−12 s−1, and a neutrino luminosity of the order of +𝐿𝜈(103 − 104GeV) ≃ 10−16 − 10−15 erg cm−2 s−1, indicating that +the emission from the forward shock is expected to be sub-dominant +compared to the one from the wind termination shock happening at +later times. +Late stage UFOs, as discussed in Ajello et al. (2021), could be +luminous enough to be detected in the GeV range by space based +telescopes such as Fermi-LAT. However, this would require the cen- +tral engine to be active for a time 𝑡 ≫ 103 yr. On the other hand, +high acceleration efficiency at the wind termination shock can be +expected to be found soon after the deceleration phase has begun. +5.2 Discussion on NGC1068 +UFOs are characterized by a prominent gamma-ray (up to ∼ 10−102 +GeV) and neutrino emission (typically up to ∼ 102 TeV) resulting +from pp interactions. Therefore, in the standard test particle regime +considered in this work, the resulting gamma-ray and neutrino spectra +will feature roughly the spectral slope (∼ 𝐸−2) of their parent protons. +We observe that from the energetic point of view, the level of neutrino +flux observed is compatible with the amount of power that UFOs can +supply on average suggesting the AGN as a plausible origin for such +an emission. However, the strongest limitation comes from the level +of gamma rays measured by Fermi-LAT. In fact, even though the AGN +radiation field can absorb efficiently TeV photons, GeV gamma-rays +come basically unabsorbed. Indeed, as pointed out by several authors +(see e.g. Murase et al. 2020; Inoue et al. 2020; Kheirandish et al. +2021; Inoue et al. 2022; Eichmann et al. 2022; Murase 2022), such +level of neutrinos is not compatible with a pp scenario with a ∼ 𝐸−2 +spectrum unless the emission comes from a region optically thick to +GeV and sub-GeV gamma rays, namely the nearest neighbourhood +of the SMBH or AGN-corona having a size of ≲ 102 𝑅𝑠 (where 𝑅𝑠 +is the Schwarzschild radius). +Even though it is not clear whether the launching radius of a +UFO can be localized at such a small distance from the SMBH, +we explored under which conditions one could expect a UFO to be +confined close to the AGN corona for a sufficient amount of time +during its deceleration phase. We found that, in order to reproduce +the level of neutrino flux inferred by IceCube (Abbasi et al. 2022a) +without exceeding the gamma-ray flux, the energy budget would not +exceed standard values typical of UFOs ( �𝐸 ≃ 1041 erg s−1) but the +requirement in terms of average gas density of the external medium +would be 𝑛ISM ≳ 1010 cm−3. Such a density could be compatible +with the high column density inferred for this source(see e.g. Matt +et al. 1997), therefore this could be a plausible scenario to account +for the observed neutrino flux. A detailed modeling of a confined +wind expanding in the ultra-dense medium goes beyond the scope of +this work, therefore it is left for a follow-up investigation. +The recent anisotropy study carried out by the Pierre Auger Obser- +vatory suggests that Seyfert and Starburst galaxies (or objects with +a spatial distribution related to these sources) may be related to this +anisotropy (Aab et al. 2018). It is intriguing that NGC1068 is the +fourth most relevant object in such catalogs, providing a contribu- +tion of about 5 − 10%. However, our calculations suggests that if the +gamma-ray flux detected by Fermi-LAT is associated with the UFO +activity in this source, then the maximum energy is bound to be too +low to be relevant for UHECRs. Viceversa, if UHECRs are to be +produced in the UFO, a different source for the gamma rays has to +be found. +6 CONCLUSIONS +In this work we investigated the potential of diffusive shock accel- +eration at the wind termination shocks of ultra-fast outflows (UFOs) +in the core of active galaxies. We developed a model of acceleration +and transport of particles in the wind bubbles excavated by UFOs +and we studied the multimessenger implications in terms of escaping +particles and high-energy photons and neutrinos produced through +hadronic interactions. +We found that protons can be accelerated up to the EeV range and +that the far infrared photon field of the torus could play a dominant +role in setting the maximum energy in such sources. In addition, +the transport condition in the UFO environment could result in a +spectral hardening of the escaping flux of the cosmic rays at the +highest energies. Such energetic and spectral properties are crucial +in light of the recent results obtained by the Pierre Auger Observatory +which suggests that the sources of UHECRs could be characterized +by hard spectra (see e.g. Aab et al. 2017, and references therein). +UFO are extremely interesting in terms of multimessenger emis- +sion since, in standard conditions, they are expected to shine in +gamma rays up to ∼ 10 GeV while being opaque beyond 10 − 102 +GeV depending on the parametric configuration. While gamma rays +are efficiently absorbed, HE neutrinos from pp and p𝛾 interactions +will be copiously produced with a spectrum extending up to ∼ 102 +PeV featuring a spectral slope as hard as the one of their parent pro- +tons. Such a property is particularly interesting since UFOs have the +potential and possibly the number density in the Universe to simul- +taneously dominate the CR spectrum at the ankle and the diffuse +neutrino flux observed beyond ∼ 102 TeV. +We finally applied our model to the UFO that has been suggested +to be hidden in the nearby galaxy NGC1068 and we found out that, +if confirmed, a well developed spherical UFO wind bubble could +dominate the gamma-ray flux observed by Fermi-LAT while it is +unlikely to contribute more than ∼ 10% of the total neutrino flux +observed by IceCube at TeV. On the other hand the base of the +outflow could be a plausible region where protons can be injected +and produce HE neutrinos while the associated GeV gamma-ray +counterpart can be efficiently reprocessed to lower energies. +The present model focused on the acceleration and transport of +protons in UFOs while the implications for primary and secondary +electrons as well as electron-positron pairs is left for future work. +UFOs are in fact expected to be perfect electron calorimeters given +the dominant synchrotron and inverse Compton timescales compared +to the escape timesciale, and the case of leptons would require a spe- +cific treatment. We computed the typical timescales for electrons and +MNRAS 000, 1–11 (2015) + +Particle acceleration and multimessenger radiation from UFOs +9 +we found that primary electrons cool very rapidly without reach- +ing TeV energies. Secondaries as well as electron-positron pairs are +also expected to cool mostly via synchrotron losses since beyond +∼ 1 TeV the interaction with the big blue bump takes place in the +Klein-Nishina regime. In this scenario, the electromagnetic cascade +is expected to be synchrotron-dominated for standard parametric as- +sumptions (𝜖B ≳ 10−2). Therefore, one could expect the leptonic +emission to produce some possible signatures in the hard X-ray to +soft gamma-ray energy band and in radio. +Heavy nuclei should also be implemented in a future work since +their transport, similar to electrons, requires the inclusion of calori- +metric conditions, continuous energy losses as well as fragmentation +and re-injection at lower masses. While our focus has been on the +proton population, it is worth mentioning that nuclei can be expected +to be co-accelerated in UFOs with a rigidity dependence in the max- +imum energy and possibly a higher efficiency in the injection of +energetic heavier elements (see e.g. Caprioli et al. 2017). In partic- +ular, while protons only partially suffer energy losses, heavy nuclei +of electric charge 𝑍 are expected to efficiently fragment on the AGN +photon field. Therefore one can expect that at early time the majority +of heavy nuclei would be reprocessed while at later time they could +start escaping efficiently with an energy as large as ∼ 𝑍 · EeV. +With the present work we propose UFOs as candidate sources of +UHECRs and efficient high-energy neutrino emitters possibly opaque +to gamma rays beyond a few tens GeV. Such properties make UFOs +a remarkable source class not only for the UHECRs detected by the +Pierre Auger Observatory but also for to the diffuse HE gamma- +ray and neutrino fluxes observed respectively by Fermi-LAT and +IceCube. +ACKNOWLEDGEMENTS +The research activity of EP and MA was supported by Villum Fonden +(project No. 18994). EP was also supported by the European Union’s +Horizon 2020 research and innovation program under the Marie +Sklodowska-Curie grant agreement No. 847523 ‘INTERACTIONS’. +FGS acknowledges financial support from the PRIN MIUR project +“ASTRI/CTA Data Challenge” (PI: P. Caraveo), contract 298/2017. +DATA AVAILABILITY +No data has been analyzed or produced in this work. +REFERENCES +Aab A., et al., 2017, J. Cosmology Astropart. Phys., 2017, 038 +Aab A., et al., 2018, ApJ, 853, L29 +Aartsen M. G., et al., 2020, Phys. Rev. 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The location of the wind shock is described by the follow- +ing relation: +𝑅sh = 23.0 pc +� 𝑡age +1Myr +�2/5 � +�𝐸 +1038erg/s +�3/10 +× +� 𝑛ISM +1cm−3 +�−3/10 � +𝑢1 +108km/s +�−1/2 +, +(A1) +where 𝑡age is the age of the system, �𝐸 the kinetic power ( �𝐸 = �𝑀𝑢2 +1/2), +𝑛ISM the ISM density and 𝑢1 the upstream wind speed. The forward +shock radius is found as follows: +𝑅fs = 76.0 pc +� 𝑡age +1Myr +�3/5 � +�𝐸 +1038erg/s +�1/5 � 𝑛ISM +1cm−3 +�−1/5 +. +(A2) +The radiation field (top panel) and the corresponding opacity (bottom +panel) of the wind bubble are shown in Figure A1. In particular, +the typical AGN radiation field feautures a combination of thermal +and non-thermal components. At around ∼ 10 meV one can identify +the infrared thermal field of the torus, while the second thermal +component peaking at ∼ 10 eV is the thermal radiation from the +accretion disk known as big blue bump. Finally, starting from the +keV range the non-thermal tail produced by electrons in the AGN +corona is present. Such a non-thermal tail can additionally feature a +somehow different shape due to the mechanism known as Compton +reflection. In this work we decided to ignore such an effect since it +has a strong dependence on the local parameters of the AGN while +having a negligible impact on the high energy particles and their +radiation. The gamma-ray opacity shown in the bottom panel clearly +highlights the prominent impact of the big blue bump and the torus +absorbing respectively in the energy ranges 10 GeV - TeV and beyond +10 TeV. +APPENDIX B: SOLUTION TO THE TRANSPORT +EQUATION +The transport equations (1) are solved separately in the upstream +and the downstream regions and joined at the wind shock location +as described in Morlino et al. (2021); Peretti et al. (2022). In what +follows we highlight the main analytical steps required to obtain the +formal solution and the key aspects of the iterative algorithm we +adopt to find the numerical solution. +10 +12 +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +E (GeV) +1038 +1040 +1042 +1044 +1046 +1048 +L (erg s +1) +LX=1042 erg s +1 +LX=1044 erg s +1 +LX=1046 erg s +1 +10 +1 +101 +103 +105 +107 +109 +E (GeV) +10 +3 +10 +2 +10 +1 +100 +101 +102 +103 +Benchmark +A +B +C +Figure A1. Top Panel: the AGN photon field for three different X-ray lu- +minosity. Dotted red, dashed green and dot-dashed blue represent the typical +spectral energy distribution of an AGN having an X-ray luminosity of 1042, +1044 and 1046 erg s−1 respectively. Bottom panel: gamma-gamma opacity of +the wind bubble where the different curves highlight the scenarios discussed +in Sec. § 3. In particular: the thick black line represents the benchmark sce- +nario while the thin lines describe the absorption in the Scenario A, B and +C. +B1 Upstream region +By integrating Eq. (1) from 𝑟 = 0 to 𝑟 < 𝑅fs one obtains: +𝑟2𝑢1 𝑓1(𝑟, 𝑝) = 𝑟2𝐷1(𝑟, 𝑝)𝜕𝑟 𝑓1(𝑟, 𝑝) − 𝐺1(𝑟, 𝑝) − 𝐻1(𝑟, 𝑝) , (B1) +where the subscript “1” refers to the upstream region, while the +functions 𝐺1 and 𝐻1 have the following expressions: +𝐺1(𝑟, 𝑝) = 1 +3 +∫ 𝑟 +0 +𝑑𝑟′ +� +− 𝜕ln[𝑝3 𝑓1(𝑟′, 𝑝)] +𝜕ln𝑝 +� +𝑓1(𝑟′, 𝑝) 𝜕𝑟′ [𝑟′2𝑢1] , +(B2) +𝐻1(𝑟, 𝑝) = +∫ 𝑟 +0 +𝑑𝑟′ 𝜆1(𝑟′, 𝑝) 𝑟′2 𝑓1(𝑟′, 𝑝) , +(B3) +where 𝜆1 = 𝜏−1 +pp + 𝜏−1 +p𝛾 + 𝜏−1 +BH is the CR energy loss accounting for +pion production in p𝛾 and pp interactions (see Kelner & Aharonian +2008) as well as Bethe-Heitler (BH) pair-production (see, e.g., Gao +et al. 2012). Defining the effective upstream velocity as: +𝑉eff,1(𝑟, 𝑝) = 𝑢1 +� +1 + 𝐺1(𝑟, 𝑝) + 𝐻1(𝑟, 𝑝) +𝑢1𝑟2 𝑓1(𝑟, 𝑝) +� +, +(B4) +MNRAS 000, 1–11 (2015) + +Particle acceleration and multimessenger radiation from UFOs +11 +the solution of Eq. (B1) is straightforwardly obtained: +𝑓1(𝑟, 𝑝) = 𝑓sh(𝑝)exp +� +− +∫ 𝑅sh +𝑟 +𝑑𝑟′𝑉eff,1(𝑟′, 𝑝) +𝐷1(𝑟′, 𝑝) +� +. +(B5) +B2 Downstream region +Similarly to the upstream case, also in the downstream Eq. (1) is +first approached through a spatial integral exploiting the boundary +condition. Integrating from from 𝑟 > 𝑅fs to 𝑟 = 𝑅fs one obtains: +𝑟2𝑢2(𝑟) 𝑓2(𝑟, 𝑝) = 𝑟2𝐷2(𝑟, 𝑝)𝜕𝑟 𝑓2(𝑟, 𝑝) + 𝑅2 +fs 𝑗esc(𝑝) + 𝐻2(𝑟, 𝑝) , +(B6) +where the subscript “2” refers to the downstream region, while the +function 𝐻2 reads: +𝐻2(𝑟, 𝑝) = +∫ 𝑅fs +𝑟 +𝑑𝑟′ 𝜆2(𝑟′, 𝑝) 𝑟′2 𝑓2(𝑟′, 𝑝) . +(B7) +As in the upstream region, it is convenient to define the effective +velocity in the downstream region as: +𝑉eff,2(𝑟, 𝑝) = 𝑢2(𝑟) +� +1 − +𝐻2(𝑟, 𝑝) +𝑟2𝑢2(𝑟) 𝑓2(𝑟, 𝑝) +� +. +(B8) +It is also useful to define the integral function 𝐼2 as: +𝐼2(𝑟, 𝑝) = +∫ 𝑟 +𝑅sh +𝑑𝑟′ +𝑟′2 𝑒−𝜑2(𝑟′,𝑝) , +(B9) +where 𝜑2(𝑟, 𝑝) has the following expression: +𝜑2(𝑟, 𝑝) = +∫ 𝑟 +𝑅sh +𝑑𝑟′𝑉eff,2(𝑟′, 𝑝) +𝐷2(𝑝) +. +(B10) +Integrating Eq. (B6) one obtains the expression for the escaping flux +and the downstream solution as: +𝑓2(𝑟, 𝑝) = 𝑓sh(𝑝) 𝑒𝜑2(𝑟,𝑝) +� +1 − +𝐼2(𝑟, 𝑝) +𝐼2(𝑅fs, 𝑝) +� +(B11) +𝑗esc(𝑝) = 𝑓sh(𝑝) +𝐷2(𝑝) +𝑅2 +fs𝐼2(𝑅fs, 𝑝) +. +(B12) +We finally notice that the escaping flux 𝑗esc can be rewritten as +𝑗esc(𝑝) = 𝜂loss +𝑢2 𝑓sh(𝑝) +1 − exp[−𝑅sh𝑢2(1 − 𝑅sh/𝑅fs)/𝐷2(𝑝)] +𝑅2 +sh +𝑅2 +fs +(B13) +where 𝜂loss is a parameter ≲ 1 accounting for energy losses. +B3 Solution at the shock +The shock solution is obtained by integrating Eq. (1) across an in- +finitely small layer embedding the wind shock. The result is the +following: +[𝐷2𝜕𝑟 𝑓2 −𝐷1𝜕𝑟 𝑓1]𝑟=𝑅sh − 𝑢1 − 𝑢2 +3 +𝑝𝜕𝑝 𝑓sh(𝑝) +𝑄0(𝑝) = 0 . (B14) +Substituting Eq. (B1) and Eq. (B6) in the first term on the left-hand +side, Eq. (B14) can be rewritten as: +𝑠𝑄0(𝑝) +𝑢1 += 𝑝𝜕𝑝 𝑓sh(𝑝) + 𝑠 𝑓sh(𝑝) + 𝑠Ψ𝑙(𝑝) 𝑓sh(𝑝) + 𝑠Ψ𝑒(𝑝) 𝑓sh(𝑝) , +(B15) +where the functions Ψ𝑘 (𝑘 = 𝑙, 𝑒) are defined as: +Ψ𝑙(𝑝) = 𝐺1(𝑅sh, 𝑝) + 𝐻1(𝑅sh, 𝑝) + 𝐻2(𝑅sh, 𝑝) +𝑢1𝑅2 +sh 𝑓sh(𝑝) +, +(B16) +Ψ𝑒(𝑝) = +[𝐷2𝐼−1 +2 (𝑅sh, 𝑝) − 𝑅2 +sh𝑢2] +𝑢1𝑅2 +sh +. +(B17) +Here the subscripts 𝑙 and 𝑒 stands for loss and escape respectively. +Finally, by recognizing a total derivative on the right hand side of +Eq. (B15), the solution at the shock can be obtained as: +𝑓sh(𝑝) = 𝑠𝜂𝑛1 +4𝜋𝑝3 +inj +� 𝑝inj +𝑝 +�𝑠 +𝑒−Γ𝑙 ( 𝑝)𝑒−Γ𝑒 ( 𝑝) , +(B18) +where: +Γ𝑙(𝑒) (𝑝) = 𝑠 +∫ 𝑝 +𝑝inj +𝑑𝑝′ +𝑝′ Ψ𝑙(𝑒) (𝑝′) . +(B19) +Notice that Eq. (B18) can be re-written in the compact form (see +Eq. (3)) 𝑓sh(𝑝) = 𝐶 𝑝−𝑠exp[−Γcut(𝑝)], where 𝐶 = 𝑠 𝜂 𝑛1 𝑝𝑠−3 +inj /(4𝜋) +and Γcut = Γ𝑙 + Γ𝑒. +B4 Iteration algorithm +The solution to the transport equation on the two sides of the shock, +𝑓1 (Eq. (B5)) and 𝑓2 (Eq. (B11)), and at the shock, 𝑓sh (Eq. (B18)), +do not have a simple analytic form since they depend on each other +through the functions𝑉eff,1,𝑉eff,2 and Ψ𝑙(𝑒), respectively. A solution +can be found found via an iterative algorithm. +We initialize the solutions for the set of functions ( 𝑓 (0) +sh , 𝑓 (0) +1 +, 𝑓 (0) +2 +) +by the solutions resulting from thefollowing no-loss conditions: +𝐺 (0) +1 += 𝐻(0) +1 += 𝐻(0) +2 += 0; 𝑉 (0) +eff,1(𝑟, 𝑝) = 𝑢1; 𝑉 (0) +eff,2(𝑟, 𝑝) = 𝑢2(𝑟). +This results in Ψ(0) +𝑙 += 0 while Ψ(0) +𝑒 +reduces to the following analytic +form: +Ψ(0) +𝑒 (𝑝) = +𝑢2/𝑢1 +exp +� 𝑅sh𝑢2 +𝐷2( 𝑝) +� +1 − 𝑅sh +𝑅fs +�� +− 1 +. +(B20) +We start from this initial approximation and find iterative solutions +by re-computing all functions with the set of solution of the previous +iteration, namely: +� +𝑓 (𝑖) +sh , 𝑓 (𝑖) +1 +, 𝑓 (𝑖) +2 +� +→ +� +𝐺 (𝑖+1) +1 +, 𝐻(𝑖+1) +1 +, 𝐻(𝑖+1) +2 +� +→ +� +𝑉 (𝑖+1) +eff,1 ,𝑉 (𝑖+1) +eff,2 , Ψ(𝑖+1) +𝑙 +, Ψ(𝑖+1) +𝑒 +� +→ 𝑓 (𝑖+1) +sh +→ +� +𝑓 (𝑖+1) +1 +, 𝑓 (𝑖+1) +2 +� +where (𝑖) and (𝑖 + 1) indicate the i-th and (i+1)-th iteration. This +algorithm is repeated until the phase space density at the n-th iteration +𝑓 (𝑛) is indistinguishable from the solution found at the iteration +(n-1)-th, 𝑓 (𝑛−1), namely when a convergence condition has been +obtained. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–11 (2015) + diff --git a/W9FRT4oBgHgl3EQf9zjY/content/tmp_files/load_file.txt b/W9FRT4oBgHgl3EQf9zjY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..08c40161d8e68d30cd5d2f4e040d23caa0beb9b2 --- /dev/null +++ b/W9FRT4oBgHgl3EQf9zjY/content/tmp_files/load_file.txt @@ -0,0 +1,994 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf,len=993 +page_content='MNRAS 000, 1–11 (2015) Preprint 1 February 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='0 Diffusive shock acceleration at EeV and associated multimessenger flux from ultra-fast outflows driven by Active Galactic Nuclei Enrico Peretti1★,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Alessandra Lamastra2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Francesco Gabriele Saturni2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Markus Ahlers1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Pasquale Blasi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Giovanni Morlino6 and Pierre Cristofari7 1 Niels Bohr International Academy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Niels Bohr Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='University of Copenhagen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Blegdamsvej 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' DK-2100 Copenhagen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Denmark 2 INAF – Osservatorio Astronomico di Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Via Frascati 33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' I-00078 Monte Porzio Catone (RM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Italy 3 ASI – Space Science Data Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Via del Politecnico snc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' I-00133 Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Italy 4 Gran Sasso Science Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Via F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Crispi 7, 67100, L’Aquila, Italy 5 INFN/Laboratori Nazionali del Gran Sasso, Via G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Acitelli 22, Assergi (AQ), Italy 6 INAF, Osservatorio Astrofisico di Arcetri, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='go E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Fermi 5, I-50125 Firenze, Italy 7 Observatoire de Paris, PSL Research University, LUTH, 5 Place J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Janssen, 92195 Meudon, France Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Active galactic nuclei (AGNi) can launch and sustain powerful winds featuring mildly relativistic velocity and wide opening angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Such winds, known as ultra-fast outflows (UFOs), can develop a bubble structure characterized by a forward shock expanding in the host galaxy and a wind termination shock separating the fast cool wind from the hot shocked wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In this work we explore whether diffusive shock acceleration can take place efficiently at the wind termination shock of UFOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We calculate the spectrum of accelerated particles and find that protons can be energized up to the EeV range promoting UFOs to promising candidates for accelerating ultra-high energy cosmic rays (UHECRs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We also compute the associated gamma-ray and neutrino fluxes and compare them with available data in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We observe that high-energy (HE) neutrinos are efficiently produced up to hundreds of PeV while the associated gamma rays are efficiently absorbed beyond a few tens of GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' By assuming a typical source density of non-jetted AGNi we expect that UFO could play a dominant role as diffuse sources of UHECRs and HE neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We finally apply our model to the recently observed NGC1068 and we find out that an obscured UFO could provide a sizeable contribution to the observed gamma-ray flux while only contributing up to ∼ 10% to the associated neutrino flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Key words: cosmic rays – Active Galactic Nuclei – particle acceleration – gamma rays – neutrinos 1 INTRODUCTION Fast wide angle winds are one of the most intriguing feedback mech- anisms of Active Galactic Nuclei (AGNi) (Silk & Rees 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Their impact on the host galaxies has long been considered to affect the dynamical evolution of the interstellar medium (ISM) and act as a regulator of star formation (Crenshaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The discovery of blue-shifted Fe K absorption lines in X-ray spectra of AGNi (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Chartas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2002) brought compelling evidence of mildly rel- ativistic velocities typically ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='1 𝑐 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='3 𝑐 (where 𝑐 is the speed of light) in such winds, which thereafter have been of- ten referred to as ultra-fast outflows (UFOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Recently UFOs have been systematically detected in both radio-quiet and radio-loud AGNi (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Markowitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Braito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Cappi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Tombesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2010a,b, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The number of observations keeps increasing with time, despite of the observational challenges, also thanks to high resolution grating spectra in the soft X-ray (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Pounds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2003) and the discovery of ultraviolet (UV) lines in addition to those already known in the X-ray (Kriss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' ★ E-mail: peretti@nbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='dk Mehdipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The search for a launching mechanism of the UFOs has not found a definitive answer yet, although there is a gen- eral agreement to ascribe this phenomenon to the accretion activity of the super massive black hole (SMBH) (see Laha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2021, for an updated review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The dynamics of the wind and the associated feedback on the host galaxy depend on whether the outflow conserves energy or momentum (King & Pounds 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In particular, if the wind plasma does not cool efficiently when it shocks with the surrounding ISM, the system is energy-conserving and the momentum flux is boosted while the wind sweeps up the ISM matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In the opposite scenario, namely if the wind radiates most of its thermal energy, it evolves conserving its momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In spite of the strong radiation fields that could strongly affect the cooling of electrons, Faucher-Giguère & Quataert (2012) showed that two-temperature plasma effects are likely to slow down radiative losses for protons thereby favoring an energy-conserving dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' A recent Fermi-LAT analysis (Ajello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2021) showed that UFOs are a new class of gamma-ray emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In this analysis the average gamma-ray emission from a sample of 11 nearby (z<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='1) radio-quiet AGNi with an UFO is derived by adopting a stacking © 2015 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='13689v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='HE] 31 Jan 2023 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Peretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The average best-fit gamma-ray spectral slope is measured to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='3, and the gamma-ray luminosity is found to scale with the AGN bolometric luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' AGN-driven outflows, similar to stellar winds (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Weaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Koo & McKee 1992a,b), are expected to develop a struc- ture characterized by an inner wind termination shock (hereafter wind shock), a contact discontinuity and an outer forward shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The forward shock has been proposed as a plausible site for particle acceleration (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Lamastra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Wang & Loeb 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Lamastra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Ajello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2021) where ideal conditions for efficient production of gamma rays and high-energy (HE) neutri- nos are expected (see also McDaniel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2023, for a recent study on molecular outflows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Wang & Loeb (2017) highlighted also the possibility that in somewhat extreme conditions a fast AGN-driven wind could have the energy budget to accelerate CRs up to the ultra- high-energy (UHE) range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The associated cumulative contribution of AGN-driven winds to the diffuse gamma-ray and neutrino flux has been also explored (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Wang & Loeb 2016a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Lamastra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In addition, the amplitude of the recently observed spectrum of the diffuse neutrino flux (Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2020), in light of the constraints imposed by the diffuse gamma-ray flux observed by Fermi-LAT (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2015), suggests that there could be a class of sources at least partially opaque to gamma rays (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Murase et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Indeed, a search for time-integrated point-like neutrino sources (Aartsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2020) highlighted an excess in the direction of the Seyfert galaxy NGC1068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The most intriguing aspect of the emission of such galaxy is the lack of gamma rays in the TeV band (Acciari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2019), where the neutrino flux is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The natural implication of an highly opaque cosmic particle accelerator triggered several studies on the multimessenger implications of HE particles populating the innermost region of AGN such as disks and accretion flows (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Gutiérrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2021) and AGN corona (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Murase et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Inoue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Kheirandish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Inoue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Eichmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Murase 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Interestingly, we recently witnessed a growth in the statistical significance of the signal from NGC1068 (Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Even though there is an increasing evidence for HE particles pop- ulating the innermost regions of active galaxies, our understanding of the acceleration mechanisms at play in such environments is still incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The mechanisms capable of energizing HE particles in the vicinity of SMBH is indeed a partially unexplored field we aim to assess in this work together with its multimessenger consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Therefore, we develop a model for particle acceleration explor- ing the diffusive shock acceleration (DSA) mechanism at the wind shock of UFOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' At this shock, unique conditions for acceleration of protons at energies as high as ∼ 1018 eV can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In addition, the medium property could make such sources bright in HE neutri- nos while being partially opaque to gamma rays beyond 10 − 102 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We discuss the role of UFOs as UHE cosmic ray (UHECR) sources in light of the spectral behavior of the particle flux escaping the AGN wind bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In particular, we show that at the highest energies spectral features harder than 𝐸−2 could appear in the spec- trum of escaping particles due to the interplay of diffusion-advection and energy losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' This result is of great interest in light of the phe- nomenological results aiming at modeling the spectrum and mass composition observed by the Pierre Auger Observatory which sug- gest UHECRs to be characterized by hard spectra (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=', Unger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2015, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Moreover, if UFOs were com- mon in galaxies this would produce important implications for their diffuse multimessenger emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We provide here an order of mag- nitude estimate of the potential role of UFOs for a diffuse flux of HE Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Structure of the wind bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The SMBH (BH) responsible for the wind launching is located on the left of the sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The blue (red) arrow corresponds to the cool (shocked) wind of the upstream (downstream) region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The wind shock (𝑅sh) separates these two regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The SAM is located between the contact discontinuity (𝑅cd) and the forward shock (𝑅fs) which bounds the system (credit: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Peretti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' neutrinos and CR at the ankle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Finally we explore whether an UFO could play a role in the neutrino emission observed in NGC1068 discussing possible realizations of such a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The manuscript is organized as follows: in § 2 we describe the model for the wind bubble and the associated particle acceleration and transport formalism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' in § 3 we discuss our results in terms of spectra of accelerated and escaping particles, we perform a parameter space scan focusing on the maximum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In § 4 we discuss the multimessenger implications in terms of HE photons and neutrinos and in § 5, we specialize our model by applying it to the case of NGC1068 and discuss possible model improvements and alternative scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We draw our conclusions in § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2 MODEL FOR PARTICLE ACCELERATION AND MULTIMESSENGER EMISSION IN UFO The fast wind launched and sustained by an AGN expands with large opening angle and mildly relativistic velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' At such speed the outflow is supersonic, therefore it drives a forward shock expanding in the external medium, while a contact discontinuity separates the shocked wind (SW) material from the shocked ambient medium (SAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The impact of the wind on the external medium creates a shock inside the wind material, the wind shock, which is trailing behind the contact discontinuity and it is oriented towards the central engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The outflow, characterized by these three discontinuities, features a bubble structure as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In a uniform medium of density 𝑛0 the wind expands with constant velocity up to the radius at which the swept-up mass roughly balances the whole mass of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' After the swept-up mass becomes dy- namically relevant, the outflow starts decelerating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' During the decel- eration phase the forward shock 𝑅fs and the wind shock 𝑅sh evolve self-similarly according to different scaling laws: 𝑅fs ∼ 𝑡3/5 and 𝑅sh ∼ 𝑡2/5 (see also Weaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Koo & McKee 1992a,b, for detailed discussions and Appendix A for additional details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Since the wind shock decelerates faster than the forward shock, the hot bubble, namely the spherical shell between the wind shock and the contact discontinuity, grows with time while remaining approximately adia- MNRAS 000, 1–11 (2015) upstream downstream BH accretion R R R disk sh cd fsParticle acceleration and multimessenger radiation from UFOs 3 batic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In this context, the wind bubble evolution can be considered as energy-conserving (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=', Faucher-Giguère & Quataert 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' On the other hand, losses in the SAM can be efficient (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Nims et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2015), so that the whole swept-up mass is eventually compressed into a relatively thin layer between the contact discontinuity, 𝑅cd, and 𝑅fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' While the wind bubble slows down its expansion, the relative veloc- ity between the plasma and the wind shock remains high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We refer to the innermost region of free expanding wind and to the shocked wind respectively as upstream and downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Different from the wind shock, the Mach number of the forward shock strongly depends on the temperature and conditions of the surrounding ISM medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Therefore, it is not guaranteed that the forward shock is strong for a sufficient amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We assume a spherically symmetric geometry for the outflow and, since the wind launching region has a negligible size compared to the whole bubble, we also assume a constant upstream velocity 𝑢1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The shocked wind is adiabatic, therefore the velocity profile reads: 𝑢2(𝑟) = 𝑢2(𝑅sh/𝑟)2, where 𝑢2 = 𝑢1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In agreement with observa- tions of UFOs, we limit our investigation to a plasma velocity lower than ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='3𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We thus neglect relativistic effects due to the mildly rel- ativistic motion of the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The target density in the upstream re- gion scales as 𝑛1(𝑟) = �𝑀/[4𝜋𝑟2𝑢1𝑚 𝑝], while in the downstream re- gion it is constant and equal to 𝑛2 = 4𝑛1(𝑅sh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The target density be- tween 𝑅cd and 𝑅fs depends on the amount of matter accumulated dur- ing the outflow evolution, 𝑛SAM = 𝑛0𝑅3 fs/(𝑅3 fs − 𝑅3 cd), where we as- sume 𝑅cd ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='9 𝑅fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We postulate a turbulent nature for the magnetic field and we estimate its amplitude in the upstream region under the assumption that a fraction 𝜖𝐵 ≲ 10% of the ram pressure is converted into magnetic field energy density, namely 𝑈𝐵(𝑟) = 𝜖𝐵𝑚 𝑝𝑛1(𝑟)𝑢2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' At the wind shock we assume that the magnetic field gets com- pressed by a factor √ 11, typical of strong shocks, and remains con- stant throughout the whole downstream region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We adopt the quasi- linear theory of diffusion, 𝐷(𝑟, 𝑝) = 𝑣(𝑝)𝑟2−𝛿 𝐿 (𝑟, 𝑝)𝑙 𝛿−1 𝑐 /3, where 𝑣 is the particle velocity, 𝑟𝐿 is the Larmor radius, 𝛿 is the slope of the turbulence power spectrum and 𝑙𝑐 is the coherence length of the magnetic field that we assume to be comparable in size with the launching radius of the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In addition, we account for the small angle scattering regime of diffusion, namely 𝐷 ∝ 𝑟2 𝐿, which takes place when 𝑟𝐿 > 𝑙𝑐 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Subedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Dundovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The average lifetime of AGNi is inferred to be ≲107 yr (Yu & Tremaine 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' During this time, the AGN is expected to show multiple episodes of activity with duty cycles of ≲105 yr duration (Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' This suggests that UFOs could have a lifetime ranging from hundreds to several thousands of years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In this context, the dynamical evolution of the system becomes soon slower than all relevant timescales involving HE particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Hence the process of particle acceleration and transport can be treated as stationary (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' § 3 where the typical timescales for HE particles in a prototype UFO are discussed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We assume a spherically symmetric transport where particles are injected via DSA at the wind shock whereas, once they reach the forward shock location, they freely escape the wind bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The transport equation reads: 𝑟2𝑢𝜕𝑟 𝑓 = 𝜕𝑟 [𝑟2𝐷𝜕𝑟 𝑓 ] + 𝑝 3 𝜕𝑝 𝑓 𝜕𝑟 [𝑟2𝑢] + 𝑟2[𝑄 − 𝜆 𝑓 ] , (1) where 𝑢 = 𝑢(𝑟) is the wind velocity profile, 𝐷 = 𝐷(𝑟, 𝑝) is the diffu- sion coefficient, 𝑄 = 𝑄(𝑟, 𝑝) is the injection term and 𝜆 = 𝜆(𝑟, 𝑝) is the loss rate accounting for pp and p𝛾 interactions (see Appendix B for additional details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' As boundary conditions, consistently with the spherical symmetry, we assume a null net flux at the center of the system, 𝑢 𝑓 − 𝐷𝜕𝑟 𝑓 |𝑟=0 = 0, while, as mentioned earlier, we regard the forward shock as a free escape boundary, 𝑓 (𝑅fs, 𝑝) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Parameters of the benchmark UFO and of the three alternative scenario considered for the multimessenger emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Parameter benchmark A B C 𝑢1/𝑐 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='2 �𝑀 [M⊙ yr−1] 10−1 𝜉CR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='12 𝜖B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='05 𝑙𝑐 [pc] 10−2 𝛿 3/2 𝐿𝑋 [erg s−1] 1044 𝑛ISM [cm−3] 104 2 · 103 5 · 102 2 · 102 𝑡age [yr] 103 3 · 103 104 2 · 104 injection term reads: 𝑄(𝑟, 𝑝) = 𝑄0(𝑝)𝛿[𝑟 − 𝑅sh] = 𝜂CR𝑢1𝑛1 4𝜋𝑝2 𝛿[𝑝 − 𝑝inj]𝛿[𝑟 − 𝑅sh], (2) where 𝑝inj = 1 GeV/c is the injection momentum of particles (picked up from the plasma) that enter the DSA process and 𝜂CR is the efficiency factor, normalized such that the CR pressure at the shock is a small fraction (≲ 10%) of the plasma ram pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (1) following the same procedure developed in Mor- lino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (2021) and Peretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (2022) with the modification that, in the present work, energy losses in the downstream region are also accounted for due to their possible dynamical relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In fact the relative small distance from an active SMBH makes the environment potentially hostile for HE particles where energy losses could affect the acceleration and limit the escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The details of the calculation are reported in Appendix B while the general form of the solution at the wind shock reads: 𝑓sh(𝑝) = 𝐶𝑝−𝑠e−Γcut( 𝑝), (3) where 𝐶 is a constant (see Appendix B), 𝑠 is the spectral index (𝑠 = 4 in strong shocks) and Γcut is a HE cut-off function (Γcut = Γ𝑙 + Γ𝑒 as detailed in Appendix B) which increases with momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' As the background photon field, we use the spectral energy dis- tribution (SED) model shown in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' A1 provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Such a SED is characterized by the big blue bump and an X-ray power-law component as described in Marconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Such field is assumed to decrease with the second power of distance from the central engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Consistently with the AGN SED we also account for the far infrared (FIR) component produced by a dusty torus (Mullaney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Gamma rays and neutrinos from pp and p𝛾 interactions are com- puted following Kelner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (2006) and Kelner & Aharonian (2008), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The energy losses due to Bethe-Heitler pair-production are taken into account as energy loss mechanism taking place at a rate 𝑡−1 BH (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Gamma-gamma absorption on the AGN SED including the torus is also accounted for by adopting the cross section appropriate for the case of an isotropic photon field (see Aharonian 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Finally, the associated flux-density of HE protons escaping the system is computed self-consistently from the solution to the transport equation as 𝑗esc(𝑝) = −𝐷2𝜕𝑟 𝑓 |𝑟=𝑅 𝑓 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The calculations illustrated here have been carried out in the con- text of the thin shell approximation, in which the SW is perfectly separated from the SAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' This implies that the cold gas acting as a target for pp interactions (see below) is all located close to the 𝑅fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' It is worth pointing out that the spatial distribution of the gas in the SW might be affected by instabilities and mixing that may lead to a more MNRAS 000, 1–11 (2015) 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Peretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 106 107 108 109 Energy [GeV] 10 1 100 101 102 103 104 105 Time [yr] age acc adv diff p BH Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Typical timescales regulating the transport of HE particles com- pared with the age of the system (thick black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The blue dashed line represents the acceleration timescale, while energy losses via photomeson and Bethe-Heitler pair-production are represented respectively by red and magenta dot-dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Advective and diffusive escape are represented by orange and green dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' pervasive, though clumpy structure of the gas, which in turn might affect the spatial and spectral properties of the secondary emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' These effects will be investigated in a future dedicated work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 3 RESULTS In order to present our model and discuss its physical implications we assume a set of typical parameters, hereafter referred to as our benchmark scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The parameters defining our benchmark sce- nario, summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 1, have been chosen according to the following criteria: we assume 𝑢1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='2 𝑐 as the average value for the terminal wind speed of UFOs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' �𝑀 = 10−1 M⊙ yr−1 has been chosen such that the total kinetic power �𝐸 = �𝑀𝑢2 1/2 matches about ∼ 3% of the total AGN bolometric luminosity (Fiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2017) charac- terized by an X-ray luminosity 𝐿𝑋 = 1044 erg s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑙𝑐 = 10−2 pc is compatible with the launching radius of the wind as predicted by accretion disk wind models (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' the age of the system 𝑡age = 103 yr has been chosen in order to assure stationary conditions, which are not guaranteed for much younger systems (the resulting shock radii at such an age are 𝑅sh ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='8 pc and 𝑅fs ≈ 3 pc);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝜖𝐵 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='05 guarantees a minor dynamical impact of the turbulent magnetic field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝛿 = 3/2 is motivated by an MHD-like (Kraichnan) turbulence cascade;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' finally, we assume 𝑛ISM = 104 cm−3 as a typical value for the external ambient medium that can be found in the core of luminous infrared galaxies (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Downes & Solomon 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Faucher-Giguère & Quataert 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We assume such a density also as the representative value for the medium of broad line regions of active galaxies which is characterized by dense clouds (of density up to 𝑛𝑐 ≲ 1010 cm−3) embedded in a hot and pressurized gas of lower density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Even though we study particle acceleration and transport by solv- ing the stationary transport equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (1), a more direct under- standing of the physics property of the solution can be obtained by analyzing the typical timescales of the different competing pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2 illustrates the typical timescales for HE particles as computed at 𝑅sh for the benchmark scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Here, the age of the system (𝜏age, thick black line) is compared with the following timescales: acceleration (𝜏acc ≈ 𝑠𝐷1/𝑢2 1, blue dashed line), diffusion (𝜏diff = (𝑅esc − 𝑅sh)2/𝐷2, green dotted line the diffusion), advection (𝜏adv = (𝑅esc − 𝑅sh)/< 𝑢2 >, orange dotted), the 𝑝𝛾 photomeson (𝜏𝑝𝛾, red dot-dashed line) and Bethe-Heitler pair-production (𝜏BH, magenta dot-dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Inelastic pp collisions were also taken into ac- count, however at the wind shock the target density is of the order of 20 cm−3, so that the associated timescale exceed 105 yr, the upper limit of the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Therefore, pp interactions are dynamically irrele- vant for the acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' On the other hand, this does not exclude them as relevant loss mechanism in the SAM where the density is orders of magnitude higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' From the interplay between the differ- ent timescales it is possible to to observe: 1) 𝜏acc ≪ 𝜏age and the minimum between losses and escape is also smaller than the age supporting the stationary assumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2) 𝜏acc as well as 𝜏diff feature a break between 102 PeV and 1 EeV due to the transition in the diffu- sion coefficient from the QLT behavior (𝑟L < 𝑙c) to the small angle scattering regime (𝑟L > 𝑙c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 3) energy losses via 𝑝𝛾 photomeson pro- duction play a dominant role at the highest energies and are expected to set the maximum energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 4) 𝜏𝑝𝛾 and 𝜏BH increase with the second power of the distance moving outward from 𝑅sh to 𝑅fs, therefore the transport in the downstream region is characterized by a close competition between energy losses and escape;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 5) the Bethe-Heitler pair production does not play a dominant role since at low energy the transport is advection-dominated while at the highest energies is regulated by the photomeson production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The spatial transport of particles is regulated by advection at low energy and by diffusion at the highest energies while energy losses, both adiabatic and inelastic collisions, can affect the normalization and/or introduce spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 3 shows the spatial distribution for three different CR energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The upstream region (𝑅/𝑅sh < 1) is characterized by the competition between dif- fusion, which tries to spatially homogenized particles, and advection which prevents low energy particles to diffuse upwind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In particu- lar, one can see that the higher the energy the stronger the impact of diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The red dotted line illustrates the spatial distribution at low energies while the blue and black lines show results for an intermediate energy and near the exponential cut off, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In the downstream region (𝑅/𝑅sh > 1) one can see that, different from the upstream one, advection-dominated transport leads to a spatially homogenized solution whereas diffusion-dominated transport leads to a number suppression while approaching the free escape bound- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' This behavior is a natural result of the spherical geometry of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 3 illustrates the spectrum of accelerated particles at the shock (thick green line), at different radii in the down- stream region (dotted curves where the red one is the closest to 𝑅sh while the blue one approaches 𝑅fs) and the associated spectrum of the escaping flux (purple dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The spectrum of accelerated particles, as suggested by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (3) and as naturally predicted by DSA in a finite system, is a power-law of index 𝑠 with maximum energy 𝐸max ≃ 1 EeV and does not show any relevant additional spectral feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' On the other hand, the particle spectrum gradually steepens in the downstream region moving from the wind shock to the forward shock as a result of escape and 𝑝𝛾 energy losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In the downstream region, energy losses play a crucial role in shaping the spectrum of particles escaping the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In particular, the photomeson interac- tions on the big blue bump occur faster than escape at ∼ 1017 eV as one can also deduce from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' This results in a dip in the spectrum at such energy, whereas at higher energies the escape is more efficient so that the spectrum hardens at the highest energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' A comment on the maximum energy is in order: the exponential function regulating the cutoff, Γcut, accounts for the geometry of the MNRAS 000, 1–11 (2015) Particle acceleration and multimessenger radiation from UFOs 5 100 2 × 100 3 × 100 4 × 100 R / Rsh 10 4 10 3 10 2 10 1 100 F (r, p) [arbitrary units] F (r, 10 PeV) F (r, 102 PeV) F (r, 103 PeV) 10 3 10 2 10 1 100 101 E [1018 eV] 106 107 108 ps F(r, p) [arbitrary units] Fsh Jesc F2 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Top panel: Spatial distribution of the CR phase space density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Low energy particles behave in the system as illustrated by the red dotted line, high energy particle behavior is represented by the blue dot-dashed curve while the black curve shows the behavior of particles at the maximum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Bottom panel: Spectrum of particles at the shock (thick green line) compared to the spectral shape of the escaping flux (dashed magenta line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The dotted curves represent the particle spectra in the downstream region from the wind shock (red) to the escape radius (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' system and loss mechanisms, so that it cannot be simplified as a ratio 𝐸/𝐸max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Therefore, here we define 𝐸max as the energy where 𝑝𝑠 𝑓sh is suppressed by one 𝑒-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In what follows we describe in detail the impact of different realizations of the system to the maximum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='1 Impact of parameters on the maximum energy A qualitative estimate of the maximum energy set by the geometry of the system can be obtained by comparing the upstream diffusion length, 𝐷1/𝑢1, with the size of such region, 𝑅sh (see also Morlino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Peretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2022, for additional discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Since at the highest energies 𝑟L is already larger than 𝑙𝑐 one can write the maximum energy as follows: 𝐸max = 𝑞𝐵 √︂ 6 𝑐 � 𝜖𝐵 �𝑀𝑙𝑐 𝑅sh �1/2 𝑢1 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='4 EeV � 𝜖𝐵 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='05 �𝑀 10−1M⊙yr−1 𝑙𝑐 10−2 pc 1 pc 𝑅sh �1/2 𝑢1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='2 𝑐 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (4) As one can see from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (4), the maximum energy for DSA at the wind shock of UFOs turns out to be of the order of EeV for standard values of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2 highlights the impact of different parametric assumptions on the maximum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In particular, we see that, according to Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Impact on the maximum energy of a parameter variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' All parameters are set to the benchmark UFO values shown in Table 1 except for those indicated in the first two columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The last row shows the result for benchmark values for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Parameter(s) Variation(s) 𝐸max [EeV] 𝑢1 [𝑐] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='03 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='1 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='03 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='31 / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='86 �𝑀 [𝑀⊙ yr−1] 10−2 / 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='29 / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='82 𝜖B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='01 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='53 / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='41 𝑙𝑐 [pc] 3 · 10−3 / 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='81 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='24 𝑡age [yr] 102 / 104 / 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='58 / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='12 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='88 𝛿 5/3 (Kolmogorov) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='02 𝑛ISM [cm−3] 103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='11 𝑈rad none / double 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='04 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='77 ( �𝑀, 𝑢1) pessimistic / optimistic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='01 / 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='53 no variations (benchmark) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='06 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (4), 𝐸max scales roughly linearly with 𝑢1 and with the square root of �𝑀 and 𝜖𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The impact of 𝑙𝑐 on 𝐸max can be understood as follows: when 𝑙𝑐 ≫ 10−2 pc, the diffusion coefficient is much larger than the benchmark scenario so that the diffusion length reaches the size of the system at lower energies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' when 𝑙𝑐 ≪ 10−2 pc the energy at which the diffusion coefficient changes regime (from the stan- dard quasi-linear theory ∼ 𝐸2−𝛿 to the small pitch-angle scattering regime ∼ 𝐸2) shifts to lower energies thereby resulting in a larger value of 𝐷 at the highest energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Therefore, since at the highest energies, diffusion dominates, a local maximum in 𝐸max appears for 𝑙𝑐 ≃ 10−2 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The age of the system does not have a strong impact on 𝐸max which is affected by less than a factor 2 for the wide range of alternatives considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Similarly, different assumptions on the slope of the turbulence cascade (Kolmogorov-like) and the external medium density have a negligible impact on 𝐸max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In fact, while the former is irrelevant because at the highest energies 𝑟L > 𝑙𝑐 and diffusion has changed regime, the latter impacts mostly the dynamics of the bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Interestingly, as also highlighted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2, different assumptions in the photon field highlight a trend which suggests that the 𝑝𝛾 interactions on the infrared field of the torus regulate the max- imum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In particular, 𝐸max increases by a factor 2 when the photon field is removed, while it decreases when a stronger photon field is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' This suggests that the infrared field of the torus could play a crucial role in regulating the maximum energy achiev- able in UFOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We finally explore the combined effect of maximum (minimum) values of 𝑢1 and �𝑀 corresponding to a plausible opti- mistic (pessimistic) scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In this context one can see that UFOs can be responsible for particle acceleration with 𝐸max ranging from 10 PeV up to 5 EeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In particular, the objects in the high luminosity end of a hypothetical luminosity function of UFOs are candidate acceleration sites of UHECRs where protons could reach a few EeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Heavier nuclei could be accelerated to higher total energies provided they survive photodisintegration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The latter possibility depends on the photon background present at the acceleration site and on the relative distance between 𝑅sh and 𝑅fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 4 GAMMA-RAYS AND HE NEUTRINOS FROM UFOS AND CONSTRAINTS TO THEIR LOCAL DENSITY The gas swept-up from the dense environment of the SMBH as well as the strong radiation field of the AGN can make hadronic interac- tions dynamically relevant in UFOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Since interactions are copiously MNRAS 000, 1–11 (2015) 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Peretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 100 102 104 106 108 E (GeV) 10 14 10 13 E2dN/dE (erg cm 2 s 1) UFO SED SAM (pp) SAM (pp) SW+SAM (p ) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Gamma-ray (thick blue line) and HE neutrino flux produced in the benchmark UFO (𝑅sh ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='8 pc) via pp (dotted red line) and p𝛾 (orange dot-dashed line) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The acronym SAM (SW) refers to the shocked ambient medium (shocked wind).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The thin blue-to-cyan lines represents the gamma-ray flux computed respectively for scenario A, B and C (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 1) in order to illustrate the dependence of the gamma-ray absorption on the bubble expansion (where 𝑅sh ≃ 2, 5 and 8 pc respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The UFO is assumed to be located at z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='013 in order to be directly compared with the best-fit UFO SED provided in Ajello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (2021), where the gray band represent the 1 𝜎 uncertainty band of such a best-fit UFO SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' taking place in the UFO wind bubble, a high level of gamma-ray and HE neutrino emission can be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Figure 4 illustrates the gamma-ray (thick blue line) and the single-flavor neutrino flux (dot- ted red for pp and dot-dashed orange for p𝛾) flux expected from the benchmark UFO placed at a redshift z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In particular, the gamma-ray flux is compared with the typical UFO spectral energy distribution (SED) as found in Ajello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Despite the fact that the benchmark scenario represents an average UFO in terms of power and maximum energy, the gamma-ray flux of the benchmark UFO (𝑅sh ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='8 pc and 𝑅fs ≃ 3 pc) cannot in fact be representative of the whole class due to the strong radial-dependence of the 𝛾𝛾 absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Therefore, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 4 we also compare it with the expected gamma-ray fluxes as predicted from scenario A (𝑅sh ≃ 2 pc and 𝑅fs ≃ 8 pc), B (𝑅sh ≃ 5 pc and 𝑅fs ≃ 22 pc) and C (𝑅sh ≃ 8 pc and 𝑅fs ≃ 40 pc) as described in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Scenarios A, B and C do not differ from the benchmark scenario in terms of total power but illustrate alternative realizations of it, having a larger size resulting from a longer evolution in a less dense environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' One can observe that these scenarios enhance the gamma-ray emission above ∼ 10 GeV due to a weaker gamma-gamma absorption and allow a better agreement with the UFO sample observed by Ajello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Regardless of the age of the system, gamma-rays of energy greater than a few TeV are completely absorbed by the infrared radiation field of the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Therefore, pp neutrinos in the 10 TeV - 1 PeV band as well as p𝛾 neutrinos in the energy band 102 TeV - 102 PeV would be produced without their gamma-ray counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' UFOs are thus expected to be bright neutrino sources featuring spectra as hard as ∼ 𝐸−2, while being highly opaque to TeV (and possibly 10 − 102 GeV) gamma-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' UFOs could be common in nearby luminous infrared galaxies (LIRGs) such as active starburst galaxies and Seyfert galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' How- ever, the abundance and distribution of these objects throughout the Universe as well as their luminosity function are poorly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Therefore, in what follows we estimate the order of magnitude of their diffuse multimessenger emission in terms of EeV cosmic rays and associated HE neutrinos and gamma rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Since the horizon for the Bethe-Heitler pair-production suffered by UHECRs on the cosmic microwave background (CMB) is placed beyond 𝑧 > 2 we neglect such a loss mechanism in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We first assume as a prototype UFO the EeV-atron described by the benchmark scenario presented in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' As discussed in Appendix B, the escaping flux is regulated by the interplay between diffusion, advection and energy losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' However, despite its complex analytic expression, assuming an ∼ 𝐸−2 spectrum the power contained by the escaping particles can be approximated as follows: 𝐿CR = ∫ 𝑑𝑝 4𝜋𝑝2[𝑝𝑐]4𝜋𝑅2 esc 𝑗esc ≃ 3 4 𝜉CR 𝜂loss �𝑀 𝑢2 1 ≃ 2 · 1043 𝜂loss 𝜉CR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='05 �𝑀 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='1 M⊙/yr � 𝑢1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='2 𝑐 �2 erg s , (5) where 𝑗esc is the escaping flux of protons as defined in Appendix B, 𝜂loss ≤ 1 is an age-dependent parameter which accounts for the relative reduction in the escaping flux due to energy losses, while the other parameters are normalized to the values shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The CR luminosity is related to the CR spectral injection rate QCR (units of GeV−1s−1) as 𝐿CR = ∫ d𝐸 𝐸QCR(𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' For simplicity, we assume in the following that the CR emission follows 𝐸−2 from GeV to EeV, such that 𝐸2𝑝QCR ≃ 𝜒𝐿CR with 𝜒 ≡ 1/ln(EeV/GeV) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In general, the locally observed CR spectrum 𝜙CR (units of GeV−1cm−2s−1sr−1) is related to the spectral emission rate of ex- tragalactic sources via a set of transport equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' For CR protons in the EeV energy range we can assume that the transport is domi- nated by continuous energy loss due to the expansion of the Universe while we neglect the effect of intergalactic magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Following the notation of Ahlers & Halzen (2018), we can estimate the local contribution of UFO EeV-atrons as: [𝐸2 𝑝𝜙CR]EeV ≃ 𝜉𝑧 4𝜋 𝑐 𝐻0 𝜌0[𝐸2 𝑝QCR]EeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (6) The factor 𝜉𝑧 is of order unity accounting for the integral in redshift of the source distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In particular, 𝜉𝑧 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='5(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='6) for a flat (star- formation rate) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The parameter 𝜌0 represents the local comoving density of sources for which we assume 𝜌0 = 10−5 Mpc−3 as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Such a value is often quoted as a typical density of AGNi with X-ray luminosity of the order of 𝐿𝑋 ≃ 1044 erg s−1 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Fiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Ueda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Murase & Waxman 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' It is worth mentioning that a source density ∼ 10−4 − 10−5 Mpc−3 matches also the number density inferred for powerful starbursts such as luminous and ultra-luminous infrared galaxies (see Gruppioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Peretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Condorelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2022) which are currently also considered to be plausible hosts for UHECR accelerators (Aab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Expressing the contribution of EeV CR protons to the spectral emission as [𝐸2𝑝QCR]EeV = 𝜒𝐿CR with 𝜒 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='05 we arrive at: [𝐸2 𝑝𝜙CR]EeV ≃ 3 · 10−7 𝜌0 10−5 Mpc−3 𝜉𝑧 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='6 𝐿CR 1043erg s−1 GeV cm2s sr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (7) In comparison, the observed CR spectrum has value of [𝐸2𝜙CR]EeV ≃ 2·10−7GeVcm−2s−1sr−1, very close to our estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The associated all-flavor neutrino spectral injection rate Q𝜈 result- ing from pp interactions can be related to QCR as: 𝐸2 𝜈Qall 𝜈 ≃ 1 2 𝜅 𝜎pp 𝑐 𝑡age 𝑛ISM 𝜂−1 loss 𝐸2 𝑝QCR ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='07 𝑛ISM,b 104cm−3 𝑡age 103yr 𝜂−1 loss 𝐸2 𝑝QCR , (8) where 𝜅 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='5 is the inelasticity of pp interactions with cross section MNRAS 000, 1–11 (2015) Particle acceleration and multimessenger radiation from UFOs 7 𝜎pp ≃ 3 · 10−26cm2 and the CR proton energy is related to neutrino energy as 𝐸𝜈 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='05𝐸p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Notice that in this order of magnitude estimate we consider only neutrinos resulting from pp interaction since they dominate up to at least 102 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Neutrinos from p𝛾 as previously shown, do not feature a different order of magnitude in the HE SED therefore the pp neutrino flux can be assumed as a good approximation also for the order of magnitude flux reached by the p𝛾 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Again, following Ahlers & Halzen (2018), we can now estimate the isotropic neutrino flux as: 𝐸2 𝜈𝜙all𝜈 = 𝜉𝑧 4𝜋 𝑐 𝐻0 𝜌0𝐸2 𝜈Qall𝜈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (9) Adopting in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (9) the expressions for the CR and neutrino luminos- ity as described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (8) together with 𝐸2𝑝QCR ≃ 𝜒𝐿CR one obtains an estimate of the total flux of EeV CRs and single flavor neutrino flux at Earth: 𝐸2 𝜈𝜙all𝜈 ≃ 2 · 10−8 GeV cm2s sr × 𝜌0 10−5Mpc−3 𝜉𝑧 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='6 𝑛ISM,b 104cm−3 𝑡age 103yr 𝐿CR 1043erg s−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (10) This flux is comparable to the level of the diffuse neutrino flux observed by IceCube (see Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2022b) for energies larger than 100 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' These estimates indicate that if UFOs were as abundant as typi- cal non-jetted AGNi and powerful starbursts they could potentially be the dominant sources of cosmic rays at EeV while also strongly contributing to the diffuse neutrino flux observed by IceCube above 100 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In this context, the associated gamma-ray flux would not be expected to contribute substantially to the diffuse gamma-ray flux observed by Fermi-LAT (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2015) due to the flat spec- tral shape and the strong absorption taking place inside the source environment above ∼ 102 GeV that would limit the energy budget of the electromagnetic cascade in the propagation of the gamma rays to the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In a scenario of minimal absorption (scenario C) one can expect at most a gamma-ray flux at ≲ 1 TeV at the same level of the neutrino one, namely 𝐸2𝛾𝜙𝛾(𝐸𝛾) ∼ 𝐸2𝜈𝜇 𝜙𝜈𝜇 (𝐸𝜈𝜇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Directly observed UFOs as well as X-ray AGNi and luminous in- frared galaxies where a UFO can be obscured, represent a promising source class where UHECRs could be accelerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The detection of a flat (∼ 𝐸−2) neutrino spectrum extending in the PeV range from the innermost core of these galaxies could serve as a hint to discriminate whether DSA is taking place as we propose in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 5 APPLICATION TO NGC1068 In what follows we specialize our calculations to the nearby Seyfert galaxy NGC1068 located at a distance 𝐷𝐿 = 14 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Mizu- moto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (2019) reported some indications of an UFO hosted in the core of NGC1068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We compute the gamma-ray and associ- ated neutrino emission through pp and p𝛾 interactions by adopting �𝑀 = 2 · 10−1 M⊙ yr−1, 𝑢1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='1 𝑐 and 𝑙𝑐 = 10 pc while assuming all other parameters as in the benchmark scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In particular, the coherence length assumed here to be as large as the system size (2 𝑅fs ≃ 𝑙𝑐 ≃ 10 pc) result in a maximum energy of approximately 𝐸max = 5 PeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' This is found in agreement with the trend presented in the previous section (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Notice that the value of 𝑙𝑐 required for this calculation is considerably larger than in our benchmark scenario discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In fact, assuming smaller values of 𝑙𝑐 leads to a neutrino spectrum extending up to several PeV with an approximately ∼ 𝐸−2 spectrum, in contradiction with the IceCube sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 10 2 100 102 104 106 108 E (GeV) 10 15 10 14 10 13 10 12 10 11 10 10 E2dN/dE (erg cm 2 s 1) Fermi MAGIC IceCube SAM (pp) SAM (pp) SW (pp) SW (pp) SW+SAM (p ) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Multimessenger emission for the case of NGC1068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The thick blue line represents the gamma-ray emission dominated by pp interactions in the shocked ambient medium (SAM), while the red dotted line is the associated neutrino flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The emission from the shocked wind (SW) (gamma-ray in cyan dashed and neutrino in magenta dot-dot-dashed) as well as the photomeson (orange dot-dashed) emission are subdominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The model prediction is com- pared with Fermi-LAT (Abdollahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2020), MAGIC (Acciari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2019) and IceCube (Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2022a) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Figure 5 illustrates the multimessenger flux produced by the ac- celerated particles in the system and under the assumption that the pressure of accelerated particles is ∼ 5% of the ram pressure at the wind termination shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' One can see that the gamma-ray emission (thick lines) is dominated by the pp contribution in the shocked am- bient medium (blue line), whereas the emission from the shocked wind (purple line) is more than two order of magnitude dimmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' § 4, the strong photon field associated with the ac- cretion disk and torus makes the source opaque to gamma rays above a few tens GeV, so that all TeV photons are completely absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The neutrino flux is dominated by the pp channel in the range from GeV up to ∼ 102 TeV where it features a reduction due to the maximum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Photomeson interactions take also place in the UFO environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' However, they produce a negligible impact on the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Overall the neutrino flux shows a remarkable flat spectrum over more than five orders of magnitude where the associated gamma-ray coun- terpart gets absorbed beyond ∼ 10 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' It is possible to notice that 1) a UFO could dominate the gamma-ray flux observed by Fermi-LAT and 2) in the TeV range the UFO could contribute from a few up to ∼ 10% of the flux measured by IceCube (Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2022a) leaving room for other possible sources such as the AGN corona, the molecular outflow or the starburst ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' A standard UFO seems there- fore disfavoured for explaining the level of neutrino flux observed by IceCube in light of the stringent upper limits imposed by MAGIC as well as the detected Fermi-LAT flux at lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='1 Forward shock scenario for NGC1068 As an alternative scenario to the acceleration at the wind termination shock one could explore the same UFO at an early stage during which the forward shock is expected to play the most relevant role in terms of particle acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The forward shock expanding in the unperturbed ISM of density 𝑛ISM ∼ 104 cm−3 can indeed foster particle acceleration through DSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Let us consider a young UFO which has not entered the de- celeration phase yet, namely it would be expanding with constant velocity ∼ 𝑢1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In this context, the forward shock is expected to be strong (M ≫ 1) thereby efficient in accelerating particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The MNRAS 000, 1–11 (2015) 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Peretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' spectrum of accelerated particles at such shock can be written as 𝑓 (𝑝) = 𝐴(𝑝/𝑝0)−𝛼 exp(−𝑝/𝑝max), where the normalization 𝐴 is estimated assuming that a fraction 𝜉fs ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='1 of the ram pressure 𝑚 𝑝𝑛ISM𝑢2 1 is converted into energized particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In the test-particle limit, 𝛼 = 4, while an upper limit on the maximum momentum can be estimated with the Hillas criterion, leading to: 𝑝max,fs ≃ 𝜉fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='1 𝑅fs pc 𝑢1 108cm/s 𝐵 𝜇G103 GeV c−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (11) An acceleration site with 𝑅fs ∼ 10−2 pc and 𝐵 ∼ 102 − 103 𝜇G allows for 𝑝max ≳ 104 − 105 GeV c−1, in principle sufficient for the pp and p𝛾 processes to produce gamma rays and neutrinos in an energy range accessible to current instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Assuming that the accelerated protons fill at most a volume 𝑉 ∼ 4 · 10−6(𝑅fs/10−2pc)3 pc3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' in which the interactions with the gas and the AGN photon field are taking place,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' the UFO is expected to produce a gamma-ray luminosity (before any absorption effects are taken into account) of the order of 𝐿𝛾(103 − 104GeV) ≃ 10−15 − 10−14 erg cm−12 s−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' and a neutrino luminosity of the order of 𝐿𝜈(103 − 104GeV) ≃ 10−16 − 10−15 erg cm−2 s−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' indicating that the emission from the forward shock is expected to be sub-dominant compared to the one from the wind termination shock happening at later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Late stage UFOs, as discussed in Ajello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (2021), could be luminous enough to be detected in the GeV range by space based telescopes such as Fermi-LAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' However, this would require the cen- tral engine to be active for a time 𝑡 ≫ 103 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' On the other hand, high acceleration efficiency at the wind termination shock can be expected to be found soon after the deceleration phase has begun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='2 Discussion on NGC1068 UFOs are characterized by a prominent gamma-ray (up to ∼ 10−102 GeV) and neutrino emission (typically up to ∼ 102 TeV) resulting from pp interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Therefore, in the standard test particle regime considered in this work, the resulting gamma-ray and neutrino spectra will feature roughly the spectral slope (∼ 𝐸−2) of their parent protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We observe that from the energetic point of view, the level of neutrino flux observed is compatible with the amount of power that UFOs can supply on average suggesting the AGN as a plausible origin for such an emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' However, the strongest limitation comes from the level of gamma rays measured by Fermi-LAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In fact, even though the AGN radiation field can absorb efficiently TeV photons, GeV gamma-rays come basically unabsorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Indeed, as pointed out by several authors (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Murase et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Inoue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Kheirandish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Inoue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Eichmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Murase 2022), such level of neutrinos is not compatible with a pp scenario with a ∼ 𝐸−2 spectrum unless the emission comes from a region optically thick to GeV and sub-GeV gamma rays, namely the nearest neighbourhood of the SMBH or AGN-corona having a size of ≲ 102 𝑅𝑠 (where 𝑅𝑠 is the Schwarzschild radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Even though it is not clear whether the launching radius of a UFO can be localized at such a small distance from the SMBH, we explored under which conditions one could expect a UFO to be confined close to the AGN corona for a sufficient amount of time during its deceleration phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We found that, in order to reproduce the level of neutrino flux inferred by IceCube (Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2022a) without exceeding the gamma-ray flux, the energy budget would not exceed standard values typical of UFOs ( �𝐸 ≃ 1041 erg s−1) but the requirement in terms of average gas density of the external medium would be 𝑛ISM ≳ 1010 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Such a density could be compatible with the high column density inferred for this source(see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Matt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 1997), therefore this could be a plausible scenario to account for the observed neutrino flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' A detailed modeling of a confined wind expanding in the ultra-dense medium goes beyond the scope of this work, therefore it is left for a follow-up investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The recent anisotropy study carried out by the Pierre Auger Obser- vatory suggests that Seyfert and Starburst galaxies (or objects with a spatial distribution related to these sources) may be related to this anisotropy (Aab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' It is intriguing that NGC1068 is the fourth most relevant object in such catalogs, providing a contribu- tion of about 5 − 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' However, our calculations suggests that if the gamma-ray flux detected by Fermi-LAT is associated with the UFO activity in this source, then the maximum energy is bound to be too low to be relevant for UHECRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Viceversa, if UHECRs are to be produced in the UFO, a different source for the gamma rays has to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 6 CONCLUSIONS In this work we investigated the potential of diffusive shock accel- eration at the wind termination shocks of ultra-fast outflows (UFOs) in the core of active galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We developed a model of acceleration and transport of particles in the wind bubbles excavated by UFOs and we studied the multimessenger implications in terms of escaping particles and high-energy photons and neutrinos produced through hadronic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We found that protons can be accelerated up to the EeV range and that the far infrared photon field of the torus could play a dominant role in setting the maximum energy in such sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In addition, the transport condition in the UFO environment could result in a spectral hardening of the escaping flux of the cosmic rays at the highest energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Such energetic and spectral properties are crucial in light of the recent results obtained by the Pierre Auger Observatory which suggests that the sources of UHECRs could be characterized by hard spectra (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Aab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2017, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' UFO are extremely interesting in terms of multimessenger emis- sion since, in standard conditions, they are expected to shine in gamma rays up to ∼ 10 GeV while being opaque beyond 10 − 102 GeV depending on the parametric configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' While gamma rays are efficiently absorbed, HE neutrinos from pp and p𝛾 interactions will be copiously produced with a spectrum extending up to ∼ 102 PeV featuring a spectral slope as hard as the one of their parent pro- tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Such a property is particularly interesting since UFOs have the potential and possibly the number density in the Universe to simul- taneously dominate the CR spectrum at the ankle and the diffuse neutrino flux observed beyond ∼ 102 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We finally applied our model to the UFO that has been suggested to be hidden in the nearby galaxy NGC1068 and we found out that, if confirmed, a well developed spherical UFO wind bubble could dominate the gamma-ray flux observed by Fermi-LAT while it is unlikely to contribute more than ∼ 10% of the total neutrino flux observed by IceCube at TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' On the other hand the base of the outflow could be a plausible region where protons can be injected and produce HE neutrinos while the associated GeV gamma-ray counterpart can be efficiently reprocessed to lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The present model focused on the acceleration and transport of protons in UFOs while the implications for primary and secondary electrons as well as electron-positron pairs is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' UFOs are in fact expected to be perfect electron calorimeters given the dominant synchrotron and inverse Compton timescales compared to the escape timesciale, and the case of leptons would require a spe- cific treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We computed the typical timescales for electrons and MNRAS 000, 1–11 (2015) Particle acceleration and multimessenger radiation from UFOs 9 we found that primary electrons cool very rapidly without reach- ing TeV energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Secondaries as well as electron-positron pairs are also expected to cool mostly via synchrotron losses since beyond ∼ 1 TeV the interaction with the big blue bump takes place in the Klein-Nishina regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In this scenario, the electromagnetic cascade is expected to be synchrotron-dominated for standard parametric as- sumptions (𝜖B ≳ 10−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Therefore, one could expect the leptonic emission to produce some possible signatures in the hard X-ray to soft gamma-ray energy band and in radio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Heavy nuclei should also be implemented in a future work since their transport, similar to electrons, requires the inclusion of calori- metric conditions, continuous energy losses as well as fragmentation and re-injection at lower masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' While our focus has been on the proton population, it is worth mentioning that nuclei can be expected to be co-accelerated in UFOs with a rigidity dependence in the max- imum energy and possibly a higher efficiency in the injection of energetic heavier elements (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Caprioli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In partic- ular, while protons only partially suffer energy losses, heavy nuclei of electric charge 𝑍 are expected to efficiently fragment on the AGN photon field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Therefore one can expect that at early time the majority of heavy nuclei would be reprocessed while at later time they could start escaping efficiently with an energy as large as ∼ 𝑍 · EeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' With the present work we propose UFOs as candidate sources of UHECRs and efficient high-energy neutrino emitters possibly opaque to gamma rays beyond a few tens GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Such properties make UFOs a remarkable source class not only for the UHECRs detected by the Pierre Auger Observatory but also for to the diffuse HE gamma- ray and neutrino fluxes observed respectively by Fermi-LAT and IceCube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The research activity of EP and MA was supported by Villum Fonden (project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 18994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' EP was also supported by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 847523 ‘INTERACTIONS’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' FGS acknowledges financial support from the PRIN MIUR project “ASTRI/CTA Data Challenge” (PI: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Caraveo), contract 298/2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' DATA AVAILABILITY No data has been analyzed or produced in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' REFERENCES Aab A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=', 2017, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': 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Castor J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=', Shapiro P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=', Moore R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=', 1977, The Astro- physical Journal, 218, 377 Yu Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=', Tremaine S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=', 2002, MNRAS, 335, 965 APPENDIX A: PROPERTIES OF THE AGN WIND BUBBLE The dynamics of the wind shock and forward shock during the decel- eration phase of a wind bubble have been studied by Koo & McKee (1992a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The location of the wind shock is described by the follow- ing relation: 𝑅sh = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='0 pc � 𝑡age 1Myr �2/5 � �𝐸 1038erg/s �3/10 × � 𝑛ISM 1cm−3 �−3/10 � 𝑢1 108km/s �−1/2 , (A1) where 𝑡age is the age of the system, �𝐸 the kinetic power ( �𝐸 = �𝑀𝑢2 1/2), 𝑛ISM the ISM density and 𝑢1 the upstream wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The forward shock radius is found as follows: 𝑅fs = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='0 pc � 𝑡age 1Myr �3/5 � �𝐸 1038erg/s �1/5 � 𝑛ISM 1cm−3 �−1/5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (A2) The radiation field (top panel) and the corresponding opacity (bottom panel) of the wind bubble are shown in Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In particular, the typical AGN radiation field feautures a combination of thermal and non-thermal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' At around ∼ 10 meV one can identify the infrared thermal field of the torus, while the second thermal component peaking at ∼ 10 eV is the thermal radiation from the accretion disk known as big blue bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Finally, starting from the keV range the non-thermal tail produced by electrons in the AGN corona is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Such a non-thermal tail can additionally feature a somehow different shape due to the mechanism known as Compton reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In this work we decided to ignore such an effect since it has a strong dependence on the local parameters of the AGN while having a negligible impact on the high energy particles and their radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The gamma-ray opacity shown in the bottom panel clearly highlights the prominent impact of the big blue bump and the torus absorbing respectively in the energy ranges 10 GeV - TeV and beyond 10 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' APPENDIX B: SOLUTION TO THE TRANSPORT EQUATION The transport equations (1) are solved separately in the upstream and the downstream regions and joined at the wind shock location as described in Morlino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Peretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In what follows we highlight the main analytical steps required to obtain the formal solution and the key aspects of the iterative algorithm we adopt to find the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 10 12 10 10 10 8 10 6 10 4 10 2 E (GeV) 1038 1040 1042 1044 1046 1048 L (erg s 1) LX=1042 erg s 1 LX=1044 erg s 1 LX=1046 erg s 1 10 1 101 103 105 107 109 E (GeV) 10 3 10 2 10 1 100 101 102 103 Benchmark A B C Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Top Panel: the AGN photon field for three different X-ray lu- minosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Dotted red, dashed green and dot-dashed blue represent the typical spectral energy distribution of an AGN having an X-ray luminosity of 1042, 1044 and 1046 erg s−1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Bottom panel: gamma-gamma opacity of the wind bubble where the different curves highlight the scenarios discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' In particular: the thick black line represents the benchmark sce- nario while the thin lines describe the absorption in the Scenario A, B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' B1 Upstream region By integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (1) from 𝑟 = 0 to 𝑟 < 𝑅fs one obtains: 𝑟2𝑢1 𝑓1(𝑟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑝) = 𝑟2𝐷1(𝑟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑝)𝜕𝑟 𝑓1(𝑟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑝) − 𝐺1(𝑟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑝) − 𝐻1(𝑟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑝) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B1) where the subscript “1” refers to the upstream region,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' while the functions 𝐺1 and 𝐻1 have the following expressions: 𝐺1(𝑟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑝) = 1 3 ∫ 𝑟 0 𝑑𝑟′ � − 𝜕ln[𝑝3 𝑓1(𝑟′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑝)] 𝜕ln𝑝 � 𝑓1(𝑟′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑝) 𝜕𝑟′ [𝑟′2𝑢1] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B2) 𝐻1(𝑟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑝) = ∫ 𝑟 0 𝑑𝑟′ 𝜆1(𝑟′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑝) 𝑟′2 𝑓1(𝑟′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑝) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B3) where 𝜆1 = 𝜏−1 pp + 𝜏−1 p𝛾 + 𝜏−1 BH is the CR energy loss accounting for pion production in p𝛾 and pp interactions (see Kelner & Aharonian 2008) as well as Bethe-Heitler (BH) pair-production (see,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=', Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Defining the effective upstream velocity as: 𝑉eff,1(𝑟, 𝑝) = 𝑢1 � 1 + 𝐺1(𝑟, 𝑝) + 𝐻1(𝑟, 𝑝) 𝑢1𝑟2 𝑓1(𝑟, 𝑝) � , (B4) MNRAS 000, 1–11 (2015) Particle acceleration and multimessenger radiation from UFOs 11 the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B1) is straightforwardly obtained: 𝑓1(𝑟, 𝑝) = 𝑓sh(𝑝)exp � − ∫ 𝑅sh 𝑟 𝑑𝑟′𝑉eff,1(𝑟′, 𝑝) 𝐷1(𝑟′, 𝑝) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B5) B2 Downstream region Similarly to the upstream case, also in the downstream Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (1) is first approached through a spatial integral exploiting the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Integrating from from 𝑟 > 𝑅fs to 𝑟 = 𝑅fs one obtains: 𝑟2𝑢2(𝑟) 𝑓2(𝑟, 𝑝) = 𝑟2𝐷2(𝑟, 𝑝)𝜕𝑟 𝑓2(𝑟, 𝑝) + 𝑅2 fs 𝑗esc(𝑝) + 𝐻2(𝑟, 𝑝) , (B6) where the subscript “2” refers to the downstream region, while the function 𝐻2 reads: 𝐻2(𝑟, 𝑝) = ∫ 𝑅fs 𝑟 𝑑𝑟′ 𝜆2(𝑟′, 𝑝) 𝑟′2 𝑓2(𝑟′, 𝑝) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B7) As in the upstream region, it is convenient to define the effective velocity in the downstream region as: 𝑉eff,2(𝑟, 𝑝) = 𝑢2(𝑟) � 1 − 𝐻2(𝑟, 𝑝) 𝑟2𝑢2(𝑟) 𝑓2(𝑟, 𝑝) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B8) It is also useful to define the integral function 𝐼2 as: 𝐼2(𝑟, 𝑝) = ∫ 𝑟 𝑅sh 𝑑𝑟′ 𝑟′2 𝑒−𝜑2(𝑟′,𝑝) , (B9) where 𝜑2(𝑟, 𝑝) has the following expression: 𝜑2(𝑟, 𝑝) = ∫ 𝑟 𝑅sh 𝑑𝑟′𝑉eff,2(𝑟′, 𝑝) 𝐷2(𝑝) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B10) Integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B6) one obtains the expression for the escaping flux and the downstream solution as: 𝑓2(𝑟, 𝑝) = 𝑓sh(𝑝) 𝑒𝜑2(𝑟,𝑝) � 1 − 𝐼2(𝑟, 𝑝) 𝐼2(𝑅fs, 𝑝) � (B11) 𝑗esc(𝑝) = 𝑓sh(𝑝) 𝐷2(𝑝) 𝑅2 fs𝐼2(𝑅fs, 𝑝) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B12) We finally notice that the escaping flux 𝑗esc can be rewritten as 𝑗esc(𝑝) = 𝜂loss 𝑢2 𝑓sh(𝑝) 1 − exp[−𝑅sh𝑢2(1 − 𝑅sh/𝑅fs)/𝐷2(𝑝)] 𝑅2 sh 𝑅2 fs (B13) where 𝜂loss is a parameter ≲ 1 accounting for energy losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' B3 Solution at the shock The shock solution is obtained by integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (1) across an in- finitely small layer embedding the wind shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' The result is the following: [𝐷2𝜕𝑟 𝑓2 −𝐷1𝜕𝑟 𝑓1]𝑟=𝑅sh − 𝑢1 − 𝑢2 3 𝑝𝜕𝑝 𝑓sh(𝑝) +𝑄0(𝑝) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B14) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B1) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B6) in the first term on the left-hand side, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B14) can be rewritten as: 𝑠𝑄0(𝑝) 𝑢1 = 𝑝𝜕𝑝 𝑓sh(𝑝) + 𝑠 𝑓sh(𝑝) + 𝑠Ψ𝑙(𝑝) 𝑓sh(𝑝) + 𝑠Ψ𝑒(𝑝) 𝑓sh(𝑝) , (B15) where the functions Ψ𝑘 (𝑘 = 𝑙, 𝑒) are defined as: Ψ𝑙(𝑝) = 𝐺1(𝑅sh, 𝑝) + 𝐻1(𝑅sh, 𝑝) + 𝐻2(𝑅sh, 𝑝) 𝑢1𝑅2 sh 𝑓sh(𝑝) , (B16) Ψ𝑒(𝑝) = [𝐷2𝐼−1 2 (𝑅sh, 𝑝) − 𝑅2 sh𝑢2] 𝑢1𝑅2 sh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B17) Here the subscripts 𝑙 and 𝑒 stands for loss and escape respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' Finally, by recognizing a total derivative on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B15), the solution at the shock can be obtained as: 𝑓sh(𝑝) = 𝑠𝜂𝑛1 4𝜋𝑝3 inj � 𝑝inj 𝑝 �𝑠 𝑒−Γ𝑙 ( 𝑝)𝑒−Γ𝑒 ( 𝑝) , (B18) where: Γ𝑙(𝑒) (𝑝) = 𝑠 ∫ 𝑝 𝑝inj 𝑑𝑝′ 𝑝′ Ψ𝑙(𝑒) (𝑝′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B19) Notice that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B18) can be re-written in the compact form (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (3)) 𝑓sh(𝑝) = 𝐶 𝑝−𝑠exp[−Γcut(𝑝)], where 𝐶 = 𝑠 𝜂 𝑛1 𝑝𝑠−3 inj /(4𝜋) and Γcut = Γ𝑙 + Γ𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' B4 Iteration algorithm The solution to the transport equation on the two sides of the shock, 𝑓1 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B5)) and 𝑓2 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B11)), and at the shock, 𝑓sh (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B18)), do not have a simple analytic form since they depend on each other through the functions𝑉eff,1,𝑉eff,2 and Ψ𝑙(𝑒), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' A solution can be found found via an iterative algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' We initialize the solutions for the set of functions ( 𝑓 (0) sh , 𝑓 (0) 1 , 𝑓 (0) 2 ) by the solutions resulting from thefollowing no-loss conditions: 𝐺 (0) 1 = 𝐻(0) 1 = 𝐻(0) 2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑉 (0) eff,1(𝑟, 𝑝) = 𝑢1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' 𝑉 (0) eff,2(𝑟, 𝑝) = 𝑢2(𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' This results in Ψ(0) 𝑙 = 0 while Ψ(0) 𝑒 reduces to the following analytic form: Ψ(0) 𝑒 (𝑝) = 𝑢2/𝑢1 exp � 𝑅sh𝑢2 𝐷2( 𝑝) � 1 − 𝑅sh 𝑅fs �� − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' (B20) We start from this initial approximation and find iterative solutions by re-computing all functions with the set of solution of the previous iteration, namely: � 𝑓 (𝑖) sh , 𝑓 (𝑖) 1 , 𝑓 (𝑖) 2 � → � 𝐺 (𝑖+1) 1 , 𝐻(𝑖+1) 1 , 𝐻(𝑖+1) 2 � → � 𝑉 (𝑖+1) eff,1 ,𝑉 (𝑖+1) eff,2 , Ψ(𝑖+1) 𝑙 , Ψ(𝑖+1) 𝑒 � → 𝑓 (𝑖+1) sh → � 𝑓 (𝑖+1) 1 , 𝑓 (𝑖+1) 2 � where (𝑖) and (𝑖 + 1) indicate the i-th and (i+1)-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' This algorithm is repeated until the phase space density at the n-th iteration 𝑓 (𝑛) is indistinguishable from the solution found at the iteration (n-1)-th, 𝑓 (𝑛−1), namely when a convergence condition has been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} +page_content=' MNRAS 000, 1–11 (2015)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9FRT4oBgHgl3EQf9zjY/content/2301.13689v1.pdf'} diff --git a/WNAzT4oBgHgl3EQfmP1C/vector_store/index.faiss b/WNAzT4oBgHgl3EQfmP1C/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..6f69ee170e37bce24fb73d9bc103f997e1277b9c --- /dev/null +++ b/WNAzT4oBgHgl3EQfmP1C/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:facd64ca8db383a9e02ce0321f87ac770a9b6750b322257f5de96dfd193ba97d +size 4325421 diff --git a/WNAzT4oBgHgl3EQfmP1C/vector_store/index.pkl b/WNAzT4oBgHgl3EQfmP1C/vector_store/index.pkl 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0000000000000000000000000000000000000000..173fbcd58729259ef0b8fddb0765f613a5c7f0d7 --- /dev/null +++ b/YNE3T4oBgHgl3EQf1wsI/content/tmp_files/2301.04748v1.pdf.txt @@ -0,0 +1,2321 @@ +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +i +LSDM: Long-Short Diffeomorphic Motion for +Weakly-Supervised Ultrasound Landmark Tracking +Zhihua Liu, Bin Yang, Yan Shen, Xuejun Ni, Huiyu Zhou +Abstract—Accurate tracking of an anatomical landmark over +time has been of high interests for disease assessment such as min- +imally invasive surgery and tumor radiation therapy. Ultrasound +imaging is a promising modality benefiting from low-cost and +real-time acquisition. However, generating a precise landmark +tracklet is very challenging, as attempts can be easily distorted +by different interference such as landmark deformation, visual +ambiguity and partial observation. In this paper, we propose +a long-short diffeomorphic motion network, which is a multi- +task framework with a learnable deformation prior to search for +the plausible deformation of landmark. Specifically, we design +a novel diffeomorphism representation in both long and short +temporal domains for delineating motion margins and reducing +long-term cumulative tracking errors. To further mitigate local +anatomical ambiguity, we propose an expectation maximisation +motion alignment module to iteratively optimize both long and +short deformation, aligning to the same directional and spatial +representation. The proposed multi-task system can be trained in +a weakly-supervised manner, which only requires few landmark +annotations for tracking and zero annotation for long-short +deformation learning. We conduct extensive experiments on two +ultrasound landmark tracking datasets. Experimental results +show that our proposed method can achieve better or competitive +landmark tracking performance compared with other state-of- +the-art tracking methods, with a strong generalization capability +across different scanner types and different ultrasound modali- +ties. +Index Terms—Medical landmark Tracking, ultrasound imag- +ing, diffeomorphic motion, long-short temporal modeling. +I. INTRODUCTION +A +CCURATE anatomical landmark tracking has attracted +significant attention in various aspects within clinical +workflows, especially in high-intensity modulated imaging +and image-guided radiation therapy (RT) [1], [2]. Real-time +accurate anatomical landmark tracking delivers precise land- +mark localization and movement estimation information in +temporal-spatial domains, which provides clinicians a mea- +surable therapy margin around clinical and surgical targets +to increase the chance of tumor control [3]. Among various +Zhihua Liu and Huiyu Zhou are with School of Computing and Mathemat- +ical Sciences, University of Leicester, Leicester LE1 7RH, U.K. +Bin Yang is with Department of Cardiovascular Sciences, College of +Life Sciences, University of Leicester, University Hospitals of Leicester +NHS Trust, Leicester LE1 9HN, U.K.; Nantong-Leicester Joint Institute of +Kidney Science, Department of Nephrology, Affiliated Hospital of Nantong +University, Nantong, 226001, China +Yan Shen, Associated Chief physician, Department of Emergency Medicine, +Affiliated Hospital of Nantong University, No.20 Xisi Road, Nantong City, +Jiangsu Province, 226001, China +Xuejun Ni, Chief physician, Department of Medical Ultrasound, Affiliated +Hospital of Nantong University, No.20 Xisi Road, Nantong City, Jiangsu +Province, 226001, China +Corresponding author: Huiyu Zhou. Email: hz143@leicester.ac.uk +ETH-05-2_2 +Frame 0 +Frame 127 +ICR-03_3 +Frame 0 +Frame 2096 +MED-01-1_3 +Frame 0 +Frame 66 +CIL-01_1 +Frame 0 +Frame 1311 +Fig. 1. Illustration of ultrasound medical landmark under different challenges. +From top to bottom: tracking under landmark deformation, acquisition noise, +visual ambiguity and partial observation. From left to right: reference frame, +example target frame, cropped region with LSDM (green) and baseline +SiamFC (yellow) tracking results. +imaging modalities, ultrasound is one of the most desirable +technique benefited from low-cost, non-invasive, and real-time +acquisition [4]. Different from high temporal-spatial resolu- +tion images such as magnetic resonance (MR) or computed +tomography (CT), ultrasound suffers from low signal-to-noise +ratio, speckle decorrelation, spatial ambiguities and aliasing, +making the small anatomical structures (such as tumor bound- +ary, vessel wall) hard to be distinguished and tracked from +surroundings [5]. +Extensive research approaches have been proposed in past +decades to track anatomical landmarks. One implementation +is using invasive artificial fiducial markers, which is limited +with regards to surgery implementation requirements and +marker migration [6], [7]. Automated learning algorithms have +been investigated and achieved significant improvements [8], +[9]. However, they cannot effectively handle the ultrasound +sequences of poor quality, where the intensity is anisotropic in +the temporal dimension, caused by sonar noise and shadowing +arXiv:2301.04748v1 [cs.CV] 11 Jan 2023 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +ii +Frame 0 +Frame 𝑡 − Δ𝑡 +Frame 𝑡 +Long Deformation +Short Deformation +Long/Short Deformation +Fig. 2. +Illustration of Long-Short Diffeomorphic Motion. Top row denotes +an example of the input image set. From top-left to top-right: Frame 0 serves +as the fixed template for long range deformation. Frame t − ∆t serves the +moving target for long range deformation, also adaptive template for short +range deformation. Frame t serves the short range target. From bottom-left to +bottom right: Long range diffeomorphism vector; Short range diffeomorphism +vector; Illustration of joint long-short diffeomorphism motion (blue: short +range deformation, yellow: long range deformation; red: cumulative error from +single range deformation). +effects. Moreover, most existing tracking methods can not +measure the topology changes of targets between frames, (e.g. +disconnected boundaries shown in Fig. 1), which varies from +patient to patient involving motions from respiratory, cardiac +or body movement. These internal and external noise greatly +influence the tracking performance, especially trackers rely on +learning similarity between the anatomical landmark template +(i.e. exemplar) and the target (i.e. instance). As long as the +ultrasound sequence becomes longer, the landmark position +estimation is of more uncertainty and the accumulated tracking +errors are larger [10]. In order to minimize the tracking errors, +a system with the capability of learning deformable shapes is +highly desirable. +In this paper, we propose a novel anatomical landmark +tracking system with a learnable motion prior. Different from +previous medical landmark tracking methods that only focused +on the tracking task only, we design a multi-task tracking +system using a novel diffeomorphism motion structure, namely +long-short diffeomorphic motion (LSDM), to learn and adapt +the non-rigid transformation between the target and the historic +reference frames, generating a landmark motion prior in both +long and short time intervals. LSDM can effectively explore +the hybrid deformation and improve the downstream tracking +task's accuracy for searching the most plausible deformed +landmark, eliminate noisy artifacts, which has not been studied +before. We further design an expectation maximization motion +alignment (EMMA) module to fully utilize the long interval +temporal directional information and short interval spatial fine- +grained information. In EMMA, the short deformation serves +as a latent motion variable. The long deformation can be +iteratively optimized by aligning to the motion representation +of the same landmark. The optimized short deformation can +also be recomposed and then form the long diffeomorphism +with updated temporal details as a close-loop regularization +to avoid over-fitting. Our contribution can be summarized as +follows: +• We propose a novel anatomical landmark tracking method +by multi-tasking both tracking and diffeomorphism mo- +tion learning. Unlike previous methods that learn two +tasks independently, we integrate the learned diffeomor- +phism as a motion prior for searching the best tracking +candidate with significant deformation, thus to improve +the tracking accuracy under challenging medical imaging +situations such as landmark tracking in noisy background +within ultrasound sequences. +• To effectively reduce the cumulative errors in long range +modeling, we design a new representation structure con- +taining diffeomorphism in both long and short time inter- +vals (LSDM). We also propose an expectation maximiza- +tion motion alignment (EMMA) module to iteratively +update both long and short time deformation utilizing +motion direction and similarities. Through EMMA, both +long and short diffeomorphism can be updated by alter- +natively constructing and optimizing the lower bound of +motion evidence, resulting in aligning deformations to the +same directional and spatial representation, which further +mitigates the motion biases brought by local anatomical +ambiguity. +• LSDM can be trained within an end-to-end weakly- +supervised fashion, which only requires few landmark an- +notations for tracking and zero annotation for long-short +motion learning. We conducted extensive experiments +on public and private ultrasound videos. Results show +that our proposed system out-performs various fully- +supervised tracking only methods with high interpretation +and strong generalization across different conditions. +The rest of this paper is structured as follows. We first +present a comprehensive review of related works (Sec.II) in +medical image tracking, motion estimation, and point out +how LSDM differs from them. In Sec.III, we detail our +proposed LSDM tracking network and EMMA module design +with theoretical analysis. Experimental setups and analysis are +presented in Sec.IV and V, respectively. Finally, we conclude +this paper in Sec.VI. +II. RELATED WORKS +We first briefly review related works in medical landmark +tracking and relevant technologies from nature scene object +tracking in Sec.II-A, which can be roughly divided into +registration and feature mapping based methods. We also +discuss some preliminary works within diffeomorphic motion +modeling in Sec.II-B. More discussion on related works can +be found in Table S1, Supplementary A. +A. Medical Landmark Tracking +Accurate anatomical landmark tracking provides fundamen- +tal and crucial information for minimizing the spatial margins +during radiation therapy, specifically using ultrasound images +with substantial benefits such as fast acquisition, low cost +and non-invasion [11]. Previous attempts on medical landmark + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +iii +1×1 +Conv +1×1 +Conv +1×1 +Conv +Kernelized Motion +E-Step +Motion Alignment +M-Step +Recompose Regularization +Expectation Maximization +Motion Alignment (EMMA) +E-Step +M-Step +Expectation Maximization +Motion Alignment (EMMA) +AlexNet +AlexNet +w/ DPN +Deformation Aware +Cross Correlation +Shared +Encoder +Shared +Encoder +Long +Decoder +Short +Decoder +Instance +Exemplar +Conv Encoder +Conv Decoder +Composition Operation +Cross Correlation +Fig. 3. Pipeline of our proposed LSDM. +tracking can be roughly divided into two categories based on +the matching formulation: registration based tracking and +feature mapping based tracking. Early attempts tried to learn +a registration framework for finding the spatial transformation +between the template frame and the rest frames, thus the +landmark position can be calculated using the affine matrix +between the template and the target frame. Banerjee et al. +[12] tracked the landmark by matching the global and local +point set between two frames. Konig et al. [13] proposed a +registration method for calculating the registered landmark +position with a normalized gradient field. These registration- +based tracking frameworks focused on minimizing the global +image registration errors, while the local anatomical errors +cannot be corrected in time. Moreover, the manually designed +features cannot fully represent the landmark deformation dur- +ing the longitude evolution [14], [15]. +Instead of implicitly calculating the landmark position using +the affine matrix from global image registration, inspired by +siamese networks [16], [17], recent works focus on automati- +cally learning the high dimensional features and measuring the +feature similarity between the exemplar and the follow-up in- +stances directly. The exemplar and instances extracted from the +same landmark should be similar in the feature space generated +from the same deep network. Bharadwaj et al. [18] applied the +basic siamese network tracking the ultrasound liver landmark. +However, the network can be confused by other regions with +similar visual representation and the fixed displacement prior +may fail when the landmark moves out of field-of-view or is +hard to be distinguished from the background. To obviate these +difficulties, Wu et al. [19] extended the siamese network by +calibrating the intermediate features. Liu et al. [20] developed +a cascaded network using two branches of a siamese network +with different sizes of inputs. However, these methods suffer +from learning the complex pair-wise relationship between the +instance and the exemplar by iterating every frame pair, which +is label intensive for medical image analysis tasks. Also, the +hierarchical feature learning network lacks meaningful insights +on minimizing the tracking error, particularly on quantifying +the safety margin required by radiation therapy treatment +planning to deliver planned doses [21], [22]. +B. Diffeomorphic Motion Modelling +Diffeomorphism describes the non-linear transformation, +measuring how topology, such as components and connected +boundaries, is deformed between two time stamps [23], [24]. +During radiation therapy, diffeomorphic motion estimation +adversely affects the planned irradiation of the target anatomy. +Recent research works use deep networks to learn the sta- +tionary velocity field parameterized as a scaling and squaring +layer to form the diffeomorphic deformation [25], [26]. These +research works greatly boosted the diffeomorphism registra- +tion performance, however, they all followed the pair-wise +deformation learning, i.e. a fixed frame is chosen as template +for the rest of the images to be pair-wise warped, which +introduces additional biases such as interpolation asymmetry +and the deformation computation complexity is proportional +to the length of the image dataset. +Different from previous attempts, in this work, we design a +simple but effective diffeomorphism structure called long-short +diffeomorphism motion (LSDM). The long diffeomorphism +serves as the pair-wise motion learning from a sparse image +set formed by the selected frames (i.e. keyframes for landmark +tracking). The short diffeomorphism plays a role as group-wise +motion to learn the deformation between the keyframe and +their previous ∆tth frame. This long-short structure can be +viewed as a hybrid structure where the long diffeomorphism +from a sparse set can reduce the deformation complexity for +a long video sequence and the short diffeomorphism from the +paired images is invertible and able to preserve anatomical +topology. Similar to the use of expectation maximization (EM) +for learning the optimized parameters of mixture models, +we propose an expectation maximization motion alignment +(EMMA) module to iteratively update and find the best long- +short deformation. Thus the drifting error caused by image + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +iv +artifacts can be effectively reduced by considering deformation +in both local and global contexts along the temporal dimen- +sion. The optimized long-short diffeomorphism motion can +be served as a complete prior for downstream tasks, where +we show the LSDM can provide meaningful deformation +information for feature extraction networks used in a relatively +challenging task such as ultrasound landmark tracking. +III. METHOD +A. Problem Formulation +We focus on anatomical landmark tracking within ultra- +sound image sequences. The goal is to generate precise +location estimation for subsequent frames given the starting +position of the landmark, which requires a stable single object- +level description with high efficacy and low complexity. Thus, +we formulate the ultrasound anatomical landmark tracking +problem following the single object tracking definition based +on similarity learning [16], [17]. The similarity-based tracking +aims to estimate the location s of landmark (exemplar) x0 +in the t-th tracking frame (instance) xs +t using an optimized +siamese network Φ parameterized by θ, resulting in minimiz- +ing the following tracking loss function: +LT r (θ) = +� +t +∥Φ(x0) ⋆ Φ(xs +t) − ys +t ∥2 +(1) +where ys +t is the associated true Gaussian confidence map +at location s in the t-th frame. ⋆ represents the correlation +operation. As we summarized in Sec.II, previous attempts for +finding the feature similarity are heavily affected by internal +and external noise. Moreover, previous tracking models cannot +construct a morphology search space to measure a precious +deformation distance between the exemplar and the instance, +missing an important prior for accurate anatomical landmark +tracking. Following multi-task setting [27], we propose a +novel framework for learning a unified filter ΦU for both +tracking landmark xs +0 at locations s = {s0, s1, ..., st} and +generating diffeomorphic deformation m = {m0, m1, ..., mt} +to update the search template with most plausible diffeo- +morphic transformation, a given dataset x with t frames +x = {x0, x1, ..., xt}: +Φ⋆ +U = arg max +ΦU +p (s, m|x, ΦU) += arg max +ΦU +p (s|m, x, ΦU) p (m|x, ΦU) +(2) +Following Eqn.(2), we formulate the landmark tracking +as an optimization problem with diffeomorphic transform +estimation as learnable equality constraints. We expect to find +the optimal unified filter Φ⋆ +U to minimize the landmark location +tracking error LT r and the diffeomorphic transform error LM +simultaneously, given a dataset x which contains t frames with +a landmark (instance) xs +F a fixed frame xF (here we refer the +first frame x0 as xF ) as transformation template: +min LT r = +� +t +∥ΦU (xs +0) ⋆ ΦU (xt) − yt∥2 +s.t LM = 0 +where LM = +� +t +∥ΦU (x0, xt) − xt∥2 +(3) +B. Long-Short Diffeomorphic Motion Structure +A na¨ıve solution for Eqn.(3) is to learn a single diffeo- +morphic motion that can estimate an accurate transforma- +tion between template frame xF and target frame xt. This +diffeomorphic prior can benefit the jointly trained tracker, +where the tracker cannot only learn the feature similarity, +but also search the most plausible location of instance xs +0 in +the target frame xt in the motion space generated from the +diffeomorphic transformation ΦU (x0, xt). However, one main +drawback of this na¨ıve setting is the cumulative error within +a long tracking sequence cannot be effectively corrected, +i.e. multiple adversarial response peaks with noisy motion +prediction cannot be erased, resulting in a high variance of +tracking trajectory along the temporal domain, which is not +ideal for minimizing the anatomical landmark location margin +during radiation therapy. Last but not least, minor motion +changes cannot be detected as the unified filter tends to learn +a complete motion for the whole sequence while the motion +shift between a few frames could be ignored. +Inspired by classical findings in video temporal analysis +[28], [29], we propose a novel diffeomorphic representation +structure called long-short diffeomorphism motion (LSDM) +to solve the issues listed above. LSDM contains estimation +of diffeomorphic motion ΦU = {φl, φs, φT r} in frame pairs +with both long time interval t and a stochastic short time +interval ∆t (∆t ≪ t). We follow the standard definition of +diffeomorphism by assuming the latent variable z of both +long and short diffeomorphism is a multivariate Gaussian +distribution with zero mean and covariance σ: +p (z) ∼ N(0, σ2) +(4) +where z is a stationary velocity field (SVF) generated by +deformation field φ within time interval t ∈ [0, 1]: +dφt +dt = v(φ(t)) = v ◦ φ(t) +(5) +where ◦ is the composition operator. Different from tradi- +tional diffeomorphism learning given a single fixed template +xF , the template of long diffeomorphism is fixed as the first +frame of the frame sequence xF += x0 and the template +of short diffeomorphism is sampled xF = xt−∆t based on +the short time interval ∆t. Thus, we can obtain a noisy +observation of the warped images set: {x0 ◦ φl, xt−∆t ◦ φs} +from a Gaussian mixture model: +p(yt−∆t, yt|z; x0, xt−∆t) ∝ N(yt−∆t; x0 ◦ φl, σ2I)+ +N(yt; xt−∆t ◦ φs, σ2I) +(6) +where yt−∆t and yt are the observations of the warped +images given x0 and xt−∆t respectively. By introducing the + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +v +Fig. 4. +Illustration of the proposed Expectation Maximization Motion +Alignment (EMMA) module. Cross means element-wise multiply. “S” means +sigmoid layer. EMMA first kernelizes the input deformation, where long +(orange) and short (blue) range deformation with iterative updates are shown +in Alg. 1. The output of EMMA is the aligned long and short range +deformation, which benefits the downstream trackers to search the tracking +instance with significant deformation. +hierarchical structure of diffeomorphism in both long and short +temporal domains, we are able to learn a reliable diffeomor- +phism combination. The long time diffeomorphism can benefit +the downstream tracker to minimize the spatial search space +for finding the exemplar xs +0. The short diffeomorphism allows +us to effectively learn the preservation of short time deforma- +tion changes based on adaptive updated template xt−∆t and +can benefit the tracker with the updated motion trends from +frame xt−∆t to frame xt. +C. Expectation Maximization Motion Alignment +The long and short diffeomorphism deformation focuses on +learning different parts of the deformation due to the choice +of the template being different. The template of the long +deformation is fixed (as x0), causing the long deformation to +focus on the generalized motion learning with a long interval +motion trajectory. The template of short diffeomorphism is +adapted, which leads the short deformation to seeking an +average mapping based on the finding of the similarity between +xt−∆t and xt. +Considering either long or short diffeomorphic motion only +may lead to biased deformation curve estimation from a +sparse partially prior (shown in Fig. 2). In practice, finding +corresponding image pixels over long diffeomorphic motion is +not trivial at all. First of all, it is extremely difficult to identify +optimal correspondences in a noisy image space over a long +interval. Second, based on Bayesian inference, physics driven +prediction models likely suggest several similar candidates in +the following image frames, which may further confuse the +downstream tracking algorithm. Finally, instead of rendering +corresponding counterparts in the next image frame over a long +interval, we rather target at a direct approximation of image +pixels over continuous short diffeomorphic motion intervals. +One of the possible solutions is to use the expectation- +maximization algorithm. In this paper, we propose an Expec- +tation Maximization Motion Alignment (EMMA) algorithm to +align both long and short deformation. Specifically, at the nth +iteration, the short motion is an extension of long motion in +both spatial-temporal domains, i.e., the direction and distance +distribution of short motion between xt−∆t and xt follows +the moving inertia of long motion between x0 and xt. Our +Algorithm 1: Expectation Maximization based Long +Short Deformation Alignment. +Input: Generated long deformation φl and short +deformation φs. +Output: Aligned deformation φN +l +and φN +s after N iterations. +1: Kernelize φs to ˜φs +2: while iteration stage < N do +3: +φN +s = φl ∗ ˜φs; +4: +z = Softmax(φN +s ) +5: +φN +l = z ∗ ˜φs +6: end while +7: φN +s = φN +l ∗ z +8: φN +s = φN +s + φs +9: φN +l = φN +l + φl +10: return Aligned long deformation φN +l +as φl, short +deformation φN +s as φs. +proposed EMMA follows this physical model by viewing the +short motion as a latent motion vector that generates the +corresponding long motion following a conditional posterior +distribution p(φs|φl, θ), parameterized by θ: +θ⋆ = arg max +θ +log p(φl|φs, θ) +(7) +where φs is the short motion used as latent variable and +together (φl, φs) is called complete motion. Our goal is that +by using the iterative optimization method EMMA, we can +find the optimal parameter θ∗ for generating the final long +motion representation, where θ∗ for Eqn. 7 can be solved by +Theorem VIII. +Theorem III.1. The expectation maximization motion align- +ment can be converged with an optimum θ∗ within N times +iteration where: +log p(φl|φs, θ∗) ≥ log p(φl|φs, θn), n = 1, . . . , N, N → ∞ +(8) +We provide the proof of Theorem VIII in Supplementary B, +which leads to the E-step and M-step of EMMA. Specifically, +EMMA first starts with random initialized parameter θ and +constructs the expectation expression at iteration n: +E-step: +p(φs|φl, θn) → Eφs|φl,θt [ log p(φl, φs|θ)] +(9) +and update the parameter for maximizing the constructed +expectation: +M-step: +θn+1 = arg max +θ +Eφs|φl,θn [ log p(φs, φl|θ)] +(10) +D. Plug-and-Play Differentiable EMMA +Fomulated from E-Step (Eqn.(9)) and M-Step (Eqn.(10)), +EMMA first kernelized short diffeomorphism φs into k seed +features ˜φs = {φ1 +s, ..., φk +s}. We model the conditional distri- +bution of φl and ˜φs as a Gaussian Mixture Model as the short + +Expectation Maximization +MotionAlignment(EMMA) +Iteration n-1 +Iterationn +Iteration n+1 +1x1Conv +Zo +1x1 Conv +Gaussian +KernelJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +vi +diffeomorphism φs is an extent of φl in the temporal domain, +parameterized by latent variable Z = {z1 +φ, ..., zk +φ}. +p(φs|φl) = +K +� +k=1 +zk +φN(φs|φl; µk, σk) +(11) +The log-likelihood of φl and φs given a fixed parameter θ +can be written as: +log p(φs, φl; θ) = log +� +zφ +p(φl, φs, zφ; θ) += log +� +zφ +Q(zφ)p(φl, φs, zφ; θ) +Q(zφ) +≥ +� +zφ +Q(zφ) log +�p(φl, φs, zφ; θ) +Q(zφ) +� +(12) +where +Q(zφ) = +p(φl, φs, zφ; θ) +� +zφ p(φl, φs, zφ; θ) += p(φl, φs, zφ; θ) +p(φl, φs; θ) += p(zφ|φl, φs; θ) +(13) +Based on the Expectation-Maximization algorithm, the E- +step of EMMA is to construct a motion evidence lower bound +(ELBO) for log p(φs, φl; θ) in Eqn.(12) and the M-step aims +to maximizing the ELBO w.r.t θ while fixing Q(zφ), which +means to maximize the likelihood of φs for the next time +step given long deformation φl generated from the history, +specifically: +Q(zi +φ) = +k +� +i=1 +p(zi +φ|φl, φs, θ) +zk +φ = +Ker(φl, ˜ +φks) +�k +j=1 Ker(φl, ˜φj +s) +(14) +where Ker is the kernel mapping function for both φl and +φs. We apply the simple inner product function as the choice +of Ker for the convenience of implementation. Following +previous works in [30], [31], we implement the E-step as a +softmax layer for the inner product of φl and ˜φs to formulate +the probabilistic distribution of Z: +Z = Softmax(φl( ˜φs)⊤) +(15) +After we have fixed the latent variable Z, the M-step of +EMMA is to maximize the probability of the appearance of +kernelized short diffeomorphism feature given a previous long +diffeomorphism: +˜φs = +zi +φφl +�k +j=1 zj +φ +(16) +After the iteration of N times, +˜φs can be updated to an +optimal representation. Following the definition of diffeomor- +phic learning in Eqn.(5), the long deformation φl can be +w/ Deformation Pyramid Network +Conv5×5 +Conv3×3 +Conv1×1 +Conv5×5 +2 Conv3×3 +Conv1×1 +Conv2×2 +Conv1×1 +(a) Tracknet(AlexNet) +Deformation Encoder +(b) Deformation Partial Decoder +(c) Deformation Complete Decoder +Long +Decoder +Short +Decoder +Shared +Decoder +Conv1×1 +Conv1×1 +Fig. 5. +Illustration of the proposed Deformation Pyramid Network (DPN). +Left (a): the deformation pyramid network transferring the motion feature +(orange) as a prior to the tracking network (gray). Right (b): Partial motion +decoder design, where long and short range deformation is modeled by two +separate decoders. Right (c): Complete motion decoder design, where long and +short range deformation is modeled by the same decoder and only decoupled +at the last layer. +re-composed using the optimized ˜φs, as the optimized short +diffeomorphism aligns with the same snapshot within a single +time slot: +φN +l = zφ ˜ +φN +s +(17) +The re-composition of φl can be viewed as a regularization +strategy. After having finished the re-composition, EMMA +achieves a close loop updating φl and φs. The algorithm of +executing EMMA for N iterations can be summarized as +Alg.1. Since the common frame xt−∆t is involved during +EMMA, we believe that the updated φN +s and φN +l +are aligned +together, following the definition of feature alignment in +similar research fields such as domain-adaptation [32] and +object re-identification [33]. +E. Deformation Pyramid Network +Inspired by Feature Pyramid Network (FPN) [34], we take +the feature from long-short diffeomorphism encoding as a prior +to learn a plausible deformation for tracking an instance for +the exemplar in subsequent frames. Specifically, a traditional +siamese network uses a shared encoder to extract features for +both the exemplar and the instance. The shared weights setting +encourages learning a high similarity, while ignoring the +exemplar's deformation along the spatial-temporal dimension, +which is significant for medical image tracking. Here, we take +the intermediate features from the deformation network as a +prior for the exemplar, fused by the proposed Deformation +Pyramid Network (DPN). Similar to FPN, DPN takes the +deformation feature hierarchy with semantics from low to high +levels, to update the exemplar features at the corresponded +levels. +As shown in Fig. 5, the DPN follows a top-down con- +struction, where the feature maps with larger size are down- +sampled by large sized kernels and smaller size feature maps +are down-sampled by small sized kernels. Note that there are +two ways to generate long and short deformation based on how +the long and short deformation share their networks' weight. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +vii +The first one is called complete network, where the long and +short deformation share the encoder and the decoder of the +entire deformation learning network. In the complete network, +the long and short deformation can only be separated at the +last layer of the decoder. Another set up is called partial +network, where the long and short deformation shares the +encoder of the deformation network. In the partial network, +the two independent decoders are used to generate long and +short deformation respectively. We test the effects of DPN +with inputs from different deformation network structures +(complete vs. partial), detailed in Sec.V-A. The structural test +does not show any preference of complete vs. partial. For +the trade-off between computational cost and performance, we +choose the complete network design. +IV. EXPERIMENTS +We demonstrate the effectiveness of boosting ultrasound +anatomical landmark tracking performance with multi-tasking +diffeomorphism prior by conducting extensive experiments +on two datasets: 2D ultrasound video dataset from public +Challenge on Liver Ultrasound Tracking (CLUST2D) and +private collected 2D kidney ultrasound video dataset from the +Affiliated Hospital of Nantong University with normal and +contrast ultrasound modality (AHNTU2D). +A. Dataset +CLUST2D is composed of 43 patients with different times +of acquisition, a total of 63 ultrasound videos collected from +4 different groups, where 24 videos are used for training and +39 videos are used for testing. In the training set, the length +of the 2D video sequence varies from 1075 to 5247 frames, +with each frame resolution varying from 393 × 457 to 524 +× 591. Landmarks to be tracked for each video subject in the +training set varies from 1 to 5. In the test set, the length of +the 2D video sequence varies from 895 to 15640, with each +frame resolution varying from 262 × 313 to 524 × 591. The +number of landmarks to be tracked for each video in the test +set varies from 1 to 4. Training labels (landmark coordinate x, +y) are provided by reliable observers. Roughly 10% of the total +sequence is annotated with random time intervals between two +entries. Test set annotation is only provided for the first frame. +Generated tracking results on the test set will be submitted and +evaluated at the official server. We suggest the reader check [2] +and official challenge website1 for more detailed information. +AHNTU2D is composed of 12 acute kidney injury patients +with different times of acquisition, totally 14 ultrasound videos +focused on both kidney collected from the Affiliated Hospi- +tal of Nantong University, China. For each video sequence, +AHNTU2D contains two modalities, i.e. normal ultrasound +(AHNTU2D-N) and contrast ultrasound (AHNTU2D-C). For +both AHNTU2D-N and AHNTU2D-C, we randomly allocate +6 videos for training, 4 videos for validating and 4 videos +for testing. The spatial resolution is 550 × 900 and the +temporal length varies from 280 to 3966. Landmarks to be +tracked for each video sequence varies from 1 to 3. All videos +1https://clust.ethz.ch/ +are manually labelled by one experienced physician from the +Affiliated Hospital of Nantong University. The data collection +followed the ethic procedure of the Affiliated Hospital of Nan- +tong University. All video sequences have been desensitized +following the standard procedure. +B. Evaluation Metric +Euclidean distance is used for evaluating the tracking error +(TE) on the ith landmark between the predicted center coor- +dinate P i +t and the ground truth coordinate GT i +t at tth frame. +TEi +t = ||P i +t –GT i +t || +(18) +TE is evaluated and summarized with mean, standard deviation +and 95th percentile from the official evaluation server. For +system evaluation with motion magnitude, no tracking error +(NoTE) is included as standard calculation: +NoTEi +t = ||P i +0–GT i +t || +(19) +C. Baseline Methods +We first the effectiveness of the deformation prior, EMMA +and DPN with ablation studies. In-House testing is conducted +as well, where the validation sequence provided by a specific +hospital is unseen during the training. We also test the stabi- +lization performance of LSDM on a randomly split training +set. +We also present the systematic tracking performance against +several state-of-the-art ultrasound landmark tracking methods, +including 2D tracking methods [20], [38], [39], [19], [40] with +leading performance on CLUST2D test set. As we built our +tracking backbone upon the similarity learning, we also re- +implement the state-of-the-art tracking method SiamFC [16] +and SiamRPN [37] for further comparison. +V. RESULTS +In this section, we present our experimental design and +result analysis. We first test the effectiveness of different +components within LSDM in Ablation Study (Sec.V-A), lead- +ing to the combination of LSDM. Then we evaluate the +generalization and robustness against different data providers +with different ultrasound scanners in In-House Test (Sec.V-B), +which has never been carried out in previous works. We also +compare our method against several representative tracking +methods on the randomly split training set with ground truth +(Sec.V-C). Finally, we report the test set result on CLUST2D, +AHNTU2D-N and AHNTU2D-C against other state-of-art +methods, human expert observation and no-tracking results +(Sec.V-D). We also add failure tracking cases and fully analyze +the reasons of mistracking (Sec.V-E). +A. Ablation Study +We evaluate the impacts of different components within +LSDM on the CLUST2D training set. The result is shown +in Supplementary C. We first evaluate the deformation quality +brought by different designs of complete and partial diffeomor- +phism learning. We then validate that EMMA can upgrade the + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +viii +MED-03-2_2 +Frame 0 +Frame 15 +Frame 491 +Frame 596 +Frame 1107 +Frame 1652 +Fig. 6. +Qualitative comparison of different baseline models and the proposed LSDM by tracking validation on the CLUST2D training set. We select a +representative dataset entry to demonstrate LSDM superior tracking performance. We show the start frame at frame 0 (top-left), landmark center coordinate +tracklet comparison (top-right) and selected frames with landmark tracking results (bottom). We can observe a stable and accurate tracking performance of +LSDM, while other baselines fail at accumulating tracking error through the whole sequence. Supplementary V summarizes other tracking results and their +visualization. Best viewed in color. +TABLE I +QUANTITATIVE COMPARISONS BETWEEN LSDM AND OTHER REPRESENTATIVE TRACKING TECHNIQUES ON CLUST2D TRAINING SET WITH RESPECT +TO MEAN, STANDARD DEVIATION, 95TH, MIN AND MAX TRACKING ERROR. IN ‘FEATURE’ COLUMN, ‘M’ MEANS ‘MANUAL EXTRACTION’ AND ‘A’ +MEANS ‘AUTOMATIC EXTRACTION’. +Method +Feature +Deformation +Long Short Memory +Mean +Std +95th +Min +Max +KCF [35] +M + + +9.35 +3.67 +19.21 +0.09 +67.92 +LCT [36] +M + + +5.12 +2.54 +8.3 +0.02 +48.82 +SiamFC [16] +A + + +4.39 +2.29 +5.33 +0.01 +33.76 +SiamRPN [37] +A + + +2.42 +1.86 +4.26 +0.01 +18.37 +LSDM(Ours) +A + + +0.81 +0.98 +1.73 +0.01 +14.92 +TABLE II +QUANTITATIVE GROUP-WISE COMPARISONS BETWEEN LSDM AND THE OTHER STATE-OF-THE-ART METHODS ON CLUST2D TEST SET WITH RESPECT +TO MEAN, STANDARD DEVIATION AND 95TH TRACKING ERROR. THE BEST RESULT IS SHOWN IN BOLD AND THE RUNNER-UP RESULT IS UNDERLINED. +CLUST2D +CIL +ETH +ICR +MED1 +MED2 +Mean +Std +TE95th +Mean +Std +TE95th +Mean +Std +TE95th +Mean +Std +TE95th +Mean +Std +TE95th +Liu et al. [20] +1.19 +1.16 +4.16 +0.59 +0.57 +1.24 +0.77 +0.78 +2.70 +0.78 +0.60 +1.81 +0.80 +0.90 +1.73 +Williamson et al. [39] +1.01 +0.84 +2.81 +0.50 +0.45 +1.13 +1.06 +2.02 +2.45 +1.04 +1.14 +2.75 +0.99 +0.78 +2.75 +Wu et al. [19] +1.41 +1.29 +4.20 +0.62 +0.79 +1.46 +0.87 +1.01 +3.30 +1.02 +1.58 +2.78 +1.10 +1.64 +3.06 +Bharadwaj et al. [18] +1.17 +0.89 +2.95 +1.65 +4.48 +2.65 +1.29 +1.83 +5.16 +1.74 +2.93 +5.80 +1.57 +1.93 +6.72 +Shen et al. [40] +1.25 +1.15 +4.03 +0.98 +0.60 +2.16 +1.02 +0.73 +2.61 +1.54 +1.49 +4.76 +1.04 +0.67 +2.18 +LSDM (Ours) +1.10 +0.79 +2.70 +1.03 +1.38 +2.19 +0.86 +0.70 +1.86 +1.08 +0.93 +2.55 +0.89 +0.59 +1.99 +lower bound of long deformation by iterative updating with +short deformation, leading to a precise motion prediction for +tasks such as landmark tracking. The iteration number within +EMMA during training and inference is also examined for +reaching a trade-off between performance and computational +cost. Finally, we argue that, instead of learning the feature +similarity directly, our proposed hybrid deformation can be +injected within the tracking network using the proposed DPN +for minimizing the tracking candidate searching space with +a plausible deformation, which further increases the tracking +performance. +Complete v.s. Partial Deformation Decoder: We first test +different designs of the deformation learning network. As we +have mentioned earlier, the deformation network takes the +concatenated batch of the reference image, long and short +interval images. The encoder extracts features and different +decoder designs reconstruct the velocity field corresponding +to different time intervals. A partial deformation network +indicates that long and short deformation is learned from two +independent decoders and a complete deformation network +means that long and short deformation is learned from the +shared decoder and only separated at the last convolution layer. +From results shown in Table S2 Supplementary C, we observe +that the complete design deformation network outperforms the +partial design with less parameters, as the complete decoder +benefits from learning the hybrid interval information not only +in the encoder but also in the decoder. +EMMA Impacts: We further test the effectiveness of +EMMA to show the benefits of iteratively optimizing the long +deformation based on short deformation as a latent motion + +350 +KCF +LCT +SiamFC +X +SiamRPN +250 +LSDM(Ours) +Manual Annotation +No Tracking +200 +500 +LO00 +1500 +2000 +2500 +300 +KCF +Y Dist +LCT +SiamFC +200 +SiamRPN +LSDM(Ours) +100 +Manual Annotation + No Tracking +500 +1000 +1500 +2000 +25000 +50 +100 +150 +200 +250 +300 +350 -0 +X +Manual Annotation +400 +0 +100 +200 +300 +400 +5000 +50-0 +100 +150 +200 -0 +250 +KCF +300 +LCT +SiamFC +SiamRPN +350-0 +LSDM(Ours) +X +ManualAnnotation +400 +0 +100 +200 +300 +400 +5000 +100 +150 ++ +200-0 +250 +300 +350 -o +400 +0 +100 +200 +300 +400 +5000 +100 +150 +200-0 +250 +300 +350 -o +400 +0 +100 +200 +300 +400 +5000 +100 +150 +200 -o +250 +300 +350-0 +400 +0 +100 +200 +300 +400 +5000 +50 +100 +150 +200-0 +250 +300 +350 -o +400 +0 +100 +200 +300 +400 +500JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +ix +variable.. From Table S2 Supplementary C, we show that +EMMA can upgrade the deformation learning under both +complete and partial motion decoder design. EMMA can +support long short deformation to reduce the negatives of +Jacobian shown in Fig. S1 Supplementary C. We further +examine how different iteration numbers in complete motion +E/M step during training and inference shown in Table S3. We +can obtain a local optimum with a 5 iteration combination for +reasonable performance without asking for further computa- +tional cost. +Tracking with DPN: Finally, we test how our proposed +system can help achieve accurate landmark tracking with +long/short deformation prior. With the feature map injected +into the tracking network, we obtain lower tracking errors in +both mean and standard deviation. Table S2 Supplementary C +shows the quantitative result of accurate tracking from LSDM +with DPN. Compared to the baseline tracking method without +deformation modeling and DPN fusion, LSDM can directly +learn the deformation between the exemplar and the follow- +up instances. By combining priors through DPN, LSDM can +minimize the search space for the tracklet and find the reliable +candidate as a tracking object with plausible deformation, +resulting in accurate tracking in spite of visual ambiguity. +B. In-House Validation +To validate the generalization of the proposed LSDM, +we manually configure the training set for in-house testing. +Specifically for CLUST2D, we select data samples from a +specific hospital provider for testing, which is unseen for the +network during the training stage. For AHNTU2D, we select +data samples from a specific ultrasound modality for testing, +which is unseen for the network during training on another +ultrasound modality. The result is shown in Supplementary +D. We aim to test whether or not a tracking method can +generalize well to track the landmark within a video from +the scanner it has never seen before. As reported in Tables S4 +and S5 in Supplementary D, LSDM generalizes well on videos +from the new ultrasound scanner during testing. LSDM our- +performs the baseline SiamFC in both mean and std in tracking +errors. By learning not only the feature similarity but also the +deformation estimation, LSDM can handle the domain shift +when testing on new scanners and new patients, showing high +robustness and great potential for various clinical workflows +such as radiation therapy. +C. Training Set Comparison +LSDM achieves accurate tracking by learning an optimized +deformation between the exemplar and follow-up instances, +which can help the downstream tracking network for finding +the best candidate with plausible deformation. As shown in +Table I, LSDM outperforms several baselines. Traditional +correlation filtering based methods such as KCF and LCT +cannot achieve satisfactory tracking performance on large- +scale datasets due to various challenges. Even though methods +like LCT incorporate historical information for updating track- +ing kernels, these baselines cannot produce accurate track- +lets, lacking high-dimensional representation. Compared with +tracking methods based on siamese networks such as SiamFC +and SiamRPN, LSDM outperforms with lower tracking error +in both mean and standard deviation. Specifically, LSDM +outperforms SiamRPN with 1.61 lower in mean and 0.88 +lower in standard deviation. Recall that SiamRPN contains an +extra branch for minimizing the regression loss of the tracking +object location directly. This indicates that only measuring the +size (such as bounding box regression in SiamRPN) of the +tracking object is not enough for accurate position estimation +while an optimized deformation can be used as a prior for +searching objects to improve tracking accuracy. We report +an example of tracklet comparison in Fig. 6 and additional +examples in Fig. S3 Supplementary E. We can observe that +our proposed LSDM can track the landmark accurately within +long time ultrasound sequences while the other baselines fail +in all different situations, e.g. distracted by other landmarks +with a high visual similarity in Fig. 6, cannot handle the +landmark deformation effectively resulted in accumulative +tracking errors in Fig. S3 Supplementary E. +D. Testset Result +We report the group wise performance of LSDM on test +sets in CLUST2D in Table II and total tracking error rank- +ing in Table S6 Supplementary F. Compared with state-of- +art methods and human labels, LSDM achieves competitive +performance. Note that the 1st place method [20] uses heavy +pre/processing techniques such as point detection and shadow +removal. On the other hand, LSDM is a simple design but +effective method without complex pre/preprocessing methods, +while the learned deformation prior provides more insightful +guiding for radiation therapy compared with other SOTA +methods and no tracking, indicating the importance of accurate +estimation of motion deformation during long time tracking. +E. Failure Case Analysis +We also report failure tracking examples caused by invalid +field-of-view and lantency in system design. Both the failure +case analysis and examples of failing tracking cases can be +found in Supplementary G. +VI. CONCLUSION +In this paper, we proposed a multi-task based tracking +method with a learnable deformation (LSDM). 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Wang, H. Lu, and X. Yang, “Alpha-refine: +Boosting tracking performance by precise bounding box estimation,” +in Proceedings of the IEEE/CVF Conference on Computer Vision and +Pattern Recognition, 2021, pp. 5289–5298. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +i +VII. SUPPLEMENTARY A +Table S1 summarizes the related works in medical landmark tracking described in Sec. II. +VIII. SUPPLEMENTARY B +Below we present the proof of Theorem III.1. We first restate Theorem III.1. +Theorem. The expectation maximization motion alignment can be converged with an optimum θ∗ within the iterations of N +times where: +log p(φl|φs, θ∗) ≥ log p(φl|φs, θn), n = 1, . . . , N, N → ∞ +(1) +Proof. Following standard Expectation Maximization, we have: +log p(φl|θ) = log +�p(φl, φs|θ) +p(φs|φl, θ) +� +(2) +by taking the expectation on both sides of Eq.(2), w.r.t p(φs|φl, θn) at iteration n, the left part of Eq.(2) equals to: +L = +� +φs +p(φs|φl, θn) log p(φl|θ)dφs += log p(φl|θ) +(3) +The right part of Eq.(2) equals to: +R = +� +φs +p(φs|φl, θn) log p(φl, φs|θ)dφs − +� +φs +p(φs|φl, θn) log p(φs|φl, θ)dφs +(4) +For the second term in Eq.(4), we have: +R = +� +φs +p(φs|φl, θn) log +�p(φs|φl, θn+1) +p(φs|φl, θn) +� +≤ log +� +φs +p(φs|φl, θn)p(φs|φl, θn+1) +p(φs|φl, θn) dφs +≤ log +� +φs +p(φs|φl, θn+1)dφs +≤ log (1) +≤ 0 +(5) +By combining Eqs.(3) and (4), we have the log-likelihood of long deformation given parameter θ: +log p(φl|θ) = Eq(φs) +� +log p(φl, φs|θ) +q(φs) +� +� +�� +� +Motion ELBO ++ +� +q(φs) log +� +q(φs) +p(φl|φs, θ) +� +dφs +� +�� +� +KL divergence +(6) +Following the structural definition in Eq.(6), EMMA first starts with randomly initialized parameter θ and construct the +expectation expression at iteration n: +E-step: +p(φs|φl, θn) → Eφs|φl,θt [ log p(φl, φs|θ)] +(7) +and update the parameter for maximizing the constructed expectation: +M-step: +θn+1 = arg max +θ +Eφs|φl,θn [ log p(φs, φl|θ)] +(8) +The ideal physical meaning of EMMA indicates that, once θ is converged to θ∗, the approximate distribution q(φs) is equal +to the true posterior p(φs|φl, θ∗). The KL divergence in Eq.(6) approaches zero and the ELBO is bounded by the maximum. +Thus, the log-likelihood of long deformation is maximized, meaning EMMA can output the most plausible long deformation. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +ii +TABLE S1 +SUMMARY OF MEDICAL LANDMARK TRACKING METHODS. IN COLUMN LEARNING, ‘R’ MEANS REGISTRATION AND ‘F’ MEANS FEATURE MAPPING. COLUMN DP STANDS FOR DEFORMATION PRIOR. +Authors +Base Model +Learning +Landmark +Modality +DP +Remarks +Banerjee et al.[12] +Grid Set +R +Liver +US +N +Pro: Rigister to reference by tracking on two scale point set. +Con: Post-process on outlier rejection needed. +Konig et al.[13] +Gradient Field +R +Liver +US +Y +Pro: Real time tracking with gradient estimation on deformation. +Con: Cannot handle cummulative errors effectively. +Yang et al.[14] +Shape Model +R +Heart +US +Y +Pro: Multi-View multi-scale heart shape estimation for registration and tracking +Con: Not end-to-end. Mannual fusion design involved. +Bharadwaj et al.[18] +Kalman Filter +F +Liver +US +N +Pro: Template update strategy from Kalman Filter output. +Con: Cannot estimate the landmark deformation. Unsatified tracking performance. +Wu et al.[19] +Siamese Network +F +Liver +US +N +Pro: Coarse-to-fine training with drift correlation based on point distance +Con: Cannot handle deformed landmark with partial observation. +Liu et al.[20] +Siamese Network +F +Liver +US +N +Pro: Multi-scale tracking network. +Con: Training performance relies on the quality of generated landmark points. +Cifor et al.[8] +Shape Model Ensemble +R +Liver +US +Y +Pro: Registration based tracking using ensemble deformation models. +Con:Lacking pricise template matching design. +Royer et al.[9] +Mechanical Simulation +R +Liver +US +N +Pro: Vertex position estimation using visual information and machenical simulation. +Con: Heavy inference workload during long time series images. +Gomariz et al.[41] +Siamese Network +F +Liver +US +N +Pro: Using previous location as a localization prior. +Con: Lacking deformation estimation. +Makhinya et al.[42] +Optical Flow +R +Liver +US +Y +Pro: Optical flow based motion estimation. +Con: Manual designed vessel features. Lacking template matching design. +Shepard et al.[38] +Block Matching +F +Liver +US +N +Pro: Multi-scale block matching method. +Con: Performance relies on local block quality. Lacking deformation estimation. +Ye et al.[?] +Motion Tracking +R +Heart +Tagged MRI +Y +Pro: Forward-Backward motion modeling. +Con: No explicit discriminal learning on landmarks. +Qin et al. [?] +Motion Tracking +R +Heart +MRI +Y +Pro: Biomechanics-informed motion modeling +Con: No historical information during motion modeling. +Rangamani et al.[?] +CNN+RNN +F +Liver +US +N +Pro: RNN location predictor based on CNN features. +Con:Heavy weight network design. No deformation learning on landmarks. +Huang et al.[?] +CNN+LSTM +F +Liver +US +N +Pro: LSTM for location refinement. +Con: Lacking interpretation during refinement process. +Ha et al.[?] +Motion Tracking +R +Abdomen +4D MRI +Y +Pro: Coupled conves optimization for real time motion estimation. +Con: Landmark tracking performance relies on the choice of block template. +Shen et al.[?] +KCF +F +Liver +US +N +Pro: Optimized KCF for real time landmark tracking. +Con: Unsatified performance. Lacking deformation learning on landmarks. +Wilms et al.[?] +Block Matching +R +Adbomen +4D MRI +Y +Pro: Coarse-to-fine training with model based regularization. +Con: Performance relies on block matching. +Williamson et al.[39] +Distance Modeling +F +Liver +US +N +Pro: Multi-template based aggregation. +Con: Performance relies on keypoint selection. +Ours +Motion aware +block matching network +F + R +Liver/Kidney +US +Y +Pro: Hybrid motion modeling for landmark deformation matching. +Con: No specific strategy handling poor quality image with limited field-of-view. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +iii +IX. SUPPLEMENTARY C: ABLATION STUDY +In this section, we show the supplemental ablation study results of LSDM, analyzed in Sec. V. Table S2 shows the component +ablation study results tested on the CLUST2D training set. Table S3 tests different impacts of different EM iteration numbers. +Fig. S1 shows the qualitative deformation comparison of LSDM based on different deformation module combinations. +TABLE S2 +QUANTITATIVE RESULTS OF ABLATION STUDIES ON LSDM REGARDING COMPLETE V.S PARTIAL DEFORMATION PRIOR NETWORK, W/O EMMA AND +FEATURE FUSION W/O DEFORMATION PYRAMID NETWORK. +Components +Metrics +Complete +Partial +EMMA +DPN +TE Mean +/- Std +Deformation Prior + +2.63 +/- 2.11 + +2.69 +/- 2.87 +EMMA + + +1.21 +/- 2.19 + + +1.56 +/- 1.73 +DPN + + + +0.92 +/- 0.76 + + + +0.81 +/- 0.98 +TABLE S3 +QUANTITATIVE RESULTS OF EMMA ITERATION NUMBER TEST ON CLUST2D TRAINING SET. +# EMMA Iteration +TE Mean +/- Std +1 +1.46 +/- 1.72 +5 +0.81 +/- 0.98 +10 +0.93 +/- 1.37 +CIL-02_1 +Pair +S +L +L+S +L+S w/EMMA +Fixed +Moving +#1075 +Fig. S1. +Qualitative deformation comparison of the proposed LSDM based on different deformation module combinations. We select one representative +dataset entriy to demonstrate superior LSDM deformation performance. For each entry, we show the fixed template at frame 0 (top-left), selected target frame +(bottom-left), warped images based on different deformation combination (bottom row) and visualization of determinant of jacobian matrix from a different +displacement field (top row), where red indicates the determinant of jacobian is greater than 1 and blue indicates the value of the determinant of jacobian is +negative. We can observe a progressive smooth setting with proposed long-short deformation and EMMA module. Together with DPN, the LSDM learns to +generate the optimal deformation for downstream tasks like tracking. Best viewed in color. + +F20 +15 +10 +5 +0 +-5 +-10JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +iv +X. SUPPLEMENTARY D: IN HOUSE TEST +In this section, we show the supplemental in-house test results of LSDM, analyzed in Sec. V-B. Table S4 shows the in-house +test result of LSDM and SiamFC [16] on the CLUST2D training set, where the test partition is never shown during the training +process. Table S5 shows the in-house test result of LSDM and SiamFC [16] on the CLUST2D training set, where the test +modality is never shown during the training process. Fig. S2 shows the qualitative response comparison between LSDM and +SiamFC during the in-house test. +TABLE S4 +QUANTITATIVE RESULT COMPARISON BETWEEN LSDM AND THE BASELINE ON THE CLUST2D TRAINING SET WITH IN-HOUSE VALIDATION SETTING, +WITH RESPECT TO MEAN, STANDARD DEVIATION AND 95TH TRACKING ERROR. +In-House Partition +Mean +Std +95th +Scanner Type +SiamFC +LSDM +SiamFC +LSDM +SiamFC +LSDM +CIL +2.01 +1.82 +3.47 +1.63 +11.49 +3.81 +Ultrasonix MDP +ETH +5.33 +1.98 +10.16 +1.21 +17.3 +4.67 +Siemens Antares +ICR +1.09 +2.19 +3.22 +1.76 +5.64 +3.76 +Elekta Clarity-Ultrasonix +MED1 +3.17 +1.35 +2.46 +1.9 +7.71 +2.91 +Zonare z.one +MED2 +4.93 +3.19 +9.27 +1.31 +19.15 +5.18 +DiPhAs Fraunhofer +TABLE S5 +QUANTITATIVE RESULT COMPARISON BETWEEN LSDM AND THE BASELINE ON THE AHNTU2D TRAINING SET WITH IN-HOUSE VALIDATION SETTING +ON DIFFERENT MODALITIES, WITH RESPECT TO MEAN, STANDARD DEVIATION AND 95TH TRACKING ERROR. +AHNTU +AHNTU-N +AHNTU-C +Mean +Std +TE95th +Mean +Std +TE95th +SiamFC [16] +2.74 +3.08 +6.24 +4.41 +6.77 +13.17 +LSDM(Ours) +1.31 +1.76 +3.72 +2.39 +4.12 +10.83 +Instance Frame +Exemplar Frame +SiamFC Response +LSDM Response +Tracking Result +MED-05-1_2 +ETH-03-2_1 +ICR-04-2_1 +Frame 0 +Frame 0 +Frame 0 +Frame 25 +Frame 55 +Frame 31 +Result Detail +Fig. S2. +Qualitative landmark tracking response comparison between the baseline model (SiamFC) and our proposed LSDM on the CLUST2D training +set. From left to right, each column represents the instance frame (frame 0), selected exemplar frame, SiamFC response, LSDM response, tracking result +visualization and zoomed-in patch. Best viewed in color. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +v +XI. SUPPLEMENTARY E: TRAINING SET COMPARISON +Fig. S3 shows additional qualitative tracklet comparision between different baseline models and our proposed LSDM tested +on CLUST2D training set, described in Sec. V-C. +CIL-02_1 +Frame 0 +Frame 49 +Frame 160 +Frame 394 +Frame 509 +Frame 850 +ETH-01-2_2 +Frame 0 +(a) +(b) +Frame 67 +Frame 970 +Frame 1803 +Frame 2555 +Frame 3282 +Fig. S3. Additional qualitative comparison of different baseline models and the proposed LSDM by tracking validation on the CLUST2D training set. We +select other two representative dataset entries to demonstrate LSDM superior tracking performance. We show the start frame at frame 0 (top-left), landmark +center coordinate tracklet comparison (top-right) and selected frames with landmark tracking results (bottom). Best viewed in color. + +0 +100 +X +200 +300 +400 +X +Manual Annotation +0 +100 +200 +300 +400 +500 +600340 +330 +320 +KCF +LCT +SiamFC +X +SiamRPN +300 +LSDM(Ours) +Manual Annotation +290 +No Tracking +280 +1000 +2000 +3000 +4000 +225 +WWW +200 +KCF +LCT +SiamFC +150 +SiamRPN +LSDM(Ours) +125 +Manual Annotation +No Tracking +100 +1000 +2000 +3000 +0 +40000 +100 +200 +X +300 +KCF +LCT +SiamFC +400 +SiamRPN +LSDM(Ours) +X +ManualAnnotation +0 +100 +200 +300 +400 +500 +6000 +100 +200- +X +300 +400 +0 +100 +200 +300 +400 +500 +6000 +100 +200 +300 +400 +0 +100 +200 +300 +400 +500 +6000 +100- +200 +X +300 - +400 +0 +100 +200 +300 +400 +500 +6000 +100 ++ +200 +X +300- +400 +0 +100 +200 +300 +400 +500 +600X340 +330 +320 +KCF +X Dist +LCT +310 +SiamFC +SiamRPN +300 +LSDM(Ours) +290 +Manual Annotation +No Tracking +280 - +200 +400 +600 +800 +1000 +VAA +200 +Dist +LC +190 +SiamFC +SiamRPN +LSDM(Ours) +180 +Manual Annotation + No Tracking +400 +600 +800 +10000 +100- +200- +300- +KCF +LCT +SiamFC +400- +SiamRPN +LSDM(Ours) +X +Manual Annotation +0 +100 +200 +300 +400 +500 +6000 +100- +200- +300- +400- +0 +100 +200 +300 +400 +500 +6000 +100 +X +200 +300 +400 +0 +100 +200 +300 +400 +500 +6000 +100- +200- +X +300 +400 +0 +100 +200 +300 +400 +500 +6000 +100- +200- +300- +400- +0 +100 +200 +300 +400 +500 +6000 +100 +200 +X +300 +400 +X +ManualAnnotation +0 +100 +200 +300 +400 +500 +600JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +vi +XII. SUPPLEMENTARY F: TEST SET RANKING RESULT +Table S6 summarizes the comparisons of tracking errors evaluated on the CLUST2D and AHNTU test sets. We observe that +our proposed framework achieves superior or competitive tracking performance against other state-of-the-art medical landmark +tracking methods. Note that LSDM is a simeple but effective multi-task design without complicated pre-/post-processings. +LSDM achieves with more stable performance during various tests, including gourp-wise test set result described in Sec. V-D. +TABLE S6 +QUANTITATIVE OVERALL TRACKING PERFORMANCE COMPARISON OF LSDM AGAINST OTHER STATE-OF-THE-ART METHODS ON CLUST2D AND +AHNTU TEST SET. +CLUST2D +Overall +Mean +Std +TE95th +Liu et al. [20] +0.69 +0.67 +1.57 +Williamson et al. [39] +0.74 +1.03 +1.85 +Wu et al. [19] +0.8 +1.16 +2.29 +LSDM (Ours) +1.01 +1.16 +2.21 +Shen et al. [40] +1.11 +0.91 +2.68 +Hallack et al. [43] +1.21 +3.17 +2.82 +Gomariz et al. [41] +1.34 +2.57 +2.95 +Makhinya and Golsel [42] +1.44 +2.8 +3.62 +Bharadwaj S., et al. [18] +1.60 +3.69 +4.21 +Kondo [44] +2.91 +10.52 +5.18 +Nouri D. & Rothberg A. [45] +3.35 +5.21 +14.19 +No Tracking +6.45 +5.11 +16.48 +AHNTU +Overall +Mean +Std +TE95th +LSDM (Ours) +1.80 +2.33 +4.17 +SiamFC [16] +4.11 +5.64 +10.49 +No Tracking +13.71 +11.6 +31.74 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +vii +XIII. SUPPLEMENTARY G: FAILURE CASE ANALYSIS +We report the examples of failed tracking cases with detailed analysis mentioned in Sec. V-E. Compared with other high- +signal-capacity modalities such as CT and MRI, Ultrasound imaging is less competent to handle the cases with low signal-to- +noise ratios. It is very difficult to extract local features for accurate tracking from such highly noisy environments. In Fig. S4, +we show a typical failure case, where the valid visible area of the input image is very narrow and a large part of the invalid +view is the shadow area caused by insufficient ultrasound gel. It is less visible even to human experts when the landmark moves +into the shadow area. The effect of LSDM on landmark matching and deformation estimation is limited, resulting in inaccurate +tracking performance. In the future work, we will extend LSDM and take the advantage of the shadow area segmentation +module as proposed in [46], [47], let LSDM be self-adaptive to the shadow areas. +Although LSDM has achieved stable and highly accurate results, we discover another type of factors affecting the tracking +performance, namely latency matching. This is because we built LSDM on the online tracking system. During the training +and inference stage, LSDM can only act on the current t-th frame and agnostic on frames after the t-th frame. This leads to +online latency in the estimation of the landmark by LSDM when the landmark changes in a nonlinear acceleration (shown in +Fig. S5). In the follow-up deployment, we will adapt the temporal receptive field of LSDM in both online and offline modes +so that LSDM can access the image sequences after time t to minimize the latency [48], [49], [50]. +Instance Frame +Exemplar Frame +ETH-11-1 +View Confidence +Valid Field +Valid Image +Fig. S4. Ultrasound image field-of-view confidence score visualization using [46]. We select a representative case from the CLUST2D test set. From left +to right: instance frame at frame 0, selected exemplar frame, generated field-of-view confidence score, valid field-of-view based on score threshold and high +visible image area within the valid field-of-view. As the landmark enters the invalid field-of-view, it is less visible even to human expert eyes, decreasing the +tracking performance to shadow agnostic methods like LSDM. Best viewed in color. +ETH-11-1 +Frame #0 +Frame #17 +Frame #44 +Frame #44 +Tracking Response +Frame #55 +Frame #55 +Tracking Response +Fig. S5. Visualization of tracking latency. We select a representative case from the CLUST2D training set. From top-left to bottom-right: instance frame at +frame 0, mid-interval frame at frame 17, one exemplar frame at frame 44, LSDM tracking response on frame 44, one exemplar frame at frame 55 and LSDM +tracking response on frame 55. We can observe a tracking latency within online tracking methods. Best viewed in color. + diff --git a/YNE3T4oBgHgl3EQf1wsI/content/tmp_files/load_file.txt b/YNE3T4oBgHgl3EQf1wsI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a7bdc9e5dc93b85af93d6a56e13772a908bf7542 --- /dev/null +++ b/YNE3T4oBgHgl3EQf1wsI/content/tmp_files/load_file.txt @@ -0,0 +1,1672 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf,len=1671 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 i LSDM: Long-Short Diffeomorphic Motion for Weakly-Supervised Ultrasound Landmark Tracking Zhihua Liu, Bin Yang, Yan Shen, Xuejun Ni, Huiyu Zhou Abstract—Accurate tracking of an anatomical landmark over time has been of high interests for disease assessment such as min- imally invasive surgery and tumor radiation therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Ultrasound imaging is a promising modality benefiting from low-cost and real-time acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' However, generating a precise landmark tracklet is very challenging, as attempts can be easily distorted by different interference such as landmark deformation, visual ambiguity and partial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In this paper, we propose a long-short diffeomorphic motion network, which is a multi- task framework with a learnable deformation prior to search for the plausible deformation of landmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Specifically, we design a novel diffeomorphism representation in both long and short temporal domains for delineating motion margins and reducing long-term cumulative tracking errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' To further mitigate local anatomical ambiguity, we propose an expectation maximisation motion alignment module to iteratively optimize both long and short deformation, aligning to the same directional and spatial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The proposed multi-task system can be trained in a weakly-supervised manner, which only requires few landmark annotations for tracking and zero annotation for long-short deformation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We conduct extensive experiments on two ultrasound landmark tracking datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Experimental results show that our proposed method can achieve better or competitive landmark tracking performance compared with other state-of- the-art tracking methods, with a strong generalization capability across different scanner types and different ultrasound modali- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Index Terms—Medical landmark Tracking, ultrasound imag- ing, diffeomorphic motion, long-short temporal modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' INTRODUCTION A CCURATE anatomical landmark tracking has attracted significant attention in various aspects within clinical workflows, especially in high-intensity modulated imaging and image-guided radiation therapy (RT) [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Real-time accurate anatomical landmark tracking delivers precise land- mark localization and movement estimation information in temporal-spatial domains, which provides clinicians a mea- surable therapy margin around clinical and surgical targets to increase the chance of tumor control [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Among various Zhihua Liu and Huiyu Zhou are with School of Computing and Mathemat- ical Sciences, University of Leicester, Leicester LE1 7RH, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Bin Yang is with Department of Cardiovascular Sciences, College of Life Sciences, University of Leicester, University Hospitals of Leicester NHS Trust, Leicester LE1 9HN, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Nantong-Leicester Joint Institute of Kidney Science, Department of Nephrology, Affiliated Hospital of Nantong University, Nantong, 226001, China Yan Shen, Associated Chief physician, Department of Emergency Medicine, Affiliated Hospital of Nantong University, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='20 Xisi Road, Nantong City, Jiangsu Province, 226001, China Xuejun Ni, Chief physician, Department of Medical Ultrasound, Affiliated Hospital of Nantong University, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='20 Xisi Road, Nantong City, Jiangsu Province, 226001, China Corresponding author: Huiyu Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Email: hz143@leicester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='uk ETH-05-2_2 Frame 0 Frame 127 ICR-03_3 Frame 0 Frame 2096 MED-01-1_3 Frame 0 Frame 66 CIL-01_1 Frame 0 Frame 1311 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Illustration of ultrasound medical landmark under different challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' From top to bottom: tracking under landmark deformation, acquisition noise, visual ambiguity and partial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' From left to right: reference frame, example target frame, cropped region with LSDM (green) and baseline SiamFC (yellow) tracking results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' imaging modalities, ultrasound is one of the most desirable technique benefited from low-cost, non-invasive, and real-time acquisition [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Different from high temporal-spatial resolu- tion images such as magnetic resonance (MR) or computed tomography (CT), ultrasound suffers from low signal-to-noise ratio, speckle decorrelation, spatial ambiguities and aliasing, making the small anatomical structures (such as tumor bound- ary, vessel wall) hard to be distinguished and tracked from surroundings [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Extensive research approaches have been proposed in past decades to track anatomical landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' One implementation is using invasive artificial fiducial markers, which is limited with regards to surgery implementation requirements and marker migration [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Automated learning algorithms have been investigated and achieved significant improvements [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' However, they cannot effectively handle the ultrasound sequences of poor quality, where the intensity is anisotropic in the temporal dimension, caused by sonar noise and shadowing arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='04748v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='CV] 11 Jan 2023 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 ii Frame 0 Frame 𝑡 − Δ𝑡 Frame 𝑡 Long Deformation Short Deformation Long/Short Deformation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Illustration of Long-Short Diffeomorphic Motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Top row denotes an example of the input image set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' From top-left to top-right: Frame 0 serves as the fixed template for long range deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Frame t − ∆t serves the moving target for long range deformation, also adaptive template for short range deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Frame t serves the short range target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' From bottom-left to bottom right: Long range diffeomorphism vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Short range diffeomorphism vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Illustration of joint long-short diffeomorphism motion (blue: short range deformation, yellow: long range deformation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' red: cumulative error from single range deformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Moreover, most existing tracking methods can not measure the topology changes of targets between frames, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' disconnected boundaries shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 1), which varies from patient to patient involving motions from respiratory, cardiac or body movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' These internal and external noise greatly influence the tracking performance, especially trackers rely on learning similarity between the anatomical landmark template (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' exemplar) and the target (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' As long as the ultrasound sequence becomes longer, the landmark position estimation is of more uncertainty and the accumulated tracking errors are larger [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In order to minimize the tracking errors, a system with the capability of learning deformable shapes is highly desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In this paper, we propose a novel anatomical landmark tracking system with a learnable motion prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Different from previous medical landmark tracking methods that only focused on the tracking task only, we design a multi-task tracking system using a novel diffeomorphism motion structure, namely long-short diffeomorphic motion (LSDM), to learn and adapt the non-rigid transformation between the target and the historic reference frames, generating a landmark motion prior in both long and short time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=" LSDM can effectively explore the hybrid deformation and improve the downstream tracking task's accuracy for searching the most plausible deformed landmark, eliminate noisy artifacts, which has not been studied before." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We further design an expectation maximization motion alignment (EMMA) module to fully utilize the long interval temporal directional information and short interval spatial fine- grained information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In EMMA, the short deformation serves as a latent motion variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The long deformation can be iteratively optimized by aligning to the motion representation of the same landmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The optimized short deformation can also be recomposed and then form the long diffeomorphism with updated temporal details as a close-loop regularization to avoid over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Our contribution can be summarized as follows: We propose a novel anatomical landmark tracking method by multi-tasking both tracking and diffeomorphism mo- tion learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Unlike previous methods that learn two tasks independently, we integrate the learned diffeomor- phism as a motion prior for searching the best tracking candidate with significant deformation, thus to improve the tracking accuracy under challenging medical imaging situations such as landmark tracking in noisy background within ultrasound sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' To effectively reduce the cumulative errors in long range modeling, we design a new representation structure con- taining diffeomorphism in both long and short time inter- vals (LSDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We also propose an expectation maximiza- tion motion alignment (EMMA) module to iteratively update both long and short time deformation utilizing motion direction and similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Through EMMA, both long and short diffeomorphism can be updated by alter- natively constructing and optimizing the lower bound of motion evidence, resulting in aligning deformations to the same directional and spatial representation, which further mitigates the motion biases brought by local anatomical ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' LSDM can be trained within an end-to-end weakly- supervised fashion, which only requires few landmark an- notations for tracking and zero annotation for long-short motion learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We conducted extensive experiments on public and private ultrasound videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Results show that our proposed system out-performs various fully- supervised tracking only methods with high interpretation and strong generalization across different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We first present a comprehensive review of related works (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='II) in medical image tracking, motion estimation, and point out how LSDM differs from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='III, we detail our proposed LSDM tracking network and EMMA module design with theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Experimental setups and analysis are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='IV and V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Finally, we conclude this paper in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' RELATED WORKS We first briefly review related works in medical landmark tracking and relevant technologies from nature scene object tracking in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='II-A, which can be roughly divided into registration and feature mapping based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We also discuss some preliminary works within diffeomorphic motion modeling in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' More discussion on related works can be found in Table S1, Supplementary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Medical Landmark Tracking Accurate anatomical landmark tracking provides fundamen- tal and crucial information for minimizing the spatial margins during radiation therapy, specifically using ultrasound images with substantial benefits such as fast acquisition, low cost and non-invasion [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Previous attempts on medical landmark JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 iii 1×1 Conv 1×1 Conv 1×1 Conv Kernelized Motion E-Step Motion Alignment M-Step Recompose Regularization Expectation Maximization Motion Alignment (EMMA) E-Step M-Step Expectation Maximization Motion Alignment (EMMA) AlexNet AlexNet w/ DPN Deformation Aware Cross Correlation Shared Encoder Shared Encoder Long Decoder Short Decoder Instance Exemplar Conv Encoder Conv Decoder Composition Operation Cross Correlation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Pipeline of our proposed LSDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' tracking can be roughly divided into two categories based on the matching formulation: registration based tracking and feature mapping based tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Early attempts tried to learn a registration framework for finding the spatial transformation between the template frame and the rest frames, thus the landmark position can be calculated using the affine matrix between the template and the target frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [12] tracked the landmark by matching the global and local point set between two frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Konig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [13] proposed a registration method for calculating the registered landmark position with a normalized gradient field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' These registration- based tracking frameworks focused on minimizing the global image registration errors, while the local anatomical errors cannot be corrected in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Moreover, the manually designed features cannot fully represent the landmark deformation dur- ing the longitude evolution [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Instead of implicitly calculating the landmark position using the affine matrix from global image registration, inspired by siamese networks [16], [17], recent works focus on automati- cally learning the high dimensional features and measuring the feature similarity between the exemplar and the follow-up in- stances directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The exemplar and instances extracted from the same landmark should be similar in the feature space generated from the same deep network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Bharadwaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [18] applied the basic siamese network tracking the ultrasound liver landmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' However, the network can be confused by other regions with similar visual representation and the fixed displacement prior may fail when the landmark moves out of field-of-view or is hard to be distinguished from the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' To obviate these difficulties, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [19] extended the siamese network by calibrating the intermediate features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [20] developed a cascaded network using two branches of a siamese network with different sizes of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' However, these methods suffer from learning the complex pair-wise relationship between the instance and the exemplar by iterating every frame pair, which is label intensive for medical image analysis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Also, the hierarchical feature learning network lacks meaningful insights on minimizing the tracking error, particularly on quantifying the safety margin required by radiation therapy treatment planning to deliver planned doses [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Diffeomorphic Motion Modelling Diffeomorphism describes the non-linear transformation, measuring how topology, such as components and connected boundaries, is deformed between two time stamps [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' During radiation therapy, diffeomorphic motion estimation adversely affects the planned irradiation of the target anatomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Recent research works use deep networks to learn the sta- tionary velocity field parameterized as a scaling and squaring layer to form the diffeomorphic deformation [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' These research works greatly boosted the diffeomorphism registra- tion performance, however, they all followed the pair-wise deformation learning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' a fixed frame is chosen as template for the rest of the images to be pair-wise warped, which introduces additional biases such as interpolation asymmetry and the deformation computation complexity is proportional to the length of the image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Different from previous attempts, in this work, we design a simple but effective diffeomorphism structure called long-short diffeomorphism motion (LSDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The long diffeomorphism serves as the pair-wise motion learning from a sparse image set formed by the selected frames (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' keyframes for landmark tracking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The short diffeomorphism plays a role as group-wise motion to learn the deformation between the keyframe and their previous ∆tth frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' This long-short structure can be viewed as a hybrid structure where the long diffeomorphism from a sparse set can reduce the deformation complexity for a long video sequence and the short diffeomorphism from the paired images is invertible and able to preserve anatomical topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Similar to the use of expectation maximization (EM) for learning the optimized parameters of mixture models, we propose an expectation maximization motion alignment (EMMA) module to iteratively update and find the best long- short deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Thus the drifting error caused by image JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 iv artifacts can be effectively reduced by considering deformation in both local and global contexts along the temporal dimen- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The optimized long-short diffeomorphism motion can be served as a complete prior for downstream tasks, where we show the LSDM can provide meaningful deformation information for feature extraction networks used in a relatively challenging task such as ultrasound landmark tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Problem Formulation We focus on anatomical landmark tracking within ultra- sound image sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The goal is to generate precise location estimation for subsequent frames given the starting position of the landmark, which requires a stable single object- level description with high efficacy and low complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Thus, we formulate the ultrasound anatomical landmark tracking problem following the single object tracking definition based on similarity learning [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The similarity-based tracking aims to estimate the location s of landmark (exemplar) x0 in the t-th tracking frame (instance) xs t using an optimized siamese network Φ parameterized by θ, resulting in minimiz- ing the following tracking loss function: LT r (θ) = � t ∥Φ(x0) ⋆ Φ(xs t) − ys t ∥2 (1) where ys t is the associated true Gaussian confidence map at location s in the t-th frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' ⋆ represents the correlation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' As we summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='II, previous attempts for finding the feature similarity are heavily affected by internal and external noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Moreover, previous tracking models cannot construct a morphology search space to measure a precious deformation distance between the exemplar and the instance, missing an important prior for accurate anatomical landmark tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Following multi-task setting [27], we propose a novel framework for learning a unified filter ΦU for both tracking landmark xs 0 at locations s = {s0, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=', st} and generating diffeomorphic deformation m = {m0, m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=', mt} to update the search template with most plausible diffeo- morphic transformation, a given dataset x with t frames x = {x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=', xt}: Φ⋆ U = arg max ΦU p (s, m|x, ΦU) = arg max ΦU p (s|m, x, ΦU) p (m|x, ΦU) (2) Following Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' (2), we formulate the landmark tracking as an optimization problem with diffeomorphic transform estimation as learnable equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We expect to find the optimal unified filter Φ⋆ U to minimize the landmark location tracking error LT r and the diffeomorphic transform error LM simultaneously, given a dataset x which contains t frames with a landmark (instance) xs F a fixed frame xF (here we refer the first frame x0 as xF ) as transformation template: min LT r = � t ∥ΦU (xs 0) ⋆ ΦU (xt) − yt∥2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='t LM = 0 where LM = � t ∥ΦU (x0, xt) − xt∥2 (3) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Long-Short Diffeomorphic Motion Structure A na¨ıve solution for Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' (3) is to learn a single diffeo- morphic motion that can estimate an accurate transforma- tion between template frame xF and target frame xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' This diffeomorphic prior can benefit the jointly trained tracker, where the tracker cannot only learn the feature similarity, but also search the most plausible location of instance xs 0 in the target frame xt in the motion space generated from the diffeomorphic transformation ΦU (x0, xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' However, one main drawback of this na¨ıve setting is the cumulative error within a long tracking sequence cannot be effectively corrected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' multiple adversarial response peaks with noisy motion prediction cannot be erased, resulting in a high variance of tracking trajectory along the temporal domain, which is not ideal for minimizing the anatomical landmark location margin during radiation therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Last but not least, minor motion changes cannot be detected as the unified filter tends to learn a complete motion for the whole sequence while the motion shift between a few frames could be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Inspired by classical findings in video temporal analysis [28], [29], we propose a novel diffeomorphic representation structure called long-short diffeomorphism motion (LSDM) to solve the issues listed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' LSDM contains estimation of diffeomorphic motion ΦU = {φl, φs, φT r} in frame pairs with both long time interval t and a stochastic short time interval ∆t (∆t ≪ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We follow the standard definition of diffeomorphism by assuming the latent variable z of both long and short diffeomorphism is a multivariate Gaussian distribution with zero mean and covariance σ: p (z) ∼ N(0, σ2) (4) where z is a stationary velocity field (SVF) generated by deformation field φ within time interval t ∈ [0, 1]: dφt dt = v(φ(t)) = v ◦ φ(t) (5) where ◦ is the composition operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Different from tradi- tional diffeomorphism learning given a single fixed template xF , the template of long diffeomorphism is fixed as the first frame of the frame sequence xF = x0 and the template of short diffeomorphism is sampled xF = xt−∆t based on the short time interval ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Thus, we can obtain a noisy observation of the warped images set: {x0 ◦ φl, xt−∆t ◦ φs} from a Gaussian mixture model: p(yt−∆t, yt|z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' x0, xt−∆t) ∝ N(yt−∆t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' x0 ◦ φl, σ2I)+ N(yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' xt−∆t ◦ φs, σ2I) (6) where yt−∆t and yt are the observations of the warped images given x0 and xt−∆t respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' By introducing the JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 v Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Illustration of the proposed Expectation Maximization Motion Alignment (EMMA) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Cross means element-wise multiply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' “S” means sigmoid layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' EMMA first kernelizes the input deformation, where long (orange) and short (blue) range deformation with iterative updates are shown in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The output of EMMA is the aligned long and short range deformation, which benefits the downstream trackers to search the tracking instance with significant deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' hierarchical structure of diffeomorphism in both long and short temporal domains, we are able to learn a reliable diffeomor- phism combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The long time diffeomorphism can benefit the downstream tracker to minimize the spatial search space for finding the exemplar xs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The short diffeomorphism allows us to effectively learn the preservation of short time deforma- tion changes based on adaptive updated template xt−∆t and can benefit the tracker with the updated motion trends from frame xt−∆t to frame xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Expectation Maximization Motion Alignment The long and short diffeomorphism deformation focuses on learning different parts of the deformation due to the choice of the template being different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The template of the long deformation is fixed (as x0), causing the long deformation to focus on the generalized motion learning with a long interval motion trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The template of short diffeomorphism is adapted, which leads the short deformation to seeking an average mapping based on the finding of the similarity between xt−∆t and xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Considering either long or short diffeomorphic motion only may lead to biased deformation curve estimation from a sparse partially prior (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In practice, finding corresponding image pixels over long diffeomorphic motion is not trivial at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' First of all, it is extremely difficult to identify optimal correspondences in a noisy image space over a long interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Second, based on Bayesian inference, physics driven prediction models likely suggest several similar candidates in the following image frames, which may further confuse the downstream tracking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Finally, instead of rendering corresponding counterparts in the next image frame over a long interval, we rather target at a direct approximation of image pixels over continuous short diffeomorphic motion intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' One of the possible solutions is to use the expectation- maximization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In this paper, we propose an Expec- tation Maximization Motion Alignment (EMMA) algorithm to align both long and short deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Specifically, at the nth iteration, the short motion is an extension of long motion in both spatial-temporal domains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=', the direction and distance distribution of short motion between xt−∆t and xt follows the moving inertia of long motion between x0 and xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Our Algorithm 1: Expectation Maximization based Long Short Deformation Alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Input: Generated long deformation φl and short deformation φs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Output: Aligned deformation φN l and φN s after N iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 1: Kernelize φs to ˜φs 2: while iteration stage < N do 3: φN s = φl ∗ ˜φs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 4: z = Softmax(φN s ) 5: φN l = z ∗ ˜φs 6: end while 7: φN s = φN l ∗ z 8: φN s = φN s + φs 9: φN l = φN l + φl 10: return Aligned long deformation φN l as φl, short deformation φN s as φs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' proposed EMMA follows this physical model by viewing the short motion as a latent motion vector that generates the corresponding long motion following a conditional posterior distribution p(φs|φl, θ), parameterized by θ: θ⋆ = arg max θ log p(φl|φs, θ) (7) where φs is the short motion used as latent variable and together (φl, φs) is called complete motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Our goal is that by using the iterative optimization method EMMA, we can find the optimal parameter θ∗ for generating the final long motion representation, where θ∗ for Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 7 can be solved by Theorem VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The expectation maximization motion align- ment can be converged with an optimum θ∗ within N times iteration where: log p(φl|φs, θ∗) ≥ log p(φl|φs, θn), n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' , N, N → ∞ (8) We provide the proof of Theorem VIII in Supplementary B, which leads to the E-step and M-step of EMMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Specifically, EMMA first starts with random initialized parameter θ and constructs the expectation expression at iteration n: E-step: p(φs|φl, θn) → Eφs|φl,θt [ log p(φl, φs|θ)] (9) and update the parameter for maximizing the constructed expectation: M-step: θn+1 = arg max θ Eφs|φl,θn [ log p(φs, φl|θ)] (10) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Plug-and-Play Differentiable EMMA Fomulated from E-Step (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' (9)) and M-Step (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' (10)), EMMA first kernelized short diffeomorphism φs into k seed features ˜φs = {φ1 s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=', φk s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We model the conditional distri- bution of φl and ˜φs as a Gaussian Mixture Model as the short Expectation Maximization MotionAlignment(EMMA) Iteration n-1 Iterationn Iteration n+1 1x1Conv Zo 1x1 Conv Gaussian KernelJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 vi diffeomorphism φs is an extent of φl in the temporal domain, parameterized by latent variable Z = {z1 φ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=', zk φ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' p(φs|φl) = K � k=1 zk φN(φs|φl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' µk, σk) (11) The log-likelihood of φl and φs given a fixed parameter θ can be written as: log p(φs, φl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' θ) = log � zφ p(φl, φs, zφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' θ) = log � zφ Q(zφ)p(φl, φs, zφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' θ) Q(zφ) ≥ � zφ Q(zφ) log �p(φl, φs, zφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' θ) Q(zφ) � (12) where Q(zφ) = p(φl, φs, zφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' θ) � zφ p(φl, φs, zφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' θ) = p(φl, φs, zφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' θ) p(φl, φs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' θ) = p(zφ|φl, φs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' θ) (13) Based on the Expectation-Maximization algorithm, the E- step of EMMA is to construct a motion evidence lower bound (ELBO) for log p(φs, φl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' θ) in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' (12) and the M-step aims to maximizing the ELBO w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='t θ while fixing Q(zφ), which means to maximize the likelihood of φs for the next time step given long deformation φl generated from the history, specifically: Q(zi φ) = k � i=1 p(zi φ|φl, φs, θ) zk φ = Ker(φl, ˜ φks) �k j=1 Ker(φl, ˜φj s) (14) where Ker is the kernel mapping function for both φl and φs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We apply the simple inner product function as the choice of Ker for the convenience of implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Following previous works in [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' we implement the E-step as a softmax layer for the inner product of φl and ˜φs to formulate the probabilistic distribution of Z: Z = Softmax(φl( ˜φs)⊤) (15) After we have fixed the latent variable Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' the M-step of EMMA is to maximize the probability of the appearance of kernelized short diffeomorphism feature given a previous long diffeomorphism: ˜φs = zi φφl �k j=1 zj φ (16) After the iteration of N times,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' ˜φs can be updated to an optimal representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Following the definition of diffeomor- phic learning in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' (5), the long deformation φl can be w/ Deformation Pyramid Network Conv5×5 Conv3×3 Conv1×1 Conv5×5 2 Conv3×3 Conv1×1 Conv2×2 Conv1×1 (a) Tracknet(AlexNet) Deformation Encoder (b) Deformation Partial Decoder (c) Deformation Complete Decoder Long Decoder Short Decoder Shared Decoder Conv1×1 Conv1×1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Illustration of the proposed Deformation Pyramid Network (DPN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Left (a): the deformation pyramid network transferring the motion feature (orange) as a prior to the tracking network (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Right (b): Partial motion decoder design, where long and short range deformation is modeled by two separate decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Right (c): Complete motion decoder design, where long and short range deformation is modeled by the same decoder and only decoupled at the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' re-composed using the optimized ˜φs, as the optimized short diffeomorphism aligns with the same snapshot within a single time slot: φN l = zφ ˜ φN s (17) The re-composition of φl can be viewed as a regularization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' After having finished the re-composition, EMMA achieves a close loop updating φl and φs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The algorithm of executing EMMA for N iterations can be summarized as Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Since the common frame xt−∆t is involved during EMMA, we believe that the updated φN s and φN l are aligned together, following the definition of feature alignment in similar research fields such as domain-adaptation [32] and object re-identification [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Deformation Pyramid Network Inspired by Feature Pyramid Network (FPN) [34], we take the feature from long-short diffeomorphism encoding as a prior to learn a plausible deformation for tracking an instance for the exemplar in subsequent frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Specifically, a traditional siamese network uses a shared encoder to extract features for both the exemplar and the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=" The shared weights setting encourages learning a high similarity, while ignoring the exemplar's deformation along the spatial-temporal dimension, which is significant for medical image tracking." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Here, we take the intermediate features from the deformation network as a prior for the exemplar, fused by the proposed Deformation Pyramid Network (DPN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Similar to FPN, DPN takes the deformation feature hierarchy with semantics from low to high levels, to update the exemplar features at the corresponded levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 5, the DPN follows a top-down con- struction, where the feature maps with larger size are down- sampled by large sized kernels and smaller size feature maps are down-sampled by small sized kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=" Note that there are two ways to generate long and short deformation based on how the long and short deformation share their networks' weight." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 vii The first one is called complete network, where the long and short deformation share the encoder and the decoder of the entire deformation learning network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In the complete network, the long and short deformation can only be separated at the last layer of the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Another set up is called partial network, where the long and short deformation shares the encoder of the deformation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In the partial network, the two independent decoders are used to generate long and short deformation respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We test the effects of DPN with inputs from different deformation network structures (complete vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' partial), detailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The structural test does not show any preference of complete vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' partial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' For the trade-off between computational cost and performance, we choose the complete network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' EXPERIMENTS We demonstrate the effectiveness of boosting ultrasound anatomical landmark tracking performance with multi-tasking diffeomorphism prior by conducting extensive experiments on two datasets: 2D ultrasound video dataset from public Challenge on Liver Ultrasound Tracking (CLUST2D) and private collected 2D kidney ultrasound video dataset from the Affiliated Hospital of Nantong University with normal and contrast ultrasound modality (AHNTU2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Dataset CLUST2D is composed of 43 patients with different times of acquisition, a total of 63 ultrasound videos collected from 4 different groups, where 24 videos are used for training and 39 videos are used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In the training set, the length of the 2D video sequence varies from 1075 to 5247 frames, with each frame resolution varying from 393 × 457 to 524 × 591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Landmarks to be tracked for each video subject in the training set varies from 1 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In the test set, the length of the 2D video sequence varies from 895 to 15640, with each frame resolution varying from 262 × 313 to 524 × 591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The number of landmarks to be tracked for each video in the test set varies from 1 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Training labels (landmark coordinate x, y) are provided by reliable observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Roughly 10% of the total sequence is annotated with random time intervals between two entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Test set annotation is only provided for the first frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Generated tracking results on the test set will be submitted and evaluated at the official server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We suggest the reader check [2] and official challenge website1 for more detailed information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' AHNTU2D is composed of 12 acute kidney injury patients with different times of acquisition, totally 14 ultrasound videos focused on both kidney collected from the Affiliated Hospi- tal of Nantong University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' For each video sequence, AHNTU2D contains two modalities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' normal ultrasound (AHNTU2D-N) and contrast ultrasound (AHNTU2D-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' For both AHNTU2D-N and AHNTU2D-C, we randomly allocate 6 videos for training, 4 videos for validating and 4 videos for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The spatial resolution is 550 × 900 and the temporal length varies from 280 to 3966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Landmarks to be tracked for each video sequence varies from 1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' All videos 1https://clust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='ch/ are manually labelled by one experienced physician from the Affiliated Hospital of Nantong University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The data collection followed the ethic procedure of the Affiliated Hospital of Nan- tong University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' All video sequences have been desensitized following the standard procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Evaluation Metric Euclidean distance is used for evaluating the tracking error (TE) on the ith landmark between the predicted center coor- dinate P i t and the ground truth coordinate GT i t at tth frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' TEi t = ||P i t –GT i t || (18) TE is evaluated and summarized with mean, standard deviation and 95th percentile from the official evaluation server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' For system evaluation with motion magnitude, no tracking error (NoTE) is included as standard calculation: NoTEi t = ||P i 0–GT i t || (19) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Baseline Methods We first the effectiveness of the deformation prior, EMMA and DPN with ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In-House testing is conducted as well, where the validation sequence provided by a specific hospital is unseen during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We also test the stabi- lization performance of LSDM on a randomly split training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We also present the systematic tracking performance against several state-of-the-art ultrasound landmark tracking methods, including 2D tracking methods [20], [38], [39], [19], [40] with leading performance on CLUST2D test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' As we built our tracking backbone upon the similarity learning, we also re- implement the state-of-the-art tracking method SiamFC [16] and SiamRPN [37] for further comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' RESULTS In this section, we present our experimental design and result analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We first test the effectiveness of different components within LSDM in Ablation Study (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='V-A), lead- ing to the combination of LSDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Then we evaluate the generalization and robustness against different data providers with different ultrasound scanners in In-House Test (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='V-B), which has never been carried out in previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We also compare our method against several representative tracking methods on the randomly split training set with ground truth (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='V-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Finally, we report the test set result on CLUST2D, AHNTU2D-N and AHNTU2D-C against other state-of-art methods, human expert observation and no-tracking results (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='V-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We also add failure tracking cases and fully analyze the reasons of mistracking (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='V-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Ablation Study We evaluate the impacts of different components within LSDM on the CLUST2D training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The result is shown in Supplementary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We first evaluate the deformation quality brought by different designs of complete and partial diffeomor- phism learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We then validate that EMMA can upgrade the JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 viii MED-03-2_2 Frame 0 Frame 15 Frame 491 Frame 596 Frame 1107 Frame 1652 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Qualitative comparison of different baseline models and the proposed LSDM by tracking validation on the CLUST2D training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We select a representative dataset entry to demonstrate LSDM superior tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We show the start frame at frame 0 (top-left), landmark center coordinate tracklet comparison (top-right) and selected frames with landmark tracking results (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We can observe a stable and accurate tracking performance of LSDM, while other baselines fail at accumulating tracking error through the whole sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Supplementary V summarizes other tracking results and their visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Best viewed in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' TABLE I QUANTITATIVE COMPARISONS BETWEEN LSDM AND OTHER REPRESENTATIVE TRACKING TECHNIQUES ON CLUST2D TRAINING SET WITH RESPECT TO MEAN, STANDARD DEVIATION, 95TH, MIN AND MAX TRACKING ERROR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' IN ‘FEATURE’ COLUMN, ‘M’ MEANS ‘MANUAL EXTRACTION’ AND ‘A’ MEANS ‘AUTOMATIC EXTRACTION’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Method Feature Deformation Long Short Memory Mean Std 95th Min Max KCF [35] M \x17 \x17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='67 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='09 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='92 LCT [36] M \x17 \x13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='54 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='02 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='82 SiamFC [16] A \x17 \x17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='01 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='76 SiamRPN [37] A \x17 \x17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='86 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='01 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='37 LSDM(Ours) A \x13 \x13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='01 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='92 TABLE II QUANTITATIVE GROUP-WISE COMPARISONS BETWEEN LSDM AND THE OTHER STATE-OF-THE-ART METHODS ON CLUST2D TEST SET WITH RESPECT TO MEAN, STANDARD DEVIATION AND 95TH TRACKING ERROR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' THE BEST RESULT IS SHOWN IN BOLD AND THE RUNNER-UP RESULT IS UNDERLINED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' CLUST2D CIL ETH ICR MED1 MED2 Mean Std TE95th Mean Std TE95th Mean Std TE95th Mean Std TE95th Mean Std TE95th Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [20] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='57 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='93 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='99 lower bound of long deformation by iterative updating with short deformation, leading to a precise motion prediction for tasks such as landmark tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The iteration number within EMMA during training and inference is also examined for reaching a trade-off between performance and computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Finally, we argue that, instead of learning the feature similarity directly, our proposed hybrid deformation can be injected within the tracking network using the proposed DPN for minimizing the tracking candidate searching space with a plausible deformation, which further increases the tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Complete v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Partial Deformation Decoder: We first test different designs of the deformation learning network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' As we have mentioned earlier, the deformation network takes the concatenated batch of the reference image, long and short interval images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The encoder extracts features and different decoder designs reconstruct the velocity field corresponding to different time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' A partial deformation network indicates that long and short deformation is learned from two independent decoders and a complete deformation network means that long and short deformation is learned from the shared decoder and only separated at the last convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' From results shown in Table S2 Supplementary C, we observe that the complete design deformation network outperforms the partial design with less parameters, as the complete decoder benefits from learning the hybrid interval information not only in the encoder but also in the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='EMMA Impacts: We further test the effectiveness of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='EMMA to show the benefits of iteratively optimizing the long ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 ix variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='. From Table S2 Supplementary C, we show that EMMA can upgrade the deformation learning under both complete and partial motion decoder design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' EMMA can support long short deformation to reduce the negatives of Jacobian shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' S1 Supplementary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We further examine how different iteration numbers in complete motion E/M step during training and inference shown in Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We can obtain a local optimum with a 5 iteration combination for reasonable performance without asking for further computa- tional cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Tracking with DPN: Finally, we test how our proposed system can help achieve accurate landmark tracking with long/short deformation prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' With the feature map injected into the tracking network, we obtain lower tracking errors in both mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Table S2 Supplementary C shows the quantitative result of accurate tracking from LSDM with DPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Compared to the baseline tracking method without deformation modeling and DPN fusion, LSDM can directly learn the deformation between the exemplar and the follow- up instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' By combining priors through DPN, LSDM can minimize the search space for the tracklet and find the reliable candidate as a tracking object with plausible deformation, resulting in accurate tracking in spite of visual ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In-House Validation To validate the generalization of the proposed LSDM, we manually configure the training set for in-house testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Specifically for CLUST2D, we select data samples from a specific hospital provider for testing, which is unseen for the network during the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' For AHNTU2D, we select data samples from a specific ultrasound modality for testing, which is unseen for the network during training on another ultrasound modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The result is shown in Supplementary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We aim to test whether or not a tracking method can generalize well to track the landmark within a video from the scanner it has never seen before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' As reported in Tables S4 and S5 in Supplementary D, LSDM generalizes well on videos from the new ultrasound scanner during testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' LSDM our- performs the baseline SiamFC in both mean and std in tracking errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' By learning not only the feature similarity but also the deformation estimation, LSDM can handle the domain shift when testing on new scanners and new patients, showing high robustness and great potential for various clinical workflows such as radiation therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Training Set Comparison LSDM achieves accurate tracking by learning an optimized deformation between the exemplar and follow-up instances, which can help the downstream tracking network for finding the best candidate with plausible deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' As shown in Table I, LSDM outperforms several baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Traditional correlation filtering based methods such as KCF and LCT cannot achieve satisfactory tracking performance on large- scale datasets due to various challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Even though methods like LCT incorporate historical information for updating track- ing kernels, these baselines cannot produce accurate track- lets, lacking high-dimensional representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Compared with tracking methods based on siamese networks such as SiamFC and SiamRPN, LSDM outperforms with lower tracking error in both mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Specifically, LSDM outperforms SiamRPN with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='61 lower in mean and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='88 lower in standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Recall that SiamRPN contains an extra branch for minimizing the regression loss of the tracking object location directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' This indicates that only measuring the size (such as bounding box regression in SiamRPN) of the tracking object is not enough for accurate position estimation while an optimized deformation can be used as a prior for searching objects to improve tracking accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We report an example of tracklet comparison in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 6 and additional examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' S3 Supplementary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We can observe that our proposed LSDM can track the landmark accurately within long time ultrasound sequences while the other baselines fail in all different situations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' distracted by other landmarks with a high visual similarity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 6, cannot handle the landmark deformation effectively resulted in accumulative tracking errors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' S3 Supplementary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Testset Result We report the group wise performance of LSDM on test sets in CLUST2D in Table II and total tracking error rank- ing in Table S6 Supplementary F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Compared with state-of- art methods and human labels, LSDM achieves competitive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Note that the 1st place method [20] uses heavy pre/processing techniques such as point detection and shadow removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' On the other hand, LSDM is a simple design but effective method without complex pre/preprocessing methods, while the learned deformation prior provides more insightful guiding for radiation therapy compared with other SOTA methods and no tracking, indicating the importance of accurate estimation of motion deformation during long time tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Failure Case Analysis We also report failure tracking examples caused by invalid field-of-view and lantency in system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Both the failure case analysis and examples of failing tracking cases can be found in Supplementary G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed a multi-task based tracking method with a learnable deformation (LSDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Instead of matching the feature similarity between exemplar and follow- up instances directly, we integrated the long-short diffeomor- phism temporal learning as a deformation prior to search the candidate with the most plausible deformation to achieve accu- rate landmark tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' LSDM took the full advantage of both long and short deformation by iteratively updating the estima- tion through the proposed expectation-maximization motion alignment (EMMA) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Experiments on both public and private ultrasound landmark tracking datasets demonstrated the effectiveness and generalization of LSDM for clinical workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Compared with other competing methods, LSDM achieved superior tracking accuracy with a strong generaliza- tion capability across different scanners and modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 x REFERENCES [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Verellen, M.' metadata={'source': 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Shao, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Porikli, “Clnet: A compact latent network for fast adjusting siamese trackers,” in European Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Springer, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 378–395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [50] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Yan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Lu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Yang, “Alpha-refine: Boosting tracking performance by precise bounding box estimation,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 5289–5298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 i VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' SUPPLEMENTARY A Table S1 summarizes the related works in medical landmark tracking described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' SUPPLEMENTARY B Below we present the proof of Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We first restate Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The expectation maximization motion alignment can be converged with an optimum θ∗ within the iterations of N times where: log p(φl|φs, θ∗) ≥ log p(φl|φs, θn), n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' , N, N → ∞ (1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Following standard Expectation Maximization, we have: log p(φl|θ) = log �p(φl, φs|θ) p(φs|φl, θ) � (2) by taking the expectation on both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' (2), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='t p(φs|φl, θn) at iteration n, the left part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' (2) equals to: L = � φs p(φs|φl, θn) log p(φl|θ)dφs = log p(φl|θ) (3) The right part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' (2) equals to: R = � φs p(φs|φl, θn) log p(φl, φs|θ)dφs − � φs p(φs|φl, θn) log p(φs|φl, θ)dφs (4) For the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' (4), we have: R = � φs p(φs|φl, θn) log �p(φs|φl, θn+1) p(φs|φl, θn) � ≤ log � φs p(φs|φl, θn)p(φs|φl, θn+1) p(φs|φl, θn) dφs ≤ log � φs p(φs|φl, θn+1)dφs ≤ log (1) ≤ 0 (5) By combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' (3) and (4), we have the log-likelihood of long deformation given parameter θ: log p(φl|θ) = Eq(φs) � log p(φl, φs|θ) q(φs) � � �� � Motion ELBO + � q(φs) log � q(φs) p(φl|φs, θ) � dφs � �� � KL divergence (6) Following the structural definition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' (6), EMMA first starts with randomly initialized parameter θ and construct the expectation expression at iteration n: E-step: p(φs|φl, θn) → Eφs|φl,θt [ log p(φl, φs|θ)] (7) and update the parameter for maximizing the constructed expectation: M-step: θn+1 = arg max θ Eφs|φl,θn [ log p(φs, φl|θ)] (8) The ideal physical meaning of EMMA indicates that, once θ is converged to θ∗, the approximate distribution q(φs) is equal to the true posterior p(φs|φl, θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The KL divergence in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' (6) approaches zero and the ELBO is bounded by the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Thus, the log-likelihood of long deformation is maximized, meaning EMMA can output the most plausible long deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 ii TABLE S1 SUMMARY OF MEDICAL LANDMARK TRACKING METHODS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' IN COLUMN LEARNING, ‘R’ MEANS REGISTRATION AND ‘F’ MEANS FEATURE MAPPING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' COLUMN DP STANDS FOR DEFORMATION PRIOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Authors Base Model Learning Landmark Modality DP Remarks Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [12] Grid Set R Liver US N Pro: Rigister to reference by tracking on two scale point set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: Post-process on outlier rejection needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Konig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [13] Gradient Field R Liver US Y Pro: Real time tracking with gradient estimation on deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: Cannot handle cummulative errors effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [14] Shape Model R Heart US Y Pro: Multi-View multi-scale heart shape estimation for registration and tracking Con: Not end-to-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Mannual fusion design involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Bharadwaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [18] Kalman Filter F Liver US N Pro: Template update strategy from Kalman Filter output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: Cannot estimate the landmark deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Unsatified tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [19] Siamese Network F Liver US N Pro: Coarse-to-fine training with drift correlation based on point distance Con: Cannot handle deformed landmark with partial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [20] Siamese Network F Liver US N Pro: Multi-scale tracking network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: Training performance relies on the quality of generated landmark points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Cifor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [8] Shape Model Ensemble R Liver US Y Pro: Registration based tracking using ensemble deformation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con:Lacking pricise template matching design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Royer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [9] Mechanical Simulation R Liver US N Pro: Vertex position estimation using visual information and machenical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: Heavy inference workload during long time series images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Gomariz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [41] Siamese Network F Liver US N Pro: Using previous location as a localization prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: Lacking deformation estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Makhinya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [42] Optical Flow R Liver US Y Pro: Optical flow based motion estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: Manual designed vessel features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Lacking template matching design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Shepard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [38] Block Matching F Liver US N Pro: Multi-scale block matching method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: Performance relies on local block quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Lacking deformation estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='] Motion Tracking R Heart Tagged MRI Y Pro: Forward-Backward motion modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: No explicit discriminal learning on landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='] Motion Tracking R Heart MRI Y Pro: Biomechanics-informed motion modeling Con: No historical information during motion modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Rangamani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='] CNN+RNN F Liver US N Pro: RNN location predictor based on CNN features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con:Heavy weight network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' No deformation learning on landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='] CNN+LSTM F Liver US N Pro: LSTM for location refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: Lacking interpretation during refinement process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Ha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='] Motion Tracking R Abdomen 4D MRI Y Pro: Coupled conves optimization for real time motion estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: Landmark tracking performance relies on the choice of block template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='] KCF F Liver US N Pro: Optimized KCF for real time landmark tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: Unsatified performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Lacking deformation learning on landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Wilms et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='] Block Matching R Adbomen 4D MRI Y Pro: Coarse-to-fine training with model based regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: Performance relies on block matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Williamson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [39] Distance Modeling F Liver US N Pro: Multi-template based aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: Performance relies on keypoint selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Ours Motion aware block matching network F + R Liver/Kidney US Y Pro: Hybrid motion modeling for landmark deformation matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Con: No specific strategy handling poor quality image with limited field-of-view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 iii IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' SUPPLEMENTARY C: ABLATION STUDY In this section, we show the supplemental ablation study results of LSDM, analyzed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Table S2 shows the component ablation study results tested on the CLUST2D training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Table S3 tests different impacts of different EM iteration numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' S1 shows the qualitative deformation comparison of LSDM based on different deformation module combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' TABLE S2 QUANTITATIVE RESULTS OF ABLATION STUDIES ON LSDM REGARDING COMPLETE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='S PARTIAL DEFORMATION PRIOR NETWORK, W/O EMMA AND FEATURE FUSION W/O DEFORMATION PYRAMID NETWORK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Components Metrics Complete Partial EMMA DPN TE Mean +/- Std Deformation Prior \x13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='63 +/- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='11 \x13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='69 +/- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='87 EMMA \x13 \x13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='21 +/- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='19 \x13 \x13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='56 +/- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='73 DPN \x13 \x13 \x13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='92 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='76 \x13 \x13 \x13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='81 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='98 TABLE S3 QUANTITATIVE RESULTS OF EMMA ITERATION NUMBER TEST ON CLUST2D TRAINING SET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' # EMMA Iteration TE Mean +/- Std 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='46 +/- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='72 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='81 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='98 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='93 +/- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='37 CIL-02_1 Pair S L L+S L+S w/EMMA Fixed Moving #1075 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Qualitative deformation comparison of the proposed LSDM based on different deformation module combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We select one representative dataset entriy to demonstrate superior LSDM deformation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' For each entry, we show the fixed template at frame 0 (top-left), selected target frame (bottom-left), warped images based on different deformation combination (bottom row) and visualization of determinant of jacobian matrix from a different displacement field (top row), where red indicates the determinant of jacobian is greater than 1 and blue indicates the value of the determinant of jacobian is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We can observe a progressive smooth setting with proposed long-short deformation and EMMA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Together with DPN, the LSDM learns to generate the optimal deformation for downstream tasks like tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Best viewed in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' F20 15 10 5 0 5 10JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 iv X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' SUPPLEMENTARY D: IN HOUSE TEST In this section, we show the supplemental in-house test results of LSDM, analyzed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' V-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Table S4 shows the in-house test result of LSDM and SiamFC [16] on the CLUST2D training set, where the test partition is never shown during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Table S5 shows the in-house test result of LSDM and SiamFC [16] on the CLUST2D training set, where the test modality is never shown during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' S2 shows the qualitative response comparison between LSDM and SiamFC during the in-house test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' TABLE S4 QUANTITATIVE RESULT COMPARISON BETWEEN LSDM AND THE BASELINE ON THE CLUST2D TRAINING SET WITH IN-HOUSE VALIDATION SETTING, WITH RESPECT TO MEAN, STANDARD DEVIATION AND 95TH TRACKING ERROR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In-House Partition Mean Std 95th Scanner Type SiamFC LSDM SiamFC LSDM SiamFC LSDM CIL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='82 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='63 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='49 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='81 Ultrasonix MDP ETH 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='98 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='21 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='67 Siemens Antares ICR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='76 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='64 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='76 Elekta Clarity-Ultrasonix MED1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='71 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='91 Zonare z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='one MED2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='93 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='19 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='31 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='18 DiPhAs Fraunhofer TABLE S5 QUANTITATIVE RESULT COMPARISON BETWEEN LSDM AND THE BASELINE ON THE AHNTU2D TRAINING SET WITH IN-HOUSE VALIDATION SETTING ON DIFFERENT MODALITIES, WITH RESPECT TO MEAN, STANDARD DEVIATION AND 95TH TRACKING ERROR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' AHNTU AHNTU-N AHNTU-C Mean Std TE95th Mean Std TE95th SiamFC [16] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='77 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='17 LSDM(Ours) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='76 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='72 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='83 Instance Frame Exemplar Frame SiamFC Response LSDM Response Tracking Result MED-05-1_2 ETH-03-2_1 ICR-04-2_1 Frame 0 Frame 0 Frame 0 Frame 25 Frame 55 Frame 31 Result Detail Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Qualitative landmark tracking response comparison between the baseline model (SiamFC) and our proposed LSDM on the CLUST2D training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' From left to right, each column represents the instance frame (frame 0), selected exemplar frame, SiamFC response, LSDM response, tracking result visualization and zoomed-in patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Best viewed in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 v XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' SUPPLEMENTARY E: TRAINING SET COMPARISON Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' S3 shows additional qualitative tracklet comparision between different baseline models and our proposed LSDM tested on CLUST2D training set, described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' V-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' CIL-02_1 Frame 0 Frame 49 Frame 160 Frame 394 Frame 509 Frame 850 ETH-01-2_2 Frame 0 (a) (b) Frame 67 Frame 970 Frame 1803 Frame 2555 Frame 3282 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Additional qualitative comparison of different baseline models and the proposed LSDM by tracking validation on the CLUST2D training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We select other two representative dataset entries to demonstrate LSDM superior tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We show the start frame at frame 0 (top-left), landmark center coordinate tracklet comparison (top-right) and selected frames with landmark tracking results (bottom).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='600JOURNAL OF LATEX CLASS FILES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 vi XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' SUPPLEMENTARY F: TEST SET RANKING RESULT Table S6 summarizes the comparisons of tracking errors evaluated on the CLUST2D and AHNTU test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We observe that our proposed framework achieves superior or competitive tracking performance against other state-of-the-art medical landmark tracking methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Note that LSDM is a simeple but effective multi-task design without complicated pre-/post-processings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' LSDM achieves with more stable performance during various tests, including gourp-wise test set result described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' V-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' TABLE S6 QUANTITATIVE OVERALL TRACKING PERFORMANCE COMPARISON OF LSDM AGAINST OTHER STATE-OF-THE-ART METHODS ON CLUST2D AND AHNTU TEST SET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' CLUST2D Overall Mean Std TE95th Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [20] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='57 Williamson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [39] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='85 Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [19] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='29 LSDM (Ours) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='21 Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [40] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='91 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='68 Hallack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [43] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='82 Gomariz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [41] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='34 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='57 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='95 Makhinya and Golsel [42] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='62 Bharadwaj S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [18] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='69 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='21 Kondo [44] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='91 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='18 Nouri D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' & Rothberg A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' [45] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='21 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='19 No Tracking 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='45 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='11 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='48 AHNTU Overall Mean Std TE95th LSDM (Ours) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='17 SiamFC [16] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='64 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='49 No Tracking 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='71 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='6 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content='74 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' 8, AUGUST 2015 vii XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' SUPPLEMENTARY G: FAILURE CASE ANALYSIS We report the examples of failed tracking cases with detailed analysis mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' V-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Compared with other high- signal-capacity modalities such as CT and MRI, Ultrasound imaging is less competent to handle the cases with low signal-to- noise ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' It is very difficult to extract local features for accurate tracking from such highly noisy environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' S4, we show a typical failure case, where the valid visible area of the input image is very narrow and a large part of the invalid view is the shadow area caused by insufficient ultrasound gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' It is less visible even to human experts when the landmark moves into the shadow area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' The effect of LSDM on landmark matching and deformation estimation is limited, resulting in inaccurate tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In the future work, we will extend LSDM and take the advantage of the shadow area segmentation module as proposed in [46], [47], let LSDM be self-adaptive to the shadow areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Although LSDM has achieved stable and highly accurate results, we discover another type of factors affecting the tracking performance, namely latency matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' This is because we built LSDM on the online tracking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' During the training and inference stage, LSDM can only act on the current t-th frame and agnostic on frames after the t-th frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' This leads to online latency in the estimation of the landmark by LSDM when the landmark changes in a nonlinear acceleration (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' In the follow-up deployment, we will adapt the temporal receptive field of LSDM in both online and offline modes so that LSDM can access the image sequences after time t to minimize the latency [48], [49], [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Instance Frame Exemplar Frame ETH-11-1 View Confidence Valid Field Valid Image Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Ultrasound image field-of-view confidence score visualization using [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We select a representative case from the CLUST2D test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' From left to right: instance frame at frame 0, selected exemplar frame, generated field-of-view confidence score, valid field-of-view based on score threshold and high visible image area within the valid field-of-view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' As the landmark enters the invalid field-of-view, it is less visible even to human expert eyes, decreasing the tracking performance to shadow agnostic methods like LSDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Best viewed in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' ETH-11-1 Frame #0 Frame #17 Frame #44 Frame #44 Tracking Response Frame #55 Frame #55 Tracking Response Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Visualization of tracking latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We select a representative case from the CLUST2D training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' From top-left to bottom-right: instance frame at frame 0, mid-interval frame at frame 17, one exemplar frame at frame 44, LSDM tracking response on frame 44, one exemplar frame at frame 55 and LSDM tracking response on frame 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' We can observe a tracking latency within online tracking methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} +page_content=' Best viewed in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNE3T4oBgHgl3EQf1wsI/content/2301.04748v1.pdf'} diff --git a/ctE4T4oBgHgl3EQfpA2l/content/tmp_files/2301.05189v1.pdf.txt b/ctE4T4oBgHgl3EQfpA2l/content/tmp_files/2301.05189v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..264149fd1fb986fa1ec01894aacb091f1790979e --- /dev/null +++ b/ctE4T4oBgHgl3EQfpA2l/content/tmp_files/2301.05189v1.pdf.txt @@ -0,0 +1,648 @@ +arXiv:2301.05189v1 [math-ph] 12 Jan 2023 +Complete characterization of nontrivial local conservation laws +and nonexistence of local Hamiltonian structures +for generalized Infeld–Rowlands equation +J. Vaˇs´ıˇcek +Mathematical Institute, +Silesian University in Opava, Czech Republic +E-mail: jakub.vasicek@math.slu.cz +January 12, 2023 +Abstract +We characterize all cases when a certain natural generalization of the Infeld–Rowlands equation +admits nontrivial local conservation laws of any order, and give explicit form of these conservation laws +modulo trivial ones. Furthermore, we prove that the equation under study admits no nontrivial local +Hamiltonian and symplectic structures and no nontrivial local Noether and inverse Noether operators; +the method of establishing the said nonexistence results can be readily applied to many other PDEs. +1 +Introduction +In this paper we consider a PDE in one dependent and three independent variables of the form +ut = − (uxxx + auy + f)x ≡ F. +(1) +where u = u(x, y, t), a = const, and f = f(u, ux) is a smooth function of its arguments; the subscripts +indicate partial derivatives in the usual manner. +This equation will be hereinafter referred to as the generalized Infeld–Rowlands equation, as it is a +natural generalization for the Infeld–Rowlands [12] equation that arises inter alia in the study of the +stability of the Ginzburg–Landau equation and is recovered from (1) upon setting a = 1 and f = u2 +x. +Note also that (1) with a = 1 and f = u2/2 can be seen as a weakly two-dimensional generalization of +the following form, see e.g. [10] and references therein, of the well-known spatially one-dimensional version +of the Kuramoto–Sivashinky equation +ut = −uxxxx − uxx − uux, +which is recovered from (1) upon setting a = 1 and f = u2/2 and dropping the y-dependence of u. +Below we provide inter alia a complete characterization of nontrivial local conservation laws of all +orders for (1). +Recall that conservation laws play an important role in the theory of PDEs, cf. e.g. +[13, 20, 21, 22, 23, 25, 30, 31] and references therein. For one, it is natural to require that discretizations +employed for numerical solving the PDE under study respect the known conservation laws, see e.g. [2]. +Furthermore, conservation laws can be used in the course of proving stability, existence and uniqueness +of certain kinds of solutions, cf. e.g. [3, 4], and the same applies to Hamiltonian structures and proving +stability, see e.g. [9] and references therein. +1 + +The problem of finding all inequivalent nontrivial local conservation laws for a given PDE is quite +difficult, especially in the case of more than two independent variables, and was successfully addressed +only for a rather small number of examples, cf. e.g. [7, 8, 11, 24, 27, 28] and references therein. +We are not aware of previous studies on conservation laws for the generalized Infeld–Rowlands equation +(1) or its special cases; on the other hand, the Lie point symmetries of the original Infeld–Rowlands equation +were completely characterized in [5] where it was also shown that the original Infeld–Rowlands equation +does not pass the Painlev´e test and thus is extremely unlikely to be integrable in the sense of soliton +theory; see also [26] for the differential invariant algebra associated to the above point symmetry algebra. +In the present paper we provide a complete list of all cases when (1) with a nonlinear f admits nontrivial +local conservation laws, and and for all the cases in question list these conservation laws up to the addition +of trivial ones. +Furthermore, we show that (1) admits no nontrivial local cosymmetries other than the characteristics +of local conservation laws, no local Hamiltonian structures, no local symplectic structures, and no local +Noether operators, and no local inverse Noether operators. It should be pointed out that to the best of +our knowledge, results on nonexistence of any local Hamiltonian structures or local symplectic structures +for PDEs in more than two independent variables were not encountered in the literature. Moreover, the +method of proof that we used can be readily applied to a number of other PDEs. +The rest of the article is organized as follows. In Section 2 we set the notation and recall some basic +definitions required for the rest of the text, Section 3 presents our main results whose proofs are then given +in Section 4, and Section 5 contains conclusions and discussion. +2 +Preliminaries +Here we shall present the prerequisites required in order to state and prove our main results, mostly +following [20]; cf. also [13]. +We say that a function f is local if it is smooth and depends at most on x, y, t and finitely many of +uij = ∂i+ju/∂xi∂yj where i, j ∈ { 0, 1, 2, . . .}. +From now on x, y, t and uij will be seen as coordinates on the appropriate jet space (or on the diffiety +associated with (1) in the terminology of [13]). +Let +Dx = ∂/∂x + +∞ +� +i,j=0 +ui+1,j∂/∂uij, Dy = ∂/∂y + +∞ +� +i,j=0 +ui,j+1∂/∂uij, Dt = ∂/∂t + +∞ +� +i,j=0 +Di +xDj +y(F)∂/∂uij, +(2) +denote the operators of total derivatives adapted to equation (1). +A local conservation law for (1) is an identity of the form +Dt(ρ) + Dx(σ) + Dy(ζ) = 0 +(3) +where ρ, σ, ζ are local functions, not all of which are zero. +Then ρ is called the density of the conservation law under study, and σ and ζ are known as x- and y- +flux components. +Let δ/δu denote the operator of variational derivative on local functions +δ/δu = +∞ +� +i,j=0 +(−1)i+jDi +xDj +y ◦ ∂/∂uij +Note that for any local function g the expression δg/δu contains only finitely many terms, so there are no +convergence issues. +2 + +For a local conservation law (3) its characteristic is defined as δρ/δu (in our setting this definition is +readily seen to be equivalent to the more standard one, cf. [13, 20]). +Let +Dt(˜ρ) + Dx(˜σ) + Dy(˜ζ) = 0 +(4) +be another local conservation law for (1). +Quite obviously, a linear combination of (3) and (4), namely +Dt(c1ρ + c2˜ρ) + Dx(c1σ + c2˜σ) + Dy(c1ζ + c2˜ζ) = 0 +where c1 and c2 be constants, is again a local conservation law for (1), i.e., local conservation laws for (1) +form a vector space. +A local conservation law for (1) is trivial, cf. e.g. Chapter 4 of [20], if there exist local functions α, β, γ +such that +ρ = Dx(α) − Dy(β), +σ = Dy(γ) − Dt(α), +ζ = Dt(β) − Dx(γ) +Two local conservation laws for (1) are equivalent if their difference is a trivial local conservation law. +For any local function h define its linearization, or formal Frechet derivative, as [20] +Dh = +∞ +� +i,j=0 +∂h/∂uijDi +xDj +y +(note that since h is local, the above sum is actually finite, so there are no convergence issues). +In particular, we have +DF = −D4 +x − aDyDx − Dx ◦ (fu + fuxDx), D∗ +F = −D4 +x − aDyDx + Dx ◦ (fuDx − Dx ◦ fux) +denote the linearization of the right-hand side of (1) and its formal adjoint (cf. below for the latter). +By definition G is a characteristic of local generalized symmetry for (1) if G is a local function that +satisfies +Dt(G) − DF(G) = 0 +(5) +and γ is a local cosymmetry for (1) if it is a local function that satisfies +Dt(γ) + D∗ +F(γ) = 0 +(6) +It can be easily shown that for any local conservation law (3) its characteristic is necessarily a cosym- +metry for (1) but the converse in general is not true, i.e., there could be local cosymmetries that are not +characteristics of local conservation laws. +Note that there is no loss of generality in assuming local cosymmetries, characteristics of local gen- +eralized symmetries, densities and flux components of local conservation laws, etc. for (1) to not involve +t-derivatives of u or mixed derivatives thereof involving t, cf. e.g. [20]. +For an operator of the form +L = +k +� +i=0 +l +� +j=0 +hijDi +xDj +y, +where bij are local functions, assuming that hkl ̸= 0 introduce the obvious notation k = degx L and +l = degy L, with the standard convention, cf. e.g. [20], that degx 0 = degy 0 = −∞. +For example, for F given by the right-hand side of (1) degx DF = 4 and degy DF = 1. +Here and below ◦ denotes composition of operators in total derivatives and for the above L we define +Dt(L) = +k +� +i=0 +l +� +j=0 +Dt(hij)Di +xDj +y, +3 + +while the formal adjoint L∗ for the above L is defined as +L∗ = +k +� +i=0 +l +� +j=0 +(−Dx)i(−Dy)j ◦ hij. +Definition 1 (cf. [18]). An operator of the form +N = +r +� +i=0 +s +� +j=0 +hijDi +xDj +y +where hij are local functions, is called a local Noether operator for (1), resp. a local inverse Noether +operator for (1), if +Dt(N) − DF ◦ N − N ◦ D∗ +F = 0, +resp. if +Dt(N) + DF ◦ N + N ◦ DF = 0 +The significance of these kinds of operators stems from the fact [18] that such operators map symme- +tries to cosymmetries or the other way around. More precisely, if P is a local Noether operator for (1), +then for any local cosymmetry γ of (1) we have that P(γ) is a characteristic of local generalized symmetry +for (1). Likewise, if J is a local inverse Noether operator for (1), then for any characteristic G of a local +generalized symmetry for (1) the quantity J(G) is a local cosymmetry for (1). +While any local symplectic operator for (1) is [18] automatically is a local inverse Noether operator, the +converse, generally speaking, is not true; we refer the reader to [13, 18] and references therein for further +details on symplectic operators. +Likewise, while any local Hamiltonian operator for (1) automatically is [18] a local Noether operator +for (1), the converse in general does not hold; see e.g. [13, 20, 23] and references therein for further details +on Hamiltonian operators. +Note that, just as for the recursion operators, cf. e.g. [13, 14, 18, 19, 20, 23] and references therein for +those, (inverse) Noether, Hamiltonian and symplectic operators for nonlinear PDEs are often nonlocal, +see e.g. [13, 20, 29], but in the present paper we concentrate on the local case to avoid dealing with +complicated issues of correct definition of action of such operators and passing from formal series in spirit +of [15, 16, 17, 20] to actual operators, especially since we have more than two independent variables in (1); +see, however, Remark 3. +3 +Main results +We start with the following result readily proved by straightforward computation +Proposition 1. For any smooth f(u, ux) equation (1) has infinitely many nontrivial conservation laws of +the form +Dt(Mu) + Dx((uxxx + auy + f)M) = 0 +(7) +where M is an arbitrary smooth function of y. +Moreover, in certain special cases we have additional nontrivial local conservation laws: +Proposition 2. In addition to the conservation laws from Proposition 1, Equation (1) with a ̸= 0 and +nonlinear smooth f = f(u, ux) further admits nontrivial local conservation laws not equivalent to those +from Proposition 1 if and only if f is linear in ux and one of the following holds: +4 + +i) there exist a smooth nonlinear function g = g(u) of u and constants k0 and k1 such that f = g(u)ux + +k1u + k0; +ii) there exist a smooth nonlinear function h = h(u) of u and constants c0 and c1 such that c1 ̸= 0 and +f = (c1∂h(u)/∂u + c0)ux + h(u) +The additional nontrivial local conservation laws in both cases i) and ii) have the form +Dt(ζu) + Dx (−(uxx − K1ux + q)ζx + (uxxx + auy + f − K2)ζ) + Dy(−auζx) = 0, +(8) +where for the case i) we have that K1 = 0, K2 = −k0, q = q(u) is a smooth function of u such that +∂q(u)/∂u = g(u), and +ζ = t(a∂L/∂y − k1L) + xL, +(9) +where L is an arbitrary smooth function of y, +while for the case ii) we have q(u) = c1h(u) + (c0 + 1/c2 +1)u, K1 = 1/c1, K2 = 0, and +ζ = exp(x/c1 + t(c0/c2 +1 + 1/c4 +1))F(at + c1y), +(10) +where F is an arbitrary smooth function of its argument. +It should be pointed out that even though q(u) in the case i) is defined up to the addition of an arbitrary +constant, say, K0, the constant in question shows up in (8) only in the term Dx(−K0ζx) which vanishes +since in the case under study ζ is given by (9). +Remark 1. Note that the constant k0 in the case i) is not really essential, as f in (1) stands under the +total x-derivative which annihilates the constant in question. +On the other hand, the constant c0 in the case ii) can be removed upon using the following change +of variables: pass from x to X = x − c0y/a while keeping all other independent and dependent variables +intact. +Remark 2. It is readily checked that both cases of Proposition 2 when additional conservation laws exist +can also be presented in a more uniform fashion as follows using a somewhat different notation: there exist +a smooth nonlinear function g = g(u) of u and constants ˜c0, ˜c1 and ˜c2 such that f = ux∂g/∂u+˜c1g+˜c0u+˜c2. +The (additional) nontrivial local conservation laws from Proposition 2 then still have the form (8), +where now for ˜c1 = 0 +ζ = xL + t(a∂L/∂y − ˜c0L) +where L is an arbitrary smooth function of y, and for ˜c1 ̸= 0 +ζ = exp(˜c1x + (˜c0 − ˜c3 +1)y/a)F(y/˜c1 + at), +where F is an arbitrary smooth function of its argument. +When combined, Propositions 1 and 2 provide a complete description of all cases when (1) with a ̸= 0 +and nonlinear f admits nontrivial local conservation laws, and give explicit formulas for the conservation +laws in question. +In particular, we have the following +Corollary 1. The only nontrivial local conservation laws admitted by the original Infeld–Rowlands equa- +tion, obtained from (1) upon setting a = 1 and f = u2 +x, are those from Proposition 1. +We also have two results concerning the cosymmetries; note that the first of those is stated separately +as it does not assume the nonlinearity of the equation in question +Proposition 3. All local cosymmetries of equation (1) with a ̸= 0 can depend at most on x, y and t. +5 + +Proposition 4. The only local cosymmetries admitted by (1) with a ̸= 0 and nonlinear f are characteristics +of local conservation laws listed in propositions 1 and 2. +Moreover, we have +Proposition 5. Equation (1) admits no nontrivial local Noether and inverse Noether operators. +As we have already mentioned in Section 2 any local symplectic operator for (1) is necessarily a local +inverse Noether operator and a local Hamiltonian operator for (1) is necessarily a local Noether operator, +so we can immediately establish nonexistence of local Hamiltonian and local symplectic structures for (1). +Corollary 2. Equation (1) admits no nontrivial local Hamiltonian and symplectic operators. +Remark 3. In fact, using the method of proof of Proposition 5, it is possible to show that (1) admits no +nontrivial local Noether and inverse Noether operators that can be represented as formal series of the form +r +� +i=−∞ +s +� +j=−∞ +bijDi +xDj +y +where r and s are any integers and bij are local functions. +4 +Proofs of the main results +Proof of Proposition 2 By Proposition 4, which will be proved later, the only cosymmetries admitted by +(1), are +1) M(y), where M is an arbitrary smooth function of y, for any smooth f, +2) ζ given by (9) or (10) if f is nonlinear and satisfies extra assumptions from case i) or ii) from Proposi- +tion 2. +Now recall that in our setup a characteristic of a local conservation law is necessarily a local cosymmetry +(the other way around this does not hold in general). +By the above, all cosymmetries of (1) depend at most on x, y, t, and it is immediate that to any such +cosymmetry χ = χ(x, y, t) there corresponds, up to the addition of a trivial local conservation law, a local +conservation law for (1) with the density ρ = χ(x, y, t)u, and the result readily follows upon compting the +associated flux components for the appropriate densities ρ. □ +Proof of Proposition 3 Suppose that γ is a local cosymmetry for (1), and let k = degx Dγ and l = degy Dγ. +It is immediate that to prove that γ depends at most on x, y, t is equivalent to proving that Dγ = 0. +Seeking a contradiction, assume that Dγ ̸= 0, Then obviously k ⩾ 0, and upon repeated use of (2) we +find that taking the partial derivative of (6) w.r.t. uk+4,l yields +2∂γ/∂ukl = 0, +Taking the above into account and acting by the operator of partial derivative w.r.t. uk+4,l−1 on (6) now +yields +2∂γ/∂uk,l−1 = 0, +and continuing in the same fashion we find that for all j = 0, 1, . . . , l +∂γ/∂ukj = 0, +hence we see, using the definition of Dγ, that in fact degx Dγ is at most k −1, which contradicts our initial +assumption degx Dγ = k. The only way to resolve this contradiction is to assume that Dγ = 0, and hence +γ can depend at most on x, y, t □ +6 + +Proof of Proposition 4 First of all, by Proposition 3, any local cosymmetry of (1) can depend at most on +x, y, t. +Then the condition (6) boils down to +∂γ +∂t − ∂4γ +∂x4 − a ∂2γ +∂x∂y − fux +∂2γ +∂x2 + fu +∂γ +∂x − fuxuxuxx +∂γ +∂x − fuuxux +∂γ +∂x = 0, +(11) +Differentiating (11) w.r.t. uxx yields +fuxux +∂γ +∂x = 0. +We readily see that if fuxux ̸= 0, i.e., f is not linear in ux, γ must be independent of x. Now, if γ is +independent of x, (11) becomes ∂γ/∂t = 0 and we conclude that γ is also independent of t. Hence the +only possible cosymmetry in this case is an arbitrary function M(y). +Let us now assume the function f to be linear in ux, i.e., to have the form +f = f1ux + f0, +where fi are smooth functions of u only. +The equation (11) then takes the following form: +∂γ +∂t − ∂4γ +∂x4 − a ∂2γ +∂x∂y − f1 +∂2γ +∂x2 + ∂f0 +∂u +∂γ +∂x = 0. +(12) +The structure of solutions to this equation depends on whether ∂f0/∂u, f1 and 1, considered as functions +of u, are linearly dependent or not, so we split the analysis of (12) into several cases. +Case 1: f1 is linearly independent from ∂f0/∂u and 1. +Then in order for (12) to hold we must, in particular, require that the coefficient at f1 vanishes, and +thus ∂2γ/∂x2 = 0 so +γ = γ0 + xγ1 +where γi are smooth functions of y and t. +Substituting this expression for γ back into (12) yields +x∂γ1 +∂t + ∂γ0 +∂t − a∂γ1 +∂y + ∂f0 +∂u γ1 = 0, +(13) +whence, upon equating to zero the coefficient at x, we immediately see that γ1 in fact depends on y alone. +The rest of (13) then yields +∂γ0 +∂t − a∂γ1 +∂y + ∂f0 +∂u γ1 = 0. +(14) +We now have two subcases. +Subcase 1a: ∂f0/∂u is linearly independent from 1. +Then for (14) to hold we must, in particular, equate to zero the coefficient at ∂f0/∂u in (14). Upon +doing so we have again, just as above, that γ1 = 0. Thus γ has to be independent of x and consequently, +by virtue of (14), of t. and hence the only possible cosymmetry in this case is again an arbitrary function +of y only, M(y). Thus, if ∂f0/∂u is linearly independent from 1 while f1 is linearly independent from both +∂f0/∂u and 1, there are no local cosymmetries for (1) other than those being characteristics of conservation +laws from Proposition 1. +7 + +Subcase 1b: ∂f0/∂u is linearly dependent from 1, so there are constants k0 and k1 such that f0 = k1u + k0. +Then (14) boils down to +∂γ0 +∂t − a∂γ1 +∂y + k1γ1 = 0, +As γ1 per above is independent of t, we find that the general solution of the above equation reads +γ0 = M + t +� +a∂γ1 +∂y − k1γ1 +� +where M is an arbitrary smooth function of y and so is γ1. +It is now clear that M again corresponds to the cosymmetry being the characteristic of the conservation +law from Proposition 1 while the arbitrary function γ1 which gives rise to an additional set of cosymmetries +xL + t +� +a∂L +∂y − k1L +� +(15) +where we have for convenience relabelled the arbitrary function by L instead of γ1. +It is readily checked that cosymmetries (15) are precisely characteristics (9) of the conservation laws +from case i) of Proposition 2. +Case 2: f1 is linearly dependent from 1 and ∂f0/∂u. +First of all observe that then, as (1) is nonlinear by assumption, the functions 1 and ∂f0/∂u must be +linearly independent as functions of u, otherwise f would be linear in both u and ux and so (1) would be +linear as well. +Thus, we now have +f1 = c1∂f0/∂u + c0 +(16) +where ci are arbitrary constants and ∂f0/∂u is nonconstant. +Since 1 and ∂f0/∂u must be linearly independent as per the above, we get from (12) upon separately +equating to zero the coefficients at 1 and ∂f0/∂u that +∂γ +∂t − ∂4γ +∂x4 − a ∂2γ +∂x∂y − c0 +∂2γ +∂x2 = 0. +(17) +and +∂γ +∂x = c1 +∂2γ +∂x2 . +(18) +Let us split this into two subcases : c1 = 0 and c1 ̸= 0 +Subcase 2a: c1 = 0. Then γ is independent of x by (18) and then also independent of t by (17). Thus, if +c1 = 0 then γ is an arbitrary smooth function of y alone and the only possible local cosymmetries in this +case are again those being characteristics of conservation laws from Proposition 1. +Subcase 2b: c1 ̸= 0. Then ∂2γ/∂x2 = (1/c1)∂γ/∂x, and (17) boils down to +∂γ +∂t − 1 +c3 +1 +∂γ +∂x − a ∂2γ +∂x∂y − c0 +c1 +∂γ +∂x = 0. +(19) +General smooth solution of (18) is obvious: +γ = γ0 + γ1 exp(x/c1) +where γi are arbitrary smooth functions of y and t. +8 + +Substituting this into (19) and equating to zero separately the coefficients at the two linearly indepen- +dent functions of x, 1 and exp(x/c1), yields +∂γ0/∂t = 0 +so γ0 in fact depends on y alone, and +∂γ1 +∂t − a +c1 +∂γ1 +∂y − c0c2 +1 + 1 +c4 +1 +γ1 = 0. +The general smooth solution of the above equation reads +γ1 = exp +�(c2 +1c0 + 1)t +c4 +1 +� +F (at + c1y) +where F is an arbitrary smooth function of its argument. +It is clear that γ = γ0(y) for smooth functions γ0(y) are the characteristics for the conservation laws +from Proposition 1 while +exp +� x +c1 ++ (c2 +1c0 + 1)t +c4 +1 +� +F (at + c1y) +with arbitrary smooth F are precisely characteristics (10) of conservation laws from case ii) of Proposition +2, which concludes the analysis of the subcase in question and thus of Case 2. +Summing up all the above special cases, we see that indeed all nontrivial local cosymmetries for (1) +with a ̸= 0 and nonlinear f = f(u, ux) are characteristics of nontrivial local conservation laws listed in +Propositions 1 and 2. □ +We now proceed to Proposition 5. +Proof of Proposition 5. Let P of the form +P = +r +� +i=0 +s +� +j=0 +pijDi +xDj +y +(20) +where Pij are local functions, be a local Noether operator for (1), i.e., ˜P = 0, where +˜P = Dt(P) − D∗ +F ◦ P − P ◦ DF +We readily see that the leading term of ˜P is −2prsDr+4 +x +Ds +y and since we require that ˜P = 0, this leading +term must vanish, i.e., prs = 0, and moreover prj = 0 for all j = 0, . . . , s − 1. +Continuing by replacing r by r − 1 in the above considerations and so on establishes that pij = 0 for +all i and j, so P = 0, i.e.(1) admits no nontrivial Noether operators of the form (20), which completes the +part of the proof concerning the Noether operators. +Likewise, assume that +B = +k +� +i=0 +l +� +j=0 +bijDi +xDj +y, +where bij are local functions, is a local inverse Noether operator for (1), i.e., it satisfies ˜B = 0 where +˜B = Dt(B) + D∗ +F ◦ B + B ◦ DF. +In a similar fashion as for the Noether operator case we see that bkj = 0 for all j = 0, . . . , l and then that +bij = 0 for all i and j, so B = 0 and the result follows. □ +9 + +5 +Conclusions and discussion +In the previous sections we gave a complete characterisation of all cases when the generalized Infeld– +Rowlands equation (1) with a ̸= 0 and nonlinear f admits nontrivial local conservation laws and have +listed all such inequivalent local conservation laws of all orders modulo the addition of trivial ones. +It turned out in particular that equation (1) for any a and f admits an infinite set of nontrivial local +conservation laws parameterized by an arbitrary function of y as described in Proposition 1. +If f is nonlinear then, as shown in Proposition 2, nontrivial local conservation laws beyond those from +Proposition 1 can exist only provided f is linear in ux and moreover has a special form given in case i) or +ii) of Proposition 2. +The conservation laws that we found have many potential applications including e.g. in the numerical +simulation for (1), cf. the discussion in Section 1. +In this connection we also point out that the conservation laws from Proposition 1 because of their +special form have one more possible application. Namely, using these, it is possible to introduce a nonlocal +potential, say w, for (1) defined by the formulas +wx = u, +wt = −(uxxx + auy + f) +(21) +and it could be of interest to study nonlocal symmetries, nonlocal conservation laws etc. involving this +potential; this is, however, beyond the scope of the present paper whose focus is on local objects and +structures (cf. e.g. [13] and references therein for a general introduction to nonlocal objects in the geometric +theory of PDEs). +Furthermore, we have shown that the only local cosymmetries admitted by (1) with nonlinear f are +the characteristics of the above conservation laws from Propositions 1 and 2. +Note that it could be of interest to study symmetries of (1) with nonlinear f and, in particular, to find +out whether the cases from Proposition 2 when (1) admits additional conservation laws are distinguished +in some fashion from the symmetry point of view as well. +Finally, we have shown that (1) admits no nontrivial local Noether operators and local inverse Noether +operators and hence a fortiori no nontrivial local Hamiltonian and symplectic structures. While similar +results are known for the case of evolution equations in two independent variables, see e.g. [6, 15] and +references therein, to the best of our knowledge this is the first result of the kind for the case of more than +two independent variables, and the same method can be readily applied to establish analogous results for +many other equations and systems, like e.g. the degenerate Burgers equation from [27]. +Acknowledgements +This research was supported by the Specific Research grant SGS/13/2020 of Silesian University in Opava. +I would like to thank Artur Sergyeyev for stimulating discussions and valuable comments. +A significant part of the computations in the paper was performed using the computer algebra package +Jets [1] whose use is hereby gratefully acknowledged. +References +[1] H. Baran, M. Marvan, Jets. A software for differential calculus on jet spaces and diffieties, available online at +http://jets.math.slu.cz +[2] A. Bhatt, B.E. 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Phys. 183 (2023), 104702 +11 + diff --git a/ctE4T4oBgHgl3EQfpA2l/content/tmp_files/load_file.txt b/ctE4T4oBgHgl3EQfpA2l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a206a82db0f2b5cb3da9f87532772881537568bb --- /dev/null +++ b/ctE4T4oBgHgl3EQfpA2l/content/tmp_files/load_file.txt @@ -0,0 +1,454 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf,len=453 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='05189v1 [math-ph] 12 Jan 2023 Complete characterization of nontrivial local conservation laws and nonexistence of local Hamiltonian structures for generalized Infeld–Rowlands equation J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Vaˇs´ıˇcek Mathematical Institute, Silesian University in Opava, Czech Republic E-mail: jakub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='vasicek@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='slu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='cz January 12, 2023 Abstract We characterize all cases when a certain natural generalization of the Infeld–Rowlands equation admits nontrivial local conservation laws of any order, and give explicit form of these conservation laws modulo trivial ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Furthermore, we prove that the equation under study admits no nontrivial local Hamiltonian and symplectic structures and no nontrivial local Noether and inverse Noether operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' the method of establishing the said nonexistence results can be readily applied to many other PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' 1 Introduction In this paper we consider a PDE in one dependent and three independent variables of the form ut = − (uxxx + auy + f)x ≡ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' (1) where u = u(x, y, t), a = const, and f = f(u, ux) is a smooth function of its arguments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' the subscripts indicate partial derivatives in the usual manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' This equation will be hereinafter referred to as the generalized Infeld–Rowlands equation, as it is a natural generalization for the Infeld–Rowlands [12] equation that arises inter alia in the study of the stability of the Ginzburg–Landau equation and is recovered from (1) upon setting a = 1 and f = u2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Note also that (1) with a = 1 and f = u2/2 can be seen as a weakly two-dimensional generalization of the following form, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [10] and references therein, of the well-known spatially one-dimensional version of the Kuramoto–Sivashinky equation ut = −uxxxx − uxx − uux, which is recovered from (1) upon setting a = 1 and f = u2/2 and dropping the y-dependence of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Below we provide inter alia a complete characterization of nontrivial local conservation laws of all orders for (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Recall that conservation laws play an important role in the theory of PDEs, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [13, 20, 21, 22, 23, 25, 30, 31] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' For one, it is natural to require that discretizations employed for numerical solving the PDE under study respect the known conservation laws, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Furthermore, conservation laws can be used in the course of proving stability, existence and uniqueness of certain kinds of solutions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [3, 4], and the same applies to Hamiltonian structures and proving stability, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [9] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' 1 The problem of finding all inequivalent nontrivial local conservation laws for a given PDE is quite difficult, especially in the case of more than two independent variables, and was successfully addressed only for a rather small number of examples, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [7, 8, 11, 24, 27, 28] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' We are not aware of previous studies on conservation laws for the generalized Infeld–Rowlands equation (1) or its special cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' on the other hand, the Lie point symmetries of the original Infeld–Rowlands equation were completely characterized in [5] where it was also shown that the original Infeld–Rowlands equation does not pass the Painlev´e test and thus is extremely unlikely to be integrable in the sense of soliton theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' see also [26] for the differential invariant algebra associated to the above point symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' In the present paper we provide a complete list of all cases when (1) with a nonlinear f admits nontrivial local conservation laws, and and for all the cases in question list these conservation laws up to the addition of trivial ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Furthermore, we show that (1) admits no nontrivial local cosymmetries other than the characteristics of local conservation laws, no local Hamiltonian structures, no local symplectic structures, and no local Noether operators, and no local inverse Noether operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' It should be pointed out that to the best of our knowledge, results on nonexistence of any local Hamiltonian structures or local symplectic structures for PDEs in more than two independent variables were not encountered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Moreover, the method of proof that we used can be readily applied to a number of other PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' The rest of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' In Section 2 we set the notation and recall some basic definitions required for the rest of the text, Section 3 presents our main results whose proofs are then given in Section 4, and Section 5 contains conclusions and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' 2 Preliminaries Here we shall present the prerequisites required in order to state and prove our main results, mostly following [20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' also [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' We say that a function f is local if it is smooth and depends at most on x, y, t and finitely many of uij = ∂i+ju/∂xi∂yj where i, j ∈ { 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' From now on x, y, t and uij will be seen as coordinates on the appropriate jet space (or on the diffiety associated with (1) in the terminology of [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Let Dx = ∂/∂x + ∞ � i,j=0 ui+1,j∂/∂uij, Dy = ∂/∂y + ∞ � i,j=0 ui,j+1∂/∂uij, Dt = ∂/∂t + ∞ � i,j=0 Di xDj y(F)∂/∂uij, (2) denote the operators of total derivatives adapted to equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' A local conservation law for (1) is an identity of the form Dt(ρ) + Dx(σ) + Dy(ζ) = 0 (3) where ρ, σ, ζ are local functions, not all of which are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Then ρ is called the density of the conservation law under study, and σ and ζ are known as x- and y- flux components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Let δ/δu denote the operator of variational derivative on local functions δ/δu = ∞ � i,j=0 (−1)i+jDi xDj y ◦ ∂/∂uij Note that for any local function g the expression δg/δu contains only finitely many terms, so there are no convergence issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' 2 For a local conservation law (3) its characteristic is defined as δρ/δu (in our setting this definition is readily seen to be equivalent to the more standard one, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [13, 20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Let Dt(˜ρ) + Dx(˜σ) + Dy(˜ζ) = 0 (4) be another local conservation law for (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Quite obviously, a linear combination of (3) and (4), namely Dt(c1ρ + c2˜ρ) + Dx(c1σ + c2˜σ) + Dy(c1ζ + c2˜ζ) = 0 where c1 and c2 be constants, is again a local conservation law for (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=', local conservation laws for (1) form a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' A local conservation law for (1) is trivial, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Chapter 4 of [20], if there exist local functions α, β, γ such that ρ = Dx(α) − Dy(β), σ = Dy(γ) − Dt(α), ζ = Dt(β) − Dx(γ) Two local conservation laws for (1) are equivalent if their difference is a trivial local conservation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' For any local function h define its linearization, or formal Frechet derivative, as [20] Dh = ∞ � i,j=0 ∂h/∂uijDi xDj y (note that since h is local, the above sum is actually finite, so there are no convergence issues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' In particular, we have DF = −D4 x − aDyDx − Dx ◦ (fu + fuxDx), D∗ F = −D4 x − aDyDx + Dx ◦ (fuDx − Dx ◦ fux) denote the linearization of the right-hand side of (1) and its formal adjoint (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' below for the latter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' By definition G is a characteristic of local generalized symmetry for (1) if G is a local function that satisfies Dt(G) − DF(G) = 0 (5) and γ is a local cosymmetry for (1) if it is a local function that satisfies Dt(γ) + D∗ F(γ) = 0 (6) It can be easily shown that for any local conservation law (3) its characteristic is necessarily a cosym- metry for (1) but the converse in general is not true, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=', there could be local cosymmetries that are not characteristics of local conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Note that there is no loss of generality in assuming local cosymmetries, characteristics of local gen- eralized symmetries, densities and flux components of local conservation laws, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' for (1) to not involve t-derivatives of u or mixed derivatives thereof involving t, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' For an operator of the form L = k � i=0 l � j=0 hijDi xDj y, where bij are local functions, assuming that hkl ̸= 0 introduce the obvious notation k = degx L and l = degy L, with the standard convention, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [20], that degx 0 = degy 0 = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' For example, for F given by the right-hand side of (1) degx DF = 4 and degy DF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Here and below ◦ denotes composition of operators in total derivatives and for the above L we define Dt(L) = k � i=0 l � j=0 Dt(hij)Di xDj y, 3 while the formal adjoint L∗ for the above L is defined as L∗ = k � i=0 l � j=0 (−Dx)i(−Dy)j ◦ hij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Definition 1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' An operator of the form N = r � i=0 s � j=0 hijDi xDj y where hij are local functions, is called a local Noether operator for (1), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' a local inverse Noether operator for (1), if Dt(N) − DF ◦ N − N ◦ D∗ F = 0, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' if Dt(N) + DF ◦ N + N ◦ DF = 0 The significance of these kinds of operators stems from the fact [18] that such operators map symme- tries to cosymmetries or the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' More precisely, if P is a local Noether operator for (1), then for any local cosymmetry γ of (1) we have that P(γ) is a characteristic of local generalized symmetry for (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Likewise, if J is a local inverse Noether operator for (1), then for any characteristic G of a local generalized symmetry for (1) the quantity J(G) is a local cosymmetry for (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' While any local symplectic operator for (1) is [18] automatically is a local inverse Noether operator, the converse, generally speaking, is not true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' we refer the reader to [13, 18] and references therein for further details on symplectic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Likewise, while any local Hamiltonian operator for (1) automatically is [18] a local Noether operator for (1), the converse in general does not hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [13, 20, 23] and references therein for further details on Hamiltonian operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Note that, just as for the recursion operators, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [13, 14, 18, 19, 20, 23] and references therein for those, (inverse) Noether, Hamiltonian and symplectic operators for nonlinear PDEs are often nonlocal, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [13, 20, 29], but in the present paper we concentrate on the local case to avoid dealing with complicated issues of correct definition of action of such operators and passing from formal series in spirit of [15, 16, 17, 20] to actual operators, especially since we have more than two independent variables in (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' see, however, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' 3 Main results We start with the following result readily proved by straightforward computation Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' For any smooth f(u, ux) equation (1) has infinitely many nontrivial conservation laws of the form Dt(Mu) + Dx((uxxx + auy + f)M) = 0 (7) where M is an arbitrary smooth function of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Moreover, in certain special cases we have additional nontrivial local conservation laws: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' In addition to the conservation laws from Proposition 1, Equation (1) with a ̸= 0 and nonlinear smooth f = f(u, ux) further admits nontrivial local conservation laws not equivalent to those from Proposition 1 if and only if f is linear in ux and one of the following holds: 4 i) there exist a smooth nonlinear function g = g(u) of u and constants k0 and k1 such that f = g(u)ux + k1u + k0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' ii) there exist a smooth nonlinear function h = h(u) of u and constants c0 and c1 such that c1 ̸= 0 and f = (c1∂h(u)/∂u + c0)ux + h(u) The additional nontrivial local conservation laws in both cases i) and ii) have the form Dt(ζu) + Dx (−(uxx − K1ux + q)ζx + (uxxx + auy + f − K2)ζ) + Dy(−auζx) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' (8) where for the case i) we have that K1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' K2 = −k0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' q = q(u) is a smooth function of u such that ∂q(u)/∂u = g(u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' and ζ = t(a∂L/∂y − k1L) + xL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' (9) where L is an arbitrary smooth function of y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' while for the case ii) we have q(u) = c1h(u) + (c0 + 1/c2 1)u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' K1 = 1/c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' K2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' and ζ = exp(x/c1 + t(c0/c2 1 + 1/c4 1))F(at + c1y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' (10) where F is an arbitrary smooth function of its argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' It should be pointed out that even though q(u) in the case i) is defined up to the addition of an arbitrary constant, say, K0, the constant in question shows up in (8) only in the term Dx(−K0ζx) which vanishes since in the case under study ζ is given by (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Note that the constant k0 in the case i) is not really essential, as f in (1) stands under the total x-derivative which annihilates the constant in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' On the other hand, the constant c0 in the case ii) can be removed upon using the following change of variables: pass from x to X = x − c0y/a while keeping all other independent and dependent variables intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' It is readily checked that both cases of Proposition 2 when additional conservation laws exist can also be presented in a more uniform fashion as follows using a somewhat different notation: there exist a smooth nonlinear function g = g(u) of u and constants ˜c0, ˜c1 and ˜c2 such that f = ux∂g/∂u+˜c1g+˜c0u+˜c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' The (additional) nontrivial local conservation laws from Proposition 2 then still have the form (8), where now for ˜c1 = 0 ζ = xL + t(a∂L/∂y − ˜c0L) where L is an arbitrary smooth function of y, and for ˜c1 ̸= 0 ζ = exp(˜c1x + (˜c0 − ˜c3 1)y/a)F(y/˜c1 + at), where F is an arbitrary smooth function of its argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' When combined, Propositions 1 and 2 provide a complete description of all cases when (1) with a ̸= 0 and nonlinear f admits nontrivial local conservation laws, and give explicit formulas for the conservation laws in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' In particular, we have the following Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' The only nontrivial local conservation laws admitted by the original Infeld–Rowlands equa- tion, obtained from (1) upon setting a = 1 and f = u2 x, are those from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' We also have two results concerning the cosymmetries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' note that the first of those is stated separately as it does not assume the nonlinearity of the equation in question Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' All local cosymmetries of equation (1) with a ̸= 0 can depend at most on x, y and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' 5 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' The only local cosymmetries admitted by (1) with a ̸= 0 and nonlinear f are characteristics of local conservation laws listed in propositions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Moreover, we have Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Equation (1) admits no nontrivial local Noether and inverse Noether operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' As we have already mentioned in Section 2 any local symplectic operator for (1) is necessarily a local inverse Noether operator and a local Hamiltonian operator for (1) is necessarily a local Noether operator, so we can immediately establish nonexistence of local Hamiltonian and local symplectic structures for (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Equation (1) admits no nontrivial local Hamiltonian and symplectic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' In fact, using the method of proof of Proposition 5, it is possible to show that (1) admits no nontrivial local Noether and inverse Noether operators that can be represented as formal series of the form r � i=−∞ s � j=−∞ bijDi xDj y where r and s are any integers and bij are local functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' 4 Proofs of the main results Proof of Proposition 2 By Proposition 4, which will be proved later, the only cosymmetries admitted by (1), are 1) M(y), where M is an arbitrary smooth function of y, for any smooth f, 2) ζ given by (9) or (10) if f is nonlinear and satisfies extra assumptions from case i) or ii) from Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Now recall that in our setup a characteristic of a local conservation law is necessarily a local cosymmetry (the other way around this does not hold in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' By the above, all cosymmetries of (1) depend at most on x, y, t, and it is immediate that to any such cosymmetry χ = χ(x, y, t) there corresponds, up to the addition of a trivial local conservation law, a local conservation law for (1) with the density ρ = χ(x, y, t)u, and the result readily follows upon compting the associated flux components for the appropriate densities ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' □ Proof of Proposition 3 Suppose that γ is a local cosymmetry for (1), and let k = degx Dγ and l = degy Dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' It is immediate that to prove that γ depends at most on x, y, t is equivalent to proving that Dγ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Seeking a contradiction, assume that Dγ ̸= 0, Then obviously k ⩾ 0, and upon repeated use of (2) we find that taking the partial derivative of (6) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' uk+4,l yields 2∂γ/∂ukl = 0, Taking the above into account and acting by the operator of partial derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' uk+4,l−1 on (6) now yields 2∂γ/∂uk,l−1 = 0, and continuing in the same fashion we find that for all j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' , l ∂γ/∂ukj = 0, hence we see, using the definition of Dγ, that in fact degx Dγ is at most k −1, which contradicts our initial assumption degx Dγ = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' The only way to resolve this contradiction is to assume that Dγ = 0, and hence γ can depend at most on x, y, t □ 6 Proof of Proposition 4 First of all, by Proposition 3, any local cosymmetry of (1) can depend at most on x, y, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Then the condition (6) boils down to ∂γ ∂t − ∂4γ ∂x4 − a ∂2γ ∂x∂y − fux ∂2γ ∂x2 + fu ∂γ ∂x − fuxuxuxx ∂γ ∂x − fuuxux ∂γ ∂x = 0, (11) Differentiating (11) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' uxx yields fuxux ∂γ ∂x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' We readily see that if fuxux ̸= 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=', f is not linear in ux, γ must be independent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Now, if γ is independent of x, (11) becomes ∂γ/∂t = 0 and we conclude that γ is also independent of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Hence the only possible cosymmetry in this case is an arbitrary function M(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Let us now assume the function f to be linear in ux, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=', to have the form f = f1ux + f0, where fi are smooth functions of u only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' The equation (11) then takes the following form: ∂γ ∂t − ∂4γ ∂x4 − a ∂2γ ∂x∂y − f1 ∂2γ ∂x2 + ∂f0 ∂u ∂γ ∂x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' (12) The structure of solutions to this equation depends on whether ∂f0/∂u, f1 and 1, considered as functions of u, are linearly dependent or not, so we split the analysis of (12) into several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Case 1: f1 is linearly independent from ∂f0/∂u and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Then in order for (12) to hold we must, in particular, require that the coefficient at f1 vanishes, and thus ∂2γ/∂x2 = 0 so γ = γ0 + xγ1 where γi are smooth functions of y and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Substituting this expression for γ back into (12) yields x∂γ1 ∂t + ∂γ0 ∂t − a∂γ1 ∂y + ∂f0 ∂u γ1 = 0, (13) whence, upon equating to zero the coefficient at x, we immediately see that γ1 in fact depends on y alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' The rest of (13) then yields ∂γ0 ∂t − a∂γ1 ∂y + ∂f0 ∂u γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' (14) We now have two subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Subcase 1a: ∂f0/∂u is linearly independent from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Then for (14) to hold we must, in particular, equate to zero the coefficient at ∂f0/∂u in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Upon doing so we have again, just as above, that γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Thus γ has to be independent of x and consequently, by virtue of (14), of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' and hence the only possible cosymmetry in this case is again an arbitrary function of y only, M(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Thus, if ∂f0/∂u is linearly independent from 1 while f1 is linearly independent from both ∂f0/∂u and 1, there are no local cosymmetries for (1) other than those being characteristics of conservation laws from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' 7 Subcase 1b: ∂f0/∂u is linearly dependent from 1, so there are constants k0 and k1 such that f0 = k1u + k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Then (14) boils down to ∂γ0 ∂t − a∂γ1 ∂y + k1γ1 = 0, As γ1 per above is independent of t, we find that the general solution of the above equation reads γ0 = M + t � a∂γ1 ∂y − k1γ1 � where M is an arbitrary smooth function of y and so is γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' It is now clear that M again corresponds to the cosymmetry being the characteristic of the conservation law from Proposition 1 while the arbitrary function γ1 which gives rise to an additional set of cosymmetries xL + t � a∂L ∂y − k1L � (15) where we have for convenience relabelled the arbitrary function by L instead of γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' It is readily checked that cosymmetries (15) are precisely characteristics (9) of the conservation laws from case i) of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Case 2: f1 is linearly dependent from 1 and ∂f0/∂u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' First of all observe that then, as (1) is nonlinear by assumption, the functions 1 and ∂f0/∂u must be linearly independent as functions of u, otherwise f would be linear in both u and ux and so (1) would be linear as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Thus, we now have f1 = c1∂f0/∂u + c0 (16) where ci are arbitrary constants and ∂f0/∂u is nonconstant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Since 1 and ∂f0/∂u must be linearly independent as per the above, we get from (12) upon separately equating to zero the coefficients at 1 and ∂f0/∂u that ∂γ ∂t − ∂4γ ∂x4 − a ∂2γ ∂x∂y − c0 ∂2γ ∂x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' (17) and ∂γ ∂x = c1 ∂2γ ∂x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' (18) Let us split this into two subcases : c1 = 0 and c1 ̸= 0 Subcase 2a: c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Then γ is independent of x by (18) and then also independent of t by (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Thus, if c1 = 0 then γ is an arbitrary smooth function of y alone and the only possible local cosymmetries in this case are again those being characteristics of conservation laws from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Subcase 2b: c1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Then ∂2γ/∂x2 = (1/c1)∂γ/∂x, and (17) boils down to ∂γ ∂t − 1 c3 1 ∂γ ∂x − a ∂2γ ∂x∂y − c0 c1 ∂γ ∂x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' (19) General smooth solution of (18) is obvious: γ = γ0 + γ1 exp(x/c1) where γi are arbitrary smooth functions of y and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' 8 Substituting this into (19) and equating to zero separately the coefficients at the two linearly indepen- dent functions of x, 1 and exp(x/c1), yields ∂γ0/∂t = 0 so γ0 in fact depends on y alone, and ∂γ1 ∂t − a c1 ∂γ1 ∂y − c0c2 1 + 1 c4 1 γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' The general smooth solution of the above equation reads γ1 = exp �(c2 1c0 + 1)t c4 1 � F (at + c1y) where F is an arbitrary smooth function of its argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' It is clear that γ = γ0(y) for smooth functions γ0(y) are the characteristics for the conservation laws from Proposition 1 while exp � x c1 + (c2 1c0 + 1)t c4 1 � F (at + c1y) with arbitrary smooth F are precisely characteristics (10) of conservation laws from case ii) of Proposition 2, which concludes the analysis of the subcase in question and thus of Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Summing up all the above special cases, we see that indeed all nontrivial local cosymmetries for (1) with a ̸= 0 and nonlinear f = f(u, ux) are characteristics of nontrivial local conservation laws listed in Propositions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' □ We now proceed to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Let P of the form P = r � i=0 s � j=0 pijDi xDj y (20) where Pij are local functions, be a local Noether operator for (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=', ˜P = 0, where ˜P = Dt(P) − D∗ F ◦ P − P ◦ DF We readily see that the leading term of ˜P is −2prsDr+4 x Ds y and since we require that ˜P = 0, this leading term must vanish, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=', prs = 0, and moreover prj = 0 for all j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' , s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Continuing by replacing r by r − 1 in the above considerations and so on establishes that pij = 0 for all i and j, so P = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' (1) admits no nontrivial Noether operators of the form (20), which completes the part of the proof concerning the Noether operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Likewise, assume that B = k � i=0 l � j=0 bijDi xDj y, where bij are local functions, is a local inverse Noether operator for (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=', it satisfies ˜B = 0 where ˜B = Dt(B) + D∗ F ◦ B + B ◦ DF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' In a similar fashion as for the Noether operator case we see that bkj = 0 for all j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' , l and then that bij = 0 for all i and j, so B = 0 and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' □ 9 5 Conclusions and discussion In the previous sections we gave a complete characterisation of all cases when the generalized Infeld– Rowlands equation (1) with a ̸= 0 and nonlinear f admits nontrivial local conservation laws and have listed all such inequivalent local conservation laws of all orders modulo the addition of trivial ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' It turned out in particular that equation (1) for any a and f admits an infinite set of nontrivial local conservation laws parameterized by an arbitrary function of y as described in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' If f is nonlinear then, as shown in Proposition 2, nontrivial local conservation laws beyond those from Proposition 1 can exist only provided f is linear in ux and moreover has a special form given in case i) or ii) of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' The conservation laws that we found have many potential applications including e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' in the numerical simulation for (1), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' the discussion in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' In this connection we also point out that the conservation laws from Proposition 1 because of their special form have one more possible application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Namely, using these, it is possible to introduce a nonlocal potential, say w, for (1) defined by the formulas wx = u, wt = −(uxxx + auy + f) (21) and it could be of interest to study nonlocal symmetries, nonlocal conservation laws etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' involving this potential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' this is, however, beyond the scope of the present paper whose focus is on local objects and structures (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [13] and references therein for a general introduction to nonlocal objects in the geometric theory of PDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Furthermore, we have shown that the only local cosymmetries admitted by (1) with nonlinear f are the characteristics of the above conservation laws from Propositions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Note that it could be of interest to study symmetries of (1) with nonlinear f and, in particular, to find out whether the cases from Proposition 2 when (1) admits additional conservation laws are distinguished in some fashion from the symmetry point of view as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Finally, we have shown that (1) admits no nontrivial local Noether operators and local inverse Noether operators and hence a fortiori no nontrivial local Hamiltonian and symplectic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' While similar results are known for the case of evolution equations in two independent variables, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' [6, 15] and references therein, to the best of our knowledge this is the first result of the kind for the case of more than two independent variables, and the same method can be readily applied to establish analogous results for many other equations and systems, like e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' the degenerate Burgers equation from [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' Acknowledgements This research was supported by the Specific Research grant SGS/13/2020 of Silesian University in Opava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' I would like to thank Artur Sergyeyev for stimulating discussions and valuable comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' A significant part of the computations in the paper was performed using the computer algebra package Jets [1] whose use is hereby gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' References [1] H.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} +page_content=' 183 (2023), 104702 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE4T4oBgHgl3EQfpA2l/content/2301.05189v1.pdf'} diff --git a/e9AzT4oBgHgl3EQfoP2a/content/tmp_files/2301.01594v1.pdf.txt b/e9AzT4oBgHgl3EQfoP2a/content/tmp_files/2301.01594v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..63ad84d9ee8466b0f9dd47f5f8ebd700eff415c8 --- /dev/null +++ b/e9AzT4oBgHgl3EQfoP2a/content/tmp_files/2301.01594v1.pdf.txt @@ -0,0 +1,681 @@ +Finding Needles in Haystack: Formal Generative +Models for Efficient Massive Parallel Simulations +Osama Maqbool +Institute of Man-Machine-Interaction +RWTH Aachen University +Aachen, Germany +maqbool@mmi.rwth-aachen.de +J¨urgen Roßmann +Institute for Man-Machine Interaction +RWTH Aachen University +Aachen, Germany +rossmann@mmi.rwth-aachen.de +Abstract—The increase in complexity of autonomous systems +is accompanied by a need of data-driven development and +validation strategies. Advances in computer graphics and cloud +clusters have opened the way to massive parallel high fidelity sim- +ulations to qualitatively address the large number of operational +scenarios. However, exploration of all possible scenarios is still +prohibitively expensive and outcomes of scenarios are generally +unknown apriori. To this end, the authors propose a method +based on bayesian optimization to efficiently learn generative +models on scenarios that would deliver desired outcomes (e.g. +collisions) with high probability. The methodology is integrated +in an end-to-end framework, which uses the OpenSCENARIO +standard to describe scenarios, and deploys highly configurable +digital twins of the scenario participants on a Virtual Test Bed +cluster. +Index Terms—massive parallel simulations, bayesian optimiza- +tion, virtual test beds, experimentable digital twins +I. INTRODUCTION +The advent of intelligent vehicles has brought with it +increasing levels of system complexity. Vehicles with ma- +chine learning components additionally incorporate black- +boxes and uncertainty within the system, transforming an +already difficult problem to a non-deterministic one. These +systems therefore require data-driven strategies in addition to +classical approaches to deliver statistical metrics on safety and +reliability. Simulations offer a natural supplement to real-world +tests, allowing reproduction of expensive or dangerous sce- +narios virtually and being scaled as needed. Advancements in +computer graphics has made high fidelity simulations possible, +which are especially beneficial for generating large volumes +of realistic sensor data required for machine learning based +perception systems. More recently, the availability of cloud +computing resources, e.g. Microsoft Azure, has offset the +procurement effort of on-premise compute clusters enabling +large-scale parallel simulations for a wider community. +Nevertheless, the space of all possible operational scenarios +remains prohibitively large, and usage of on-premise or cloud- +based compute clusters is additionally a cost-incurring process. +This warrants a motivation to only simulate scenarios that +would be meaningful to the actual development process. For +This work is part of the project “KImaDiZ”, supported by the German +Aerospace Center (DLR) with funds of the German Federal Ministry of +Economics and Technology (BMWi), support code 50 RA 1934. +Fig. 1: Visualizations of cut-in-from-left simulation. Transpar- +ent “ghosts” illustrate parallel variants at the same time-point. +instance, the training process of a machine learning based +accident-prevention system would require simulations either +violating a safety metric or close to the point of violation. +Assuming knowledge of the system, this can be achieved +by constraints on the type of scenarios, i.e. inputs to the +simulation [1] [2]. Intelligent vehicular systems, however are +black-box systems and outcomes of scenarios are rarely known +apriori. Although extensive literature exists on adversarial +validation of systems [3], i.e. iterative search within the +scenario space to yield failure outcomes, these are generally +falsification approaches and do not suffice to generate the large +quantity of scenarios required for comprehensive coverage of +system behaviors. +This paper proposes an efficient and flexible methodology to +learn generative models over the scenario space with respect to +given outcomes. Specifying outcome metrics as cost functions +on simulation traces, we employ bayesian optimization [4] +to learn a surrogate function as belief over the cost function +with reasonable confidence around minima. The surrogate is +then used for fitting generative models. The methodology is +arXiv:2301.01594v1 [cs.LG] 3 Jan 2023 + +formalized in an end-to-end framework which uses the Open- +SCENARIO standard [5] to specify scenarios and brings them +to life by constructing digital twins of scenario participants +in a virtual test bed (VTB) cluster. The setup is tested on a +highway scenario, some examples of which are illustrated in +Fig. 1. +The rest of the paper is structured as follows: Sec. II +provides a survey of related work followed by a formal +problem definition in Sec. III and introduction to theoretical +methods in Sec. IV. Sec. V explains the full architecture that +employs the proposed methodology, and Sec. VI applies the +methodology to an example application, finally followed by +the conclusion. +II. RELATED WORK +The scenario-based architecture used by the authors is +based on the PEGASUS project [6], which aimed towards +standardized safety qualification of autonomous vehicles and +has spawned various works for formal scenario description +[7] [8] and derivations of critical scenarios via data- [9] or +knowledge-driven [10] methods. Scenic [1] provides another +formal description language for scenarios, which is similarly +used for formal safety qualifications [11]. +Stepping outside the formalized scenario domain, many +approaches tackle the validation problem of systems with +machine learning (ML) components via falsification - typically +optimization-based search policies to find counter-examples +for systems. Among them are image generation methods, +such as domain randomization in [12] and [13], or similarly +executed point clouds generation approaches as in [14]. Such +approaches attack the object-detection or planning ML di- +rectly, but the exploited parameters lack semantic information +about the scenario and do not offer a holistic interpretation +with respect to the system. To this end, VERIFAI [15] in- +troduced a comprehensive validation tool to find falsifying +examples via semantic scenario parameters that is usable with +external simulators. [16] proposed a compositional falsification +framework that isolates feature spaces for the non-ML and ML +sub-systems to generate counter-examples separately. [17] uses +signal temporal logic for formalized safety specification and +automated falsification of ML-based systems with graph-based +bayesian optimization. +The above approaches are effective for efficient falsification +but do not offer generative models for massive parallel simu- +lations. Within this area, data-driven approaches are typically +employed [18], which use traffic databases to directly sample +[19] or learn targeted probability distribution models [20]. +[20] explicitly allows the incorporation of desired outcomes +that dictate the created distributions. [21] uses a model prior +in addition to traffic databases to synthesize both knowl- +edge and data in subsequent models. Data-driven methods, +while critical for bridging real and virtual domains, have +inherently limited potential of exploring novel scenarios via +intelligent search algorithms, unlike their rule-based counter- +parts introduced above. Our proposed methodology integrates +an efficient optimization routine in a formalized rule-based +scenario framework, which yields a generative model rather +than singular counter-examples, which can be used to generate +a large number of counter-examples and allows convenient +storage and reuse of insights achieved by the optimization run. +III. PROBLEM STATEMENT +A scenario S is composed of participating agents A and +actions U, where each action u ∈ U is defined for certain +agents a ∈ A and controlling parameters p ∈ P: +S := {A, U}, +u := f(a, p). +(1) +For instance, Sec. VI specifies a cut-in scenario with two +vehicles as agents and cut-in maneuver as action. The param- +eters such as action trigger times, relative initial positions and +velocities control the action exection and form the basis of +scenario variation. A simulation model M maps the scenario +agents to sub-models and derives initial states s0 and input tra- +jectories U(t) as dictated by scenario actions and parameters, +such that +s(t) = M(s0, U(t), t), +(2) +T = {s(t0), ..., s(tN)}, t0 ≤ t ≤ tN +(3) +where s(t) are system states, and T is a trace containing +all state trajectories. An outcome specification is defined as a +formula φ of predicates over T, such that (⊨ indicates is true +for) +φ ⊨ T ⇔ ξ(T) = 0. +(4) +ξ is a positive real-valued cost function over the trace. +The objective is to find a distribution Pθ over the scenario +parameter space P such that the resulting simulation and its +subsequent trace has a high probability to minimize ξ, +argmax +θ +pθ(ξ = 0). +(5) +IV. BACKGROUND +This section presents a theoretical overview of the methods +used in Sec. V to tackle the objective in (5). +A. Signal Temporal Logic +Signal temporal logic (STL) [22] is used within the formal +verification community for monitoring continuous temporal +signals of complex systems and is used to express the outcome +specifications in (4). A specification (formula) is expressed as +a combination of multiple predicates: +φ := true | π | ¬ φ | φ ∧ ψ | φ U[a,b]ψ, +(6) +where ψ is another formula and U[a,b] an until operator +over the time interval [a, b]. A predicate π is defined as a real +function at a single time-point of the trace. STL allows formal +and intuitive outcome specifications with complex boolean +elements such as implication, if-and-only-if, as well as timed +qualifiers such as eventually (F) or always (G) true: + +F[a,b]φ = true U[a,b] φ +(7) += ∃t′ ∈ [t + a, t + b] such that φ ⊨ t′, +(8) +G[a,b]φ = ¬(F[a,b]¬φ) +(9) += ∀t′ ∈ [t + a, t + b] such that φ ⊨ t′. +(10) +E.g. Sec. VI uses the time-to-collision based predicate +(1 − ttc > 0) to define a specification over the whole trace: +F[0,T ](1 − ttc > 0). The real value of predicates π affords a +robustness metric ρ to the specification in addition to truth- +false inferences. E.g. the robustness of the F[0,T ](1−ttc > 0) +is the maximum value of (1 − ttc) throughout the simulation. +Robustness for compound specifications can be calculated +automatically: +ρφ∧ψ(t) := min(ρφ(t), ρψ(t)), +(11) +ρφ U[a,b] ψ(t) := +max +τ∈t+[a,b] +� +min +� +ρψ(τ), min +s∈[t,τ]ρφ(s) +�� +, +(12) +where ρφ indicates robustness of specification φ. Robustness +metric impart both qualitative and quantitative semantics to +specifications and open the door for mathematical algorithms +to automatically search for falsifying traces. +B. Bayesian Optimization +Bayesian optimization (BO) is known for its sample- +efficiency and flexibility in optimizing black-box functions. +BO maintains a surrogate function as belief over the black-box +function and updates it via adaptive measurements. Typical +choice for the surrogate function is a Gaussian Process (GP), +but other methods such as random forests are also used [23]. +1) Gaussian Processes: Following the formalism in [24], +GPs model a prior function f(x) ∼ GP(m(x), k(x, x′)) as +a joint gaussian distribution over the continuous function +values. m(x) is the prior mean and k(x, x′) is a kernel that +defines covariance between any two points and encodes prior +assumptions on the target function. We use the commonly +used Matern kernel which provides explicit parameters to +control the smoothness of the fitted function. The GP prior is +conditioned on measurements y(x) to create a posterior belief, +also a GP. For a zero prior mean and non-noisy measurements +(since simulations deliver ground-truth data), the posterior GP +is +f ∗|x∗, X, y(x) ∼ N(µ(x∗), Σ(x∗, x)) +(13) +µ(x∗) = k(x∗, x)K−1 +xx y(x), +(14) +Σ(x∗, x) = k(x∗, x∗) − k(x∗, x)K−1 +xx k(x, x∗), +(15) +where the covariance Σ(x∗, x) is between the new points x∗ +and previously measured x and Kxx is the kernel matrix +whose entries Kxx(i, j) = k(xi, xj) are evaluated between +each of the previously measured points x. +Configure +Scenario +Start +Evaluate Cost +acceptable +Finish +initialize with +seed scenarios +Get Best Candidates +Update Belief +yes +no +Generate +Variants +VTB Instance +Compute Cluster +DT +Repository +VTB Instance +Simulation +Model +VTB Instance +Simulation +Model +Simulation +Model +VTB Instance +Concrete +Scenarios +Abstract +Scenario +Logical +Scenario +Logical +Scenario +Adaptive Sampling +VTB +Fig. 2: Framework for scenario-based cluster simulations +2) Acquisition Function: Given a surrogate prior GP, BO +optimizes an acquisition function iteratively to choose the +next candidate x∗ to compute a surrogate posterior. Among +the variety of acquisition functions available, we use the +Thompson Sampling (TS) [25] method which chooses the +candidate x∗ by +x∗ = argmin +x∈D +f ∗, +(16) +where f ∗ is sampled from the surrogate prior. As it is a +distribution, samples f ∗ from the same prior vary from one- +another, thus implicitly imparting both exploration and ex- +ploitation to (16). The posterior calculated with the candidate +x∗ and its measurement y(x∗) is used as prior for the next +iteration, and the process continues until f ∗ +min converges. +C. Gaussian Mixture Models +Gaussian mixture model are a weighted sum of N gaussian +distributions +p(x|θ) = +N +� +i=1 +wiN(x; µi, Σi), +(17) +commonly used as an unsupervised learning technique. The +algorithm learns a number of normal distributions based on +clusters of given data. Typically, an expectation maximization +algorithm is employed to learn the maximum likelihood esti- +mates of hyper-parameters θ (means µi, covariance σi) [26]. +We employ bayesian gaussian mixture models from the python +scikit-learn library [27] which also learns the optimal number +of distributions to fit on the data. +V. FRAMEWORK +This section presents the framework to solve (5), illustrated +in Fig. 2. The overall workflow can be reiterated as such: +given a scenario template and its parameter space, the desired +outcomes of scenario simulations are specified as formal +specifications. An optimizer learns a distribution over the pa- +rameter space via successive simulations so that sampling from +said distribution has high likelihood of conforming to given +specifications, i.e. delivering desired simulation outcomes. The +overall formalism is roughly derived from [28] with emphasis +on the scenario variation aspect. + ++ generate() +Distribution ++ sample() +VariationEngine ++ generate() +1..* +0..* +1..* +1 +Constraint ++ isSampleValid() +0..* +getSample() +isSampleValid() +Parameter ++ sample() +GMM +Fig. 3: Meta-model for logical scenario based on [2] +1) Scenarios: Scenarios specify the actors and events in a +human-readable manner independent of the simulation frame- +work. We specify the static scenario (road layout etc.) with +ASAM OpenDRIVE [29] format and the dynamic scenario +(actor behaviour) with OpenSCENARIO [5]. Our framework +additionally differentiates between the abstract, logical and +concrete scenarios. The abstract scenario, referring both the +static and dynamic scenario, is complete with respect to +actors and events but defines no values for the scenario +parameters. The logical scenario allows human designers to +specify a scenario parameter space using a formalized meta- +model illustrated in Fig. 3 from [2]. Given parameter ranges +and inter-parameter relations specified per the Constraint and +probability distributions per the Distribution template, the +Variation Engine use Markov-Chain Monte-Carlo methods to +efficiently sample the resulting parameter space. This paper +focuses on the case where the designer desires certain simula- +tion outcomes but does not know the corresponding parameter +distributions or constraints beforehand. These outcomes are +specified within the logical scenario as-are, the subsequent +adaptive sampling algorithm derives the optimal parameter +distribution and saves it as an instance of the Distribution +template (see the yellow highlighted boxes in Fig. 3). The +Variation Engine can then sample the parameter space to +generate concrete scenarios. Concrete scenarios are complete +in all aspects, and can be converted to simulation models for +the virtual test beds (VTB). +2) Outcome Specification: +An outcome specification is +defined with the STL formalism introduced in Sec. IV-A over +one or more predicates - real-valued functions that can be +evaluated over a simulation trace. For each concrete scenario +and its subsequent simulation trace, STL formalism allows +the evaluation of a robustness value, whose sign indicates +conformity and value indicates the extent of conformity. We +additionally use a cost metric ξ that affords control over the +subsequent optimization algorithms: +ξ(T) = +� +−ρφT, +if ρφT < 0 +0, +otherwise. +(18) +The cost metric offers mere convenience and is irrelevant for +the optimization routine. (18) uses it to invert robustness for +minimization and to assert that all scenarios conforming to the +specification are equally relevant. Since scenario distributions +rather than singular examples are the goal, this encourage +(a) BO surrogate estimation +(b) Samples from fitted GMM +Fig. 4: Adaptive sampling example on the Griewank function +the adaptive sampling algorithm to fit on a diverse range of +conforming scenarios rather than the most robust scenario. +3) Adaptive Sampling: Illustrated in Fig. 2, bayesian op- +timization (BO) initializes with rasterized parameter samples +and their evaluated costs (p0, ξ0) to fit a Gaussian Process (GP) +f as a surrogate function as in (13). Since ξ must be evaluated +on VTB, it is an expensive process, and the idea is not just +to find the minimum ξ, but to estimate an f that forms a +reasonable belief on ξ near minima. This is further encouraged +by a batch variant of Thompson Sampling (TS), wherein N +surrogate f are optimized simultaneously. In each successive +iteration, BO optimizes the TS acquisition function in (16) to +chose the next best candidate parameters. The candidates and +their costs (p∗ +t , ξ∗ +t ) thus create a new posterior GP as in (13). +The converged surrogate forms the basis to infer scenarios +likely to yield the desired outcome, as parameters with low +cost on a sufficiently-fit surrogate are likely to have lower +costs on the VTB as well. A large number of parameters +corresponding to minimum cost range of the surrogate are +therefore fit on a gaussian mixture model in (17). The GMM +is presented as Pθ the solution to (5), and serves as an intuitive +and lightweight description for the scenario distribution. It +is saved within the logical scenario as an instance of the +Distribution template via a set of hyper-parameter, which can +also be tuned to control the variance of concrete scenarios. +Example: An example for the adaptive sampling method is +illustrated in Fig. 4 for a two-dimensional Griewank function +[30]. Fig. 4a shows that the surrogate function regresses well +on the original function around the minima even with existence +of multiple minima. This is relevant since all parameter +regions corresponding to an outcome are typically desired. +The experiment was carried out with an initial draw of 11 +samples, and converged in 8 iterations, each with a batch size +of 5. The GMM was fitted to parameter values corresponding +to the surrogate cost range [0−0.25]. Samples from the GMM +and their actual function evaluation is illustrated in Fig. 4b. +4) Simulation on VTB Clusters: Concrete scenarios, sam- +pled from GMM and any other constraints, are imported in +VTB instances running on a compute cluster. VTB serves as a +cross-domain platform offering modular simulation algorithms +(e.g. rigid body dynamics, realistic sensor simulation). For +each concrete scenario, a VTB instance constructs a simulation + +cost +dS +dS +dV +dV +T +T +Fig. 5: (Left) Bayesian optimization iterations for cut-in-from-left scenario, initialized with a rasterized grid (not shown). (Right) +Samples from fitted GMM vs uniform samples evaluated on the VTB +model by mapping actors to digital twins from a central +repository, and actions to input trajectories and initial states +of the digital twins. We use the framework developed by [28] +within the VEROSIM platform [31]. Each concrete scenario +and subsequent VTB instance are fully independent and self- +contained, constituting a freely scalable “embarassingly paral- +lel” problem. The simulation traces are logged in an SQLite +data-bank [32] accessible throughout the framework cluster. +VI. TRANSFER IN APPLICATION +The presented application is from a highway Traffic Jam +Chauffeur (TJC) research project, where a machine-learning +based TJC module must be comprehensively validated against +its operational scenarios. We present one of the scenarios, +cut-in-from-left, wherein a test vehicle Vehicle 1 behind the +ego vehicle cuts in its lane from left, and the goal is to find +a distribution of critically dangerous scenarios that the ego +vehicle may encounter. +Scenario 1 Abstract Scenario: Cut-In-From-Left +Parameters: dS, dV, T +Init: +1: Actor ego +pos ← 1000m, vel ← 16.667ms−1, lane ← 0 +2: Actor: vehicle 1 +rel pos ← dS, rel vel ← dV , rel lane ← −1 +Story: +Action: Cut-In +3: +Actors ← vehicle 1 +4: +Trigger Time ← T +The abstract scenario is defined with OpenSCENARIO +formalism and summarized in Scenario 1. The initial relative +position and velocity of vehicle 1, and trigger time for the +cut-in maneuver are undefined parameters, whose correspond- +ing ranges are described in the Logical Scenario 2, which +also states an appropriate outcome specification for critical +scenarios: the time-to-collision between two vehicles must be +below 1s at-least once during the simulation. The parameter +distribution to achieve high conformity to the specification +Fig. 6: Vehicle tracks for 200 samples from GMM for Spec1. +is specified with a gaussian mixture model (GMM) with +unknown hyper-parameters θ. +Scenario 2 Logical Scenario: Cut-In-From-Left +Parameters: +1: dS ← range [−30, 0] +2: dV ← range [0.5, 2.0] +3: T ← range [0.5, 3.0] +Outcome Specs: +4: Spec1 : F(1 − ttc) > 0 +5: +Distribution : GMM ← θ +The adaptive sampling methodology creates a an initial set +of concrete scenarios with 64 rasterized samples of the param- +eter space. For each concrete scenario, a VTB instance delivers +a time-to-collision trace calculated at each simulation time- +step via projections of the position and velocity vectors of both +vehicles. An STL parser evaluates the outcome specification +Spec1 to calculate the cost of each concrete scenario. The +bayesian optimization (BO) routine then iteratively suggests +new samples to evaluate on the VTB while maintaining a sur- +rogate Guassian Process (GP). Fig. 5 (left) illustrates BO for +6 iterations each with a batch size of 10, suggesting samples +in optimal yet diverse regions. The converged surrogate GP is +used to infer parameter candidates with cost (time-to-collision) +less than 1, which are then fitted to a GMM to derive θ. Fig. + +5 (center) depicts 200 samples from the learned θ that were +evaluated on the VTB. The high percentage of evaluations +conforming to Spec1 (zero cost) is evident as compared to +200 uniform samples evaluated on the VTB (Fig. 5 right). +The VTB evaluations on GMM samples are also visualized +interactively using methodologies from [33] in Fig. 1 and 6. +Fig. 6 visualizes the vehicle tracks for all 200 simulations, +whereas Fig. 1 shows selected examples in detail. +VII. CONCLUSION +This paper proposed generative models to efficiently test +relevant use-cases for on-cluster massive parallel simulations. +The authors proposed a formal framework based on formal +specifications, human-readable scenarios and modular virtual +test beds to set up the optimization and simulation tool chain, +and presented a bayesian optimization based adaptive sam- +pling algorithm to learn the optimal generative models. 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Roßmann, “Interactive analysis and visualization of dig- +ital twins in high-dimensional state spaces,” in 2018 15th International +Conference on Control, Automation, Robotics and Vision (ICARCV). +IEEE, 2018, pp. 241–246. + diff --git a/e9AzT4oBgHgl3EQfoP2a/content/tmp_files/load_file.txt b/e9AzT4oBgHgl3EQfoP2a/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3033beaa9f01489b52bd0ee605d3302f7b253a54 --- /dev/null +++ b/e9AzT4oBgHgl3EQfoP2a/content/tmp_files/load_file.txt @@ -0,0 +1,475 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf,len=474 +page_content='Finding Needles in Haystack: Formal Generative Models for Efficient Massive Parallel Simulations Osama Maqbool Institute of Man-Machine-Interaction RWTH Aachen University Aachen, Germany maqbool@mmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='rwth-aachen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='de J¨urgen Roßmann Institute for Man-Machine Interaction RWTH Aachen University Aachen, Germany rossmann@mmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='rwth-aachen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='de Abstract—The increase in complexity of autonomous systems is accompanied by a need of data-driven development and validation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Advances in computer graphics and cloud clusters have opened the way to massive parallel high fidelity sim- ulations to qualitatively address the large number of operational scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' However, exploration of all possible scenarios is still prohibitively expensive and outcomes of scenarios are generally unknown apriori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' To this end, the authors propose a method based on bayesian optimization to efficiently learn generative models on scenarios that would deliver desired outcomes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' collisions) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The methodology is integrated in an end-to-end framework, which uses the OpenSCENARIO standard to describe scenarios, and deploys highly configurable digital twins of the scenario participants on a Virtual Test Bed cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Index Terms—massive parallel simulations, bayesian optimiza- tion, virtual test beds, experimentable digital twins I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' INTRODUCTION The advent of intelligent vehicles has brought with it increasing levels of system complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Vehicles with ma- chine learning components additionally incorporate black- boxes and uncertainty within the system, transforming an already difficult problem to a non-deterministic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' These systems therefore require data-driven strategies in addition to classical approaches to deliver statistical metrics on safety and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Simulations offer a natural supplement to real-world tests, allowing reproduction of expensive or dangerous sce- narios virtually and being scaled as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Advancements in computer graphics has made high fidelity simulations possible, which are especially beneficial for generating large volumes of realistic sensor data required for machine learning based perception systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' More recently, the availability of cloud computing resources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Microsoft Azure, has offset the procurement effort of on-premise compute clusters enabling large-scale parallel simulations for a wider community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Nevertheless, the space of all possible operational scenarios remains prohibitively large, and usage of on-premise or cloud- based compute clusters is additionally a cost-incurring process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' This warrants a motivation to only simulate scenarios that would be meaningful to the actual development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' For This work is part of the project “KImaDiZ”, supported by the German Aerospace Center (DLR) with funds of the German Federal Ministry of Economics and Technology (BMWi), support code 50 RA 1934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 1: Visualizations of cut-in-from-left simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Transpar- ent “ghosts” illustrate parallel variants at the same time-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' instance, the training process of a machine learning based accident-prevention system would require simulations either violating a safety metric or close to the point of violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Assuming knowledge of the system, this can be achieved by constraints on the type of scenarios, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' inputs to the simulation [1] [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Intelligent vehicular systems, however are black-box systems and outcomes of scenarios are rarely known apriori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Although extensive literature exists on adversarial validation of systems [3], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' iterative search within the scenario space to yield failure outcomes, these are generally falsification approaches and do not suffice to generate the large quantity of scenarios required for comprehensive coverage of system behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' This paper proposes an efficient and flexible methodology to learn generative models over the scenario space with respect to given outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Specifying outcome metrics as cost functions on simulation traces, we employ bayesian optimization [4] to learn a surrogate function as belief over the cost function with reasonable confidence around minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The surrogate is then used for fitting generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The methodology is arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='01594v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='LG] 3 Jan 2023 formalized in an end-to-end framework which uses the Open- SCENARIO standard [5] to specify scenarios and brings them to life by constructing digital twins of scenario participants in a virtual test bed (VTB) cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The setup is tested on a highway scenario, some examples of which are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The rest of the paper is structured as follows: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' II provides a survey of related work followed by a formal problem definition in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' III and introduction to theoretical methods in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' V explains the full architecture that employs the proposed methodology, and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' VI applies the methodology to an example application, finally followed by the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' RELATED WORK The scenario-based architecture used by the authors is based on the PEGASUS project [6], which aimed towards standardized safety qualification of autonomous vehicles and has spawned various works for formal scenario description [7] [8] and derivations of critical scenarios via data- [9] or knowledge-driven [10] methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Scenic [1] provides another formal description language for scenarios, which is similarly used for formal safety qualifications [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Stepping outside the formalized scenario domain, many approaches tackle the validation problem of systems with machine learning (ML) components via falsification - typically optimization-based search policies to find counter-examples for systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Among them are image generation methods, such as domain randomization in [12] and [13], or similarly executed point clouds generation approaches as in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Such approaches attack the object-detection or planning ML di- rectly, but the exploited parameters lack semantic information about the scenario and do not offer a holistic interpretation with respect to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' To this end, VERIFAI [15] in- troduced a comprehensive validation tool to find falsifying examples via semantic scenario parameters that is usable with external simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' [16] proposed a compositional falsification framework that isolates feature spaces for the non-ML and ML sub-systems to generate counter-examples separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' [17] uses signal temporal logic for formalized safety specification and automated falsification of ML-based systems with graph-based bayesian optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The above approaches are effective for efficient falsification but do not offer generative models for massive parallel simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Within this area, data-driven approaches are typically employed [18], which use traffic databases to directly sample [19] or learn targeted probability distribution models [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' [20] explicitly allows the incorporation of desired outcomes that dictate the created distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' [21] uses a model prior in addition to traffic databases to synthesize both knowl- edge and data in subsequent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Data-driven methods, while critical for bridging real and virtual domains, have inherently limited potential of exploring novel scenarios via intelligent search algorithms, unlike their rule-based counter- parts introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Our proposed methodology integrates an efficient optimization routine in a formalized rule-based scenario framework, which yields a generative model rather than singular counter-examples, which can be used to generate a large number of counter-examples and allows convenient storage and reuse of insights achieved by the optimization run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' PROBLEM STATEMENT A scenario S is composed of participating agents A and actions U, where each action u ∈ U is defined for certain agents a ∈ A and controlling parameters p ∈ P: S := {A, U}, u := f(a, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' (1) For instance, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' VI specifies a cut-in scenario with two vehicles as agents and cut-in maneuver as action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The param- eters such as action trigger times, relative initial positions and velocities control the action exection and form the basis of scenario variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' A simulation model M maps the scenario agents to sub-models and derives initial states s0 and input tra- jectories U(t) as dictated by scenario actions and parameters, such that s(t) = M(s0, U(t), t), (2) T = {s(t0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=', s(tN)}, t0 ≤ t ≤ tN (3) where s(t) are system states, and T is a trace containing all state trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' An outcome specification is defined as a formula φ of predicates over T, such that (⊨ indicates is true for) φ ⊨ T ⇔ ξ(T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' (4) ξ is a positive real-valued cost function over the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The objective is to find a distribution Pθ over the scenario parameter space P such that the resulting simulation and its subsequent trace has a high probability to minimize ξ, argmax θ pθ(ξ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' (5) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' BACKGROUND This section presents a theoretical overview of the methods used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' V to tackle the objective in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Signal Temporal Logic Signal temporal logic (STL) [22] is used within the formal verification community for monitoring continuous temporal signals of complex systems and is used to express the outcome specifications in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' A specification (formula) is expressed as a combination of multiple predicates: φ := true | π | ¬ φ | φ ∧ ψ | φ U[a,b]ψ, (6) where ψ is another formula and U[a,b] an until operator over the time interval [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' A predicate π is defined as a real function at a single time-point of the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' STL allows formal and intuitive outcome specifications with complex boolean elements such as implication, if-and-only-if, as well as timed qualifiers such as eventually (F) or always (G) true: F[a,b]φ = true U[a,b] φ (7) = ∃t′ ∈ [t + a, t + b] such that φ ⊨ t′, (8) G[a,b]φ = ¬(F[a,b]¬φ) (9) = ∀t′ ∈ [t + a, t + b] such that φ ⊨ t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' (10) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' VI uses the time-to-collision based predicate (1 − ttc > 0) to define a specification over the whole trace: F[0,T ](1 − ttc > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The real value of predicates π affords a robustness metric ρ to the specification in addition to truth- false inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' the robustness of the F[0,T ](1−ttc > 0) is the maximum value of (1 − ttc) throughout the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Robustness for compound specifications can be calculated automatically: ρφ∧ψ(t) := min(ρφ(t), ρψ(t)), (11) ρφ U[a,b] ψ(t) := max τ∈t+[a,b] � min � ρψ(τ), min s∈[t,τ]ρφ(s) �� , (12) where ρφ indicates robustness of specification φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Robustness metric impart both qualitative and quantitative semantics to specifications and open the door for mathematical algorithms to automatically search for falsifying traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Bayesian Optimization Bayesian optimization (BO) is known for its sample- efficiency and flexibility in optimizing black-box functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' BO maintains a surrogate function as belief over the black-box function and updates it via adaptive measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Typical choice for the surrogate function is a Gaussian Process (GP), but other methods such as random forests are also used [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 1) Gaussian Processes: Following the formalism in [24], GPs model a prior function f(x) ∼ GP(m(x), k(x, x′)) as a joint gaussian distribution over the continuous function values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' m(x) is the prior mean and k(x, x′) is a kernel that defines covariance between any two points and encodes prior assumptions on the target function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' We use the commonly used Matern kernel which provides explicit parameters to control the smoothness of the fitted function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The GP prior is conditioned on measurements y(x) to create a posterior belief, also a GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' For a zero prior mean and non-noisy measurements (since simulations deliver ground-truth data), the posterior GP is f ∗|x∗, X, y(x) ∼ N(µ(x∗), Σ(x∗, x)) (13) µ(x∗) = k(x∗, x)K−1 xx y(x), (14) Σ(x∗, x) = k(x∗, x∗) − k(x∗, x)K−1 xx k(x, x∗), (15) where the covariance Σ(x∗, x) is between the new points x∗ and previously measured x and Kxx is the kernel matrix whose entries Kxx(i, j) = k(xi, xj) are evaluated between each of the previously measured points x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Configure Scenario Start Evaluate Cost acceptable Finish initialize with seed scenarios Get Best Candidates Update Belief yes no Generate Variants VTB Instance Compute Cluster DT Repository VTB Instance Simulation Model VTB Instance Simulation Model Simulation Model VTB Instance Concrete Scenarios Abstract Scenario Logical Scenario Logical Scenario Adaptive Sampling VTB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 2: Framework for scenario-based cluster simulations 2) Acquisition Function: Given a surrogate prior GP, BO optimizes an acquisition function iteratively to choose the next candidate x∗ to compute a surrogate posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Among the variety of acquisition functions available, we use the Thompson Sampling (TS) [25] method which chooses the candidate x∗ by x∗ = argmin x∈D f ∗, (16) where f ∗ is sampled from the surrogate prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' As it is a distribution, samples f ∗ from the same prior vary from one- another, thus implicitly imparting both exploration and ex- ploitation to (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The posterior calculated with the candidate x∗ and its measurement y(x∗) is used as prior for the next iteration, and the process continues until f ∗ min converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Gaussian Mixture Models Gaussian mixture model are a weighted sum of N gaussian distributions p(x|θ) = N � i=1 wiN(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' µi, Σi), (17) commonly used as an unsupervised learning technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The algorithm learns a number of normal distributions based on clusters of given data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Typically, an expectation maximization algorithm is employed to learn the maximum likelihood esti- mates of hyper-parameters θ (means µi, covariance σi) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' We employ bayesian gaussian mixture models from the python scikit-learn library [27] which also learns the optimal number of distributions to fit on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' FRAMEWORK This section presents the framework to solve (5), illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The overall workflow can be reiterated as such: given a scenario template and its parameter space, the desired outcomes of scenario simulations are specified as formal specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' An optimizer learns a distribution over the pa- rameter space via successive simulations so that sampling from said distribution has high likelihood of conforming to given specifications, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' delivering desired simulation outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The overall formalism is roughly derived from [28] with emphasis on the scenario variation aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' + generate() Distribution + sample() VariationEngine + generate() 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='.* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='.* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='.* 1 Constraint + isSampleValid() 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='.* getSample() isSampleValid() Parameter + sample() GMM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 3: Meta-model for logical scenario based on [2] 1) Scenarios: Scenarios specify the actors and events in a human-readable manner independent of the simulation frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' We specify the static scenario (road layout etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=') with ASAM OpenDRIVE [29] format and the dynamic scenario (actor behaviour) with OpenSCENARIO [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Our framework additionally differentiates between the abstract, logical and concrete scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The abstract scenario, referring both the static and dynamic scenario, is complete with respect to actors and events but defines no values for the scenario parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The logical scenario allows human designers to specify a scenario parameter space using a formalized meta- model illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 3 from [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Given parameter ranges and inter-parameter relations specified per the Constraint and probability distributions per the Distribution template, the Variation Engine use Markov-Chain Monte-Carlo methods to efficiently sample the resulting parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' This paper focuses on the case where the designer desires certain simula- tion outcomes but does not know the corresponding parameter distributions or constraints beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' These outcomes are specified within the logical scenario as-are, the subsequent adaptive sampling algorithm derives the optimal parameter distribution and saves it as an instance of the Distribution template (see the yellow highlighted boxes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The Variation Engine can then sample the parameter space to generate concrete scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Concrete scenarios are complete in all aspects, and can be converted to simulation models for the virtual test beds (VTB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 2) Outcome Specification: An outcome specification is defined with the STL formalism introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' IV-A over one or more predicates - real-valued functions that can be evaluated over a simulation trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' For each concrete scenario and its subsequent simulation trace, STL formalism allows the evaluation of a robustness value, whose sign indicates conformity and value indicates the extent of conformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' We additionally use a cost metric ξ that affords control over the subsequent optimization algorithms: ξ(T) = � −ρφT, if ρφT < 0 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' (18) The cost metric offers mere convenience and is irrelevant for the optimization routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' (18) uses it to invert robustness for minimization and to assert that all scenarios conforming to the specification are equally relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Since scenario distributions rather than singular examples are the goal, this encourage (a) BO surrogate estimation (b) Samples from fitted GMM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 4: Adaptive sampling example on the Griewank function the adaptive sampling algorithm to fit on a diverse range of conforming scenarios rather than the most robust scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 3) Adaptive Sampling: Illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 2, bayesian op- timization (BO) initializes with rasterized parameter samples and their evaluated costs (p0, ξ0) to fit a Gaussian Process (GP) f as a surrogate function as in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Since ξ must be evaluated on VTB, it is an expensive process, and the idea is not just to find the minimum ξ, but to estimate an f that forms a reasonable belief on ξ near minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' This is further encouraged by a batch variant of Thompson Sampling (TS), wherein N surrogate f are optimized simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' In each successive iteration, BO optimizes the TS acquisition function in (16) to chose the next best candidate parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The candidates and their costs (p∗ t , ξ∗ t ) thus create a new posterior GP as in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The converged surrogate forms the basis to infer scenarios likely to yield the desired outcome, as parameters with low cost on a sufficiently-fit surrogate are likely to have lower costs on the VTB as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' A large number of parameters corresponding to minimum cost range of the surrogate are therefore fit on a gaussian mixture model in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The GMM is presented as Pθ the solution to (5), and serves as an intuitive and lightweight description for the scenario distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' It is saved within the logical scenario as an instance of the Distribution template via a set of hyper-parameter, which can also be tuned to control the variance of concrete scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Example: An example for the adaptive sampling method is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 4 for a two-dimensional Griewank function [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 4a shows that the surrogate function regresses well on the original function around the minima even with existence of multiple minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' This is relevant since all parameter regions corresponding to an outcome are typically desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The experiment was carried out with an initial draw of 11 samples, and converged in 8 iterations, each with a batch size of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The GMM was fitted to parameter values corresponding to the surrogate cost range [0−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Samples from the GMM and their actual function evaluation is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 4) Simulation on VTB Clusters: Concrete scenarios, sam- pled from GMM and any other constraints, are imported in VTB instances running on a compute cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' VTB serves as a cross-domain platform offering modular simulation algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' rigid body dynamics, realistic sensor simulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' For each concrete scenario, a VTB instance constructs a simulation cost dS dS dV dV T T Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 5: (Left) Bayesian optimization iterations for cut-in-from-left scenario, initialized with a rasterized grid (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' (Right) Samples from fitted GMM vs uniform samples evaluated on the VTB model by mapping actors to digital twins from a central repository, and actions to input trajectories and initial states of the digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' We use the framework developed by [28] within the VEROSIM platform [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Each concrete scenario and subsequent VTB instance are fully independent and self- contained, constituting a freely scalable “embarassingly paral- lel” problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The simulation traces are logged in an SQLite data-bank [32] accessible throughout the framework cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' TRANSFER IN APPLICATION The presented application is from a highway Traffic Jam Chauffeur (TJC) research project, where a machine-learning based TJC module must be comprehensively validated against its operational scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' We present one of the scenarios, cut-in-from-left, wherein a test vehicle Vehicle 1 behind the ego vehicle cuts in its lane from left, and the goal is to find a distribution of critically dangerous scenarios that the ego vehicle may encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Scenario 1 Abstract Scenario: Cut-In-From-Left Parameters: dS, dV, T Init: 1: Actor ego pos ← 1000m, vel ← 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='667ms−1, lane ← 0 2: Actor: vehicle 1 rel pos ← dS, rel vel ← dV , rel lane ← −1 Story: Action: Cut-In 3: Actors ← vehicle 1 4: Trigger Time ← T The abstract scenario is defined with OpenSCENARIO formalism and summarized in Scenario 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The initial relative position and velocity of vehicle 1, and trigger time for the cut-in maneuver are undefined parameters, whose correspond- ing ranges are described in the Logical Scenario 2, which also states an appropriate outcome specification for critical scenarios: the time-to-collision between two vehicles must be below 1s at-least once during the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The parameter distribution to achieve high conformity to the specification Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 6: Vehicle tracks for 200 samples from GMM for Spec1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' is specified with a gaussian mixture model (GMM) with unknown hyper-parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Scenario 2 Logical Scenario: Cut-In-From-Left Parameters: 1: dS ← range [−30, 0] 2: dV ← range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='0] 3: T ← range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content='0] Outcome Specs: 4: Spec1 : F(1 − ttc) > 0 5: Distribution : GMM ← θ The adaptive sampling methodology creates a an initial set of concrete scenarios with 64 rasterized samples of the param- eter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' For each concrete scenario, a VTB instance delivers a time-to-collision trace calculated at each simulation time- step via projections of the position and velocity vectors of both vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' An STL parser evaluates the outcome specification Spec1 to calculate the cost of each concrete scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The bayesian optimization (BO) routine then iteratively suggests new samples to evaluate on the VTB while maintaining a sur- rogate Guassian Process (GP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 5 (left) illustrates BO for 6 iterations each with a batch size of 10, suggesting samples in optimal yet diverse regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The converged surrogate GP is used to infer parameter candidates with cost (time-to-collision) less than 1, which are then fitted to a GMM to derive θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 5 (center) depicts 200 samples from the learned θ that were evaluated on the VTB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The high percentage of evaluations conforming to Spec1 (zero cost) is evident as compared to 200 uniform samples evaluated on the VTB (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 5 right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The VTB evaluations on GMM samples are also visualized interactively using methodologies from [33] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 6 visualizes the vehicle tracks for all 200 simulations, whereas Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' 1 shows selected examples in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' CONCLUSION This paper proposed generative models to efficiently test relevant use-cases for on-cluster massive parallel simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The authors proposed a formal framework based on formal specifications, human-readable scenarios and modular virtual test beds to set up the optimization and simulation tool chain, and presented a bayesian optimization based adaptive sam- pling algorithm to learn the optimal generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' The framework was tested on an example scenario and compared with uniform simulations, where the presented method showed substantially better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' REFERENCES [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Fremont, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} +page_content=' Dreossi, S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9AzT4oBgHgl3EQfoP2a/content/2301.01594v1.pdf'} diff --git a/edE3T4oBgHgl3EQfHQkd/content/tmp_files/2301.04321v1.pdf.txt b/edE3T4oBgHgl3EQfHQkd/content/tmp_files/2301.04321v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4d1482cf14687bb04419b5b02176f0771967b79b --- /dev/null +++ b/edE3T4oBgHgl3EQfHQkd/content/tmp_files/2301.04321v1.pdf.txt @@ -0,0 +1,1384 @@ +Draft version January 12, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +Conclusive evidence for a population of water-worlds around M-dwarfs remains elusive +James G. Rogers +,1 Hilke E. Schlichting +,1 and James E. Owen +2 +1Department of Earth, Planetary, and Space Sciences, The University of California, Los Angeles, 595 Charles E. Young Drive East, Los +Angeles, CA 90095, USA +2Astrophysics Group, Department of Physics, Imperial College London, Prince Consort Rd, London, SW7 2AZ, UK +ABSTRACT +The population of small, close-in exoplanets is bifurcated into super-Earths and sub-Neptunes. We +calculate physically motivated mass-radius relations for sub-Neptunes, with rocky cores and H/He +dominated atmospheres, accounting for their thermal evolution, irradiation and mass-loss. For planets +≲ 10 M⊕, we find that sub-Neptunes retain atmospheric mass fractions that scale with planet mass and +show that the resulting mass-radius relations are degenerate with results for ‘water-worlds’ consisting +of a 1:1 silicate-to-ice composition ratio. We further demonstrate that our derived mass-radius relation +is in excellent agreement with the observed exoplanet population orbiting M-dwarfs and that planet +mass and radii alone are insufficient to determine the composition of some sub-Neptunes. Finally, +we highlight that current exoplanet demographics show an increase in the ratio of super-Earths to +sub-Neptunes with both stellar mass (and therefore luminosity) and age, which are both indicative of +thermally driven atmospheric escape processes. Therefore, such processes should not be ignored when +making compositional inferences in the mass-radius diagram. +Keywords: planets and satellites: atmospheres - planets and satellites: physical evolution - planet star +interactions +1. INTRODUCTION +The observed population of small, close-in exoplanets +with radii ≲ 4R⊕ and orbital periods ≲ 100 days (e.g. +Borucki et al. 2011; Howard et al. 2012; Fressin et al. +2013; Silburt et al. 2015; Mulders et al. 2018; Zink et al. +2019; Petigura et al. 2022) provide an intriguing problem +in terms of their formation pathway. With no analogue +in our solar system, such planets have been observed +to bifurcate into two separate sub-populations, centred +at ∼ 1.3R⊕ (referred to as ‘super-Earths’) and ∼ 2.4R⊕ +(referred to as ‘sub-Neptunes’), with a scarcity of planets +in-between at ∼ 1.8R⊕ labelled as the ‘radius gap’ (e.g. +Fulton et al. 2017; Van Eylen et al. 2018; Berger et al. +2020a; Petigura et al. 2022). +Two categories of evolutionary models have emerged +to explain this phenomenon. +The first relies on at- +mospheric evolution, as it is known that many sub- +Neptunes require a significant H/He dominated atmo- +sphere to explain their observed mass and radius (e.g. +Weiss & Marcy 2014; Jontof-Hutter et al. 2016; Ben- +neke et al. 2019). +Under this class of models, super- +Earths are expected to have lost their primordial at- +mosphere and are thus observed at their core1 radius. +Sub-Neptunes, on the other hand, have maintained their +atmosphere, bloating their size to the observed peak at +∼ 2.4R⊕. Typically, atmospheric mass-loss is thought +to cause this bifurcation, with smaller mass, highly ir- +radiated planets losing their hydrogen atmospheres to +become super-Earths, whilst larger mass, colder plan- +ets remain as sub-Neptunes. Two successful mass-loss +models are XUV photoevaporation, which relies on high- +energy stellar flux (e.g. Owen & Wu 2013; Lopez & Fort- +ney 2013) and core-powered mass-loss, which calls upon +remnant thermal energy from formation and bolometric +stellar luminosity (e.g. Ginzburg et al. 2018; Gupta & +Schlichting 2019). Other models may also explain the +radius gap via atmospheric escape due to giant impacts +(e.g. Inamdar & Schlichting 2016; Wyatt et al. 2020) +1 The term ‘core’, as used for the remainder of this letter, refers +to the solid/liquid bulk interior of a planet, as opposed to the +geological nomenclature of the iron core. In general, the mass of +a sub-Neptune is approximately given by its core mass, as the +atmospheric mass makes up less, often much less, than 10% of +the planet’s total mass. +arXiv:2301.04321v1 [astro-ph.EP] 11 Jan 2023 + +ID2 +or through gaseous accretion of primordial atmospheres +(e.g. Lee & Connors 2021; Lee et al. 2022). +The alternative model to atmospheric mass-loss and +evolution is the ‘water-world’ hypothesis, in which the +radius gap arises due to a difference in planet composi- +tion; super-Earths consisting of a silicate-iron mixture, +sub-Neptunes consisting of an ice-silicate mixture. Since +a planetary core of a given mass increases in size for +lower bulk densities, the water-world/sub-Neptune pop- +ulation exists at a larger size and hence separated from +the rocky super-Earths (e.g. Mordasini et al. 2009; Ray- +mond et al. 2018; Zeng et al. 2019; Turbet et al. 2020; +Mousis et al. 2020). A corollary of this model is that sub- +Neptunes form exterior to the protoplanetary disc water- +ice line, in which there is an increased solid mass due to +the condensation of volatiles. Pebble accretion has been +shown to predict such planets will have silicate-to-ice +ratios of 1:1 (e.g. Lodders 2003; Bitsch et al. 2015; Ven- +turini et al. 2020; Br¨ugger et al. 2020), although no ex- +planation has been provided as to why such planets did +not accrete voluminous H/He dominated atmospheres, +as would be expected from accretion and migration in +a protoplanetary disc. Nonetheless, under this model, +water-worlds then migrate inwards to the locations at +which we observe them today, thus forming the popula- +tion of close-in sub-Neptunes. +Sub-Neptunes consisting, on a population level, of a +1:1 silicate-to-ice mixtures are in tension with the pre- +dictions from atmospheric mass-loss models since both +photoevaporation and core-powered mass-loss models +find that both super-Earths and sub-Neptunes have core +bulk densities, and hence compositions, roughly consis- +tent with that of Earth (33% iron, 67% silicate e.g. Wu +2019; Gupta & Schlichting 2019; Rogers & Owen 2021; +Rogers et al. 2022). In light of the difference between +the water-world and the atmospheric-mass loss models, +it follows that unlocking the origins of the radius gap is +of crucial importance for understanding the underlying +formation pathways of small, close-in exoplanets. +In a recent study, Luque & Pall´e (2022) asserted +that there is evidence for a population of water-worlds +among planets orbiting M-dwarfs, by comparing ob- +served planet masses and radii with various planet- +composition models in the mass-radius diagram. They +find that many planets in their sample are consistent +with a 1:1 silicate-to-ice composition ratio, as well as +population synthesis modelling from Burn et al. (2021). +They also use the mass-radius relations for rocky bodies +hosting H/He dominated atmospheres from Zeng et al. +(2019) to claim that the planet sample was inconsis- +tent with rocky cores hosting H/He atmospheres. Un- +fortunately, these adopted mass-radius models for H/He +dominated atmospheres were not appropriate for plan- +ets at fixed age i.e. analogous to stellar isochrones. In +order to do this analysis, one must consider the thermal +evolution and atmospheric mass-loss with the associated +changes in entropy of H/He atmospheres over time. +In this letter, we calculate physically motivated, self- +consistent, population-level mass-radius relations for +rocky planets hosting H/He atmospheres, which cru- +cially take into account atmospheric evolution and ir- +radiation from the host star. We show that the sample +of observed sub-Neptunes around M-dwarfs from Luque +& Pall´e (2022) is, in fact, also consistent with rocky +cores hosting H/He dominated atmospheres. We con- +clude that, currently, there is no conclusive evidence for +differentiating, on a population level, between water- +worlds and rocky cores hosting H/He atmospheres from +sub-Neptune masses and radii alone. +2. METHOD +It is commonplace for mass-radius diagrams to be used +as a visual guide to the population of observed exoplan- +ets (e.g Wu & Lithwick 2013; Weiss & Marcy 2014; Had- +den & Lithwick 2014; Rogers 2015; Dressing et al. 2015; +Wolfgang et al. 2016; Chen & Kipping 2017; Van Eylen +et al. 2021). To interpret the observations, theoretical +mass-radius relations are used to plot a planet’s size as a +function of mass for a given composition. For solid bod- +ies consisting of iron, silicate and ice mass fractions, the +models of Fortney et al. (2007); Zeng et al. (2019) are +commonly used, in which the planet radius Rp scales ap- +proximately as Rp/R⊕ ∝ (Mp/M⊕)1/4, where Mp is the +planet mass (Valencia et al. 2006). Specifically the mod- +els of Zeng et al. (2019), which were adopted in Luque +& Pall´e (2022), give the following mass-radius relation +for water-worlds consisting of a 1:1 silicate-to-ice ratio: +Rp +R⊕ +≈ 1.24 +� Mp +M⊕ +�0.27 +. +(1) +Our task is to determine the mass-radius relation for +rocky/iron-rich cores with a H/He dominated atmo- +sphere. +Crucially, we aim to calculate physically- +motivated, self-consistent mass-radius relations, which +incorporate the physics of atmospheric evolution, includ- +ing mass-loss and cooling, which strongly modifies the +mass-radius relation from the canonical H/He results of +Zeng et al. (2019). We highlight that the H/He mass- +radius models from Zeng et al. (2019) assume constant +specific entropy in a purely adiabatic atmosphere. The +assumption of constant specific entropy (which sets the +adiabat) for planets of varying mass is not accurate for +planets that have undergone thermodynamic processes +such as cooling and mass-loss, which naturally reduce + +3 +the specific entropy of a planet and depend on many +variables such as planet mass and equilibrium temper- +ature. Moreover, in the Zeng et al. (2019) mass-radius +relations, each model’s specific entropy is parameterised +with a temperature defined at a fixed pressure of 100 bar. +We note that this temperature is often mistaken for the +equilibrium temperature of a given planet. +The tem- +perature and density of a purely adiabatic atmosphere +will drop far below the equilibrium temperature within +a few scale heights of the planet’s surface. In reality, an +outer radiative layer will form as a planet comes into +radiative equilibrium with the host star (e.g. Guillot +2010; Lee et al. 2014; Piso & Youdin 2014; Ginzburg +et al. 2016). +We point the reader to the mass-radius +relations of Lopez & Fortney (2014), which account for +thermal evolution, including radiative-convective mod- +els that provide planet size at a constant age for a given +mass and H/He mass fraction. +2.1. Method +Atmospheric mass-loss sculpts the exoplanet popula- +tion such that planets with larger core masses and there- +fore deeper gravitational potential wells retain larger +atmospheric masses. +This is also true for planets at +cooler equilibrium temperatures, since they receive a +smaller integrated stellar flux, which drives hydrody- +namic escape. Whilst one can explore these basic de- +pendencies analytically (see Appendix A), the easiest +way to fully understand these effects, in conjunction +with thermodynamic cooling and contraction, is with +numerical models. Firstly we use the semi-analytic nu- +merical models for XUV photoevaporation from Owen +& Wu (2017); Owen & Campos Estrada (2020) and core- +powered mass-loss from Gupta & Schlichting (2019); +Gupta et al. (2022) to numerically model populations +of planets undergoing atmospheric-mass loss driven by +both mechanisms (see Rogers et al. 2021, for a full dis- +cussion of both models). For both models we assume +an atmospheric adiabatic index of γ = 5/3, a core heat +capacity of 107 erg g−1 K−1 (Valencia et al. 2010) and +an opacity scaling law of κ ∝ P αT β, where α = 0.68, +β = 0.45 and κ = 1.29 × 10−2 cm2 g−1 at 100 bar and +1000 K (Rogers & Seager 2010a). Both models rely on +the hydrodynamic escape of hydrogen-dominated mate- +rial, hence we expect the predicted mass-radius distri- +butions to be very similar. +Chronologically speaking, there are three dominant +atmospheric processes that small, close-in exoplanets +with H/He dominated atmospheres undergo. Firstly, at- +mospheric mass is accrued via core-nucleated accretion +whilst immersed in a protoplanetary disc (e.g. Rafikov +2006; Lee et al. 2014; Piso & Youdin 2014; Ginzburg +et al. 2016). +Then, as the disc disperses, the atmo- +spheric mass of some planets is rapidly removed through +a “boil-off” process (also referred to as “spontaneous +mass-loss”) as the confining pressure from the disc is re- +moved on timescales ∼ 105 yrs (e.g. Ikoma & Hori 2012; +Owen & Wu 2016; Ginzburg et al. 2016). This is ap- +propriate for smaller mass planets ≲ 10M⊕, since larger +mass cores may open gaps in the gaseous protoplanetary +disc, resulting in different atmospheric evolution during +disc dispersal. Finally, once the disc has completely dis- +persed, these latter processes transition into XUV pho- +toevaporation and core-powered mass-loss (e.g. Lopez & +Fortney 2013; Owen & Wu 2013; Ginzburg et al. 2018; +Gupta & Schlichting 2019) combined with thermal cool- +ing and contraction. +Since we are not explicitly incorporating gaseous ac- +cretion and boil-off, our initial conditions must encom- +pass such processes. +To account for both scenarios, +namely in which boil-off does/does not occur, we adopt +two sets of initial conditions. The first scenario assumes +that planets have undergone a boil-off phase during disc +dispersal, for which we assume that planets host an ini- +tial atmospheric mass-fraction according to: +Xinit = 0.01 +� Mc +M⊕ +�0.44 � +Teq +1000 K +�0.25 +, +(2) +which +comes +from +the +theoretical +predictions +of +Ginzburg et al. (2016), which account for core accretion +and boil-off. In the inference work from Rogers et al. +(2022), the authors showed that this relation can be ac- +curately recovered from the data by inferring the corre- +lation between core mass and atmospheric mass fraction +prior to XUV photoevaporation for a sample of Kepler, +K2 and TESS planets. +In the second scenario, we do not enforce a boil-off +phase. There is currently uncertainty as to the details of +this mechanism, as it is a non-standard escape process, +particularly at high masses whereby gaps can be opened +in the protoplanetary disc. In light of this, we provide +an additional agnostic set of initial atmospheric mass +fractions, drawn log-uniformly in the range: +log Xinit ∼ U(10−3, 0.3), +(3) +where U is a uniform distribution, where the lower limit +avoids large mass cores hosting negligible atmospheric +mass fractions (we assume planets with X ≤ 10−4 to be +completely stripped i.e. super-Earths), whilst the up- +per limit is chosen to avoid self-gravitating atmospheres +which are known to be extremely rare and the semi- +analytic models do not account for (Wolfgang & Lopez +2015; Rogers et al. 2022). In essence, this distribution +accounts for all possible initial atmospheric conditions. + +4 +GJ 3470 b +GJ 436 b +K2 18 b +GJ 1214 b +GJ 3470 b +GJ 436 b +K2 18 b +GJ 1214 b +Photoevaporation +Boil-Off Initial Conditions +Core-Powered Mass-Loss +Agnostic Initial Conditions +Observations +Figure 1. Synthetic mass-radius distributions are shown for populations of planets evolved with photoevaporation and core- +powered mass-loss in left and right-hand panels, respectively, coloured by their equilibrium temperatures. Super-Earths are +stripped of their H/He dominated atmospheres and fall onto a relation consistent with an Earth-like composition (brown-dashed), +whilst sub-Neptunes retain a significant atmosphere. In the top panels, we assume an initial distribution of atmospheric masses +appropriate for a boil-off scenario (Eq. 2), in which planets lose a significant amount of H/He mass during disc dispersal. We +characterise the resulting narrow mass-radius distribution with a median line (orange dashed, Eq. 6). In the middle panels, +we adopt agnostic initial conditions (Eq. 3) and parameterise this mass-radius relation with 2σ limits (orange dotted lines). In +the bottom panels, we compare our theoretical mass-radius distributions (orange dashed/dotted lines, Eq. 6) with the observed +sample of M-dwarf orbiting exoplanets from Luque & Pall´e (2022), together with the mass-radius relation for water-worlds (blue +solid line, Eq. 1). We find that boil-off initial conditions provide mass-radius relations that are completely degenerate with that +of water-worlds. Furthermore, even when adopting agnostic initial conditions, the observations are accurately reproduced since +the mass-radius distribution is naturally explained due to mass-loss and cooling/contraction of H/He dominated atmospheres. +We highlight planets with confirmed escaping H/He detections with blue-shaded regions (namely; K2 18 b, GJ 3470 b, GJ 436 +b and, tentatively, GJ 1214 b). + +5 +As we shall show, even this agnostic set of initial condi- +tions accurately reproduces the observations. +We assume planetary cores are of Earth-like composi- +tion, such that they have a silicate-to-iron ratio of 67:33 +(Owen & Wu 2017; Gupta & Schlichting 2019; Rogers +& Owen 2021), making use of the mass-radius relations +from Fortney et al. (2007). +To approximately match +the stellar sample from Luque & Pall´e (2022), we adopt +a Gaussian stellar mass distribution centred at 0.3M⊙ +with a standard deviation of 0.1M⊙. We evolve a popu- +lation of 105 planets for each mass-loss model for 5 Gyrs +to match the approximate ages of observed planets, al- +though this final age makes no difference to the final +mass-radius distribution2. +We randomly draw orbital +periods from a broken power law: +dN +d logP ∝ +� +� +� +P a, +P < P0 days +P b, +P > P0 days, +(4) +where a = 1.0, b = −1.5 and P0 = 8.0 are chosen to +approximately match the population of observed plan- +ets orbiting M-dwarfs from Kepler (e.g. Petigura et al. +2022). We also place an upper limit on orbital periods of +30 days, since most M dwarf orbiting planets with mea- +sured masses and radii are observed with TESS, which +has a baseline capable of observing planets out to this +orbital separation. We randomly draw the planet core +masses in a log-uniform manner, so as to evenly sample +the mass-radius diagram. +Finally, we remove planets +with an RV semi-amplitude ≤ 30 cm s−1 to approxi- +mate current RV sensitivity limits. +3. RESULTS AND DISCUSSION +3.1. Mass-radius relation for sub-Neptunes with rocky +cores and H/He atmospheres +Figure 1 demonstrates the mass-radius relations for +a population of rocky cores, initially hosting H/He +rich atmospheres, that have undergone thermal evo- +lution and atmospheric mass-loss over 5 Gyrs. +Since +both photoevaporation (see left-hand panels) and core- +powered mass-loss models (see right-hand panels) de- +rive from hydrodynamic escape mechanisms, their pre- +dictions are very similar in this plane. +The bimodal +distribution is clearly seen, with super-Earths typically +residing at orbital separations corresponding to higher +equilibrium temperatures and having been stripped of +their hydrogen-dominated atmospheres. As such, super- +Earths fall on an Earth-like composition line in the +2 This is because the trend of entropy with mass is maintained +across various ages, meaning that the slope and position of the +mass-radius plane are age-insensitive. +mass-radius diagram. Sub-Neptunes, on the other hand, +maintain, despite some atmospheric mass-loss, a H/He +atmosphere, the amount of which scales with core mass +among other variables, such that more massive cores +retain larger atmospheric mass-fractions. These H/He +atmospheres increase the radii of sub-Neptunes above +that expected for an Earth-composition core. We note +that the mass-radius relation for planets in the absence +of atmospheric mass-loss is less steep with planet mass +for a given atmospheric mass fraction (e.g. see Figure 1 +of Lopez & Fortney 2014), as opposed to the models of +Zeng et al. (2019) (see Section 3.3). Hence, the mass- +radius observations for sub-Neptunes can only be fit +with H/He atmospheric mass-fractions that scale with +planet mass, which is a natural outcome of the hydro- +dynamic atmospheric-loss processes discussed above. +In the top and middle panels, we show mass-radius +distributions for both sets of initial conditions; boil-off +(see Equation 2) and agnostic (see Equation 3) respec- +tively. One can see that the boil-off scenario produces +a narrow mass-radius distribution, whilst the agnostic +initial conditions produce a wider range in sub-Neptune +radii for a given mass, owing to the increased range in +initial atmospheric mass fractions. Note however that +even with this set of agnostic initial conditions, the wider +spread in sub-Neptune sizes shrinks as the planets cool +and contract to smaller radii. As such, the majority of +sub-Neptunes sit close to the models which started with +boil-off initial conditions. This is because thermal evo- +lution and mass-loss of H/He dominated atmospheres +naturally produce this relation, independent of initial +conditions. +To provide a useful reference for comparison with +future observations, we quantify these mass-radius re- +lations, which we highlight are appropriate for sub- +Neptunes in the range 1.0 ≲ Mp/M⊕ ≲ 30, with quartic +logarithmic functions: +Rp +R⊕ += a0 + a1 ln +� Mp +M⊕ +� ++ a2 ln +� Mp +M⊕ +�2 ++ a3 ln +� Mp +M⊕ +�3 ++ a4 ln +� Mp +M⊕ +�4 +, +(5) +where coefficients are summarised for photoevaporation +and core-powered mass-loss in Tables 1 and 2 respec- +tively. Since the boil-off scenario is extremely narrow +(top panels of Figure 1), we quantify its median value +(orange dashed line) for both models by calculating their +median planet size for bins in planet mass. We then fit +these median values to Equation 6. Similarly, for the ag- +nostic initial conditions (bottom panels of Figure 1), we +quantify this wider mass-radius distribution by finding +2σ limits in planet size for planet mass bins. + +6 +Boil-Off +Agnostic (lower) +Agnostic (upper) +a0 +1.3104 +1.2131 +1.5776 +a1 +0.2862 +0.2326 +0.7713 +a2 +0.1329 +-0.0139 +0.5921 +a3 +-0.0174 +0.0367 +-0.2325 +a4 +0.0002 +-0.0065 +0.0301 +Table 1. Coefficients for mass-radius relations for photo- +evaporation, given by a quartic logarithmic equation from +Equation 6. +Boil-off initial atmospheric conditions (see +dashed-orange line in Figure 1) are from Equation 2, agnos- +tic initial atmospheric conditions (see dotted-orange lines in +Figure 1) are from Equation 3, with upper and lower planet +size bounds given. +Boil-Off +Agnostic (lower) +Agnostic (upper) +a0 +1.3255 +1.5776 +1.2131 +a1 +0.4168 +0.7713 +0.2326 +a2 +0.1567 +0.5921 +-0.0139 +a3 +-0.07224 +-0.2325 +0.0367 +a4 +0.01092 +0.0301 +-0.0065 +Table 2. Same as Table 1, but for core-powered mass-loss +models. +In the bottom panels of Figure 1, we show our pre- +dicted mass-radius relations from Equation 6 alongside +the observed sample from Luque & Pall´e (2022), consist- +ing of 48 planets orbiting 26 M-dwarfs systems with stel- +lar masses 0.1 ≲ M∗/M⊙ ≲ 0.6. Figure 1 clearly demon- +strates that the mass-radius observations are in excellent +agreement with sub-Neptunes which have rocky interi- +ors and H/He atmospheres provided that their thermal +evolution and mass-loss histories are accounted for. +In the case of boil-off initial conditions, (orange- +dashed line) the mass-radius relation from our atmo- +spheric evolution and mass-loss models is degenerate +with bodies of a 1:1 silicate-to-ice ratio (Zeng et al. +2019) (blue solid line). In the case of agnostic initial +conditions, (orange dotted-lines) the mass-radius rela- +tion encompasses all observed planets. Finally, we also +highlight planets with blue-shaded circles that have con- +firmed escaping hydrogen/helium atmospheres. Namely, +these are K2 18 b (Benneke et al. 2019; dos Santos +et al. 2020), GJ 436 b (Bean et al. 2008; Pont et al. +2009; Knutson et al. 2011; Ehrenreich et al. 2015; Turner +et al. 2016), GJ 3470 b (Fukui et al. 2013; Nascimbeni +et al. 2013; Crossfield et al. 2013; Dragomir et al. 2015; +Awiphan et al. 2016; Bourrier et al. 2018; Ninan et al. +2020) and GJ 1214 b (although we highlight that this +is a tentative detection from Orell-Miquel et al. 2022). +K2 18 b is an interesting case, since it is close3 to the +mass-radius relations for atmospheric evolution (orange +dashed line) and water-worlds (blue solid). +However, +the direct hydrogen detection suggests it is inconsistent +with the water-world hypothesis since such planets can- +not host significant hydrogen atmospheres whilst still +being consistent with observed masses and radii. +We also note that whilst many observed intermediate +mass planets (2 ≲ Mp/M⊕ ≲ 10) are tightly clustered +around the mass-radius relations for water-worlds and +H/He atmospheres with boil-off initial conditions, there +are many high-mass planets, including those with escap- +ing H/He detections; GJ 3470 b, GJ 436 b, and GJ 1214 +b, that sit above both of these mass-radius relations. +They do however sit within the bounds of H/He mass +relations with agnostic initial conditions. As discussed +in Section 2, boil-off is likely inefficient for planets with +Mc ≳ 10M⊕ since such planets will begin to open gaps +in their protoplanetary discs, hence implying the agnos- +tic initial conditions (Equation 3) are more appropriate +for such planets. The observations appear to support +this notion. We highlight that more work is needed to +understand boil-off and that these planets provide im- +portant tests of such processes. +3.2. Verifying mass-radius relations with MESA +Whilst the semi-analytic model of atmospheric evo- +lution for photoevaporation from Owen & Wu (2017); +Owen & Campos Estrada (2020) and core-powered +mass-loss from Ginzburg et al. (2018); Gupta & Schlicht- +ing (2019) are computationally inexpensive and thus al- +low large populations of planets to be generated, they +lack complex physics such as a detailed model for convec- +tion, self-gravity and realistic equations of state. There- +fore, as in Owen & Wu (2017), we corroborate our semi- +analytical modelling from Figure 1 by comparing our +results with numerical models performed with Modules +for Experiments in Stellar Astrophysics (MESA) (Paxton +et al. 2011, 2013, 2015, 2018), which solves and evolves +the stellar structure equations with accurate H/He equa- +tions of state from Saumon et al. (1995) and dust-free +opacity tables from Freedman et al. (2008) for low- +mass and irradiated planets. These sophisticated mod- +els remove free parameters from the problem, such as +choices in adiabatic index and opacities since these are +determined self-consistently. We follow previous works +to model low-mass planets (Owen & Wu 2013, 2016; +Chen & Rogers 2016; Kubyshkina et al. 2020; Malsky & +3 In fact, other literature values would place K2 18 b precisely on +the mass-radius relations for H/He atmospheres and water-worlds +(e.g. Sarkis et al. 2018). + +7 +Rogers 2020), and evolve each model for 5 Gyrs, adopt- +ing stellar irradiation performed with the F∗ − Σ rou- +tine from MESA (Paxton et al. 2013), which injects ir- +radiative flux within a column density of Σ. For these +models, we follow Owen & Wu (2013, 2016) and assume +Σ = 250 g cm−2, appropriate for opacities to incoming +stellar irradiation of κν = 4 × 10−3 cm2 g−1 (Guillot +2010). +In Figure 2, we show populations of planets evolved +with MESA at equilibrium temperatures of 300K, 500K +and 800K in the top, middle and bottom panels respec- +tively, represented with black triangles. For simplicity, +we only compare these results with the semi-analytic +photoevaporation models since planets of different core +masses can be stripped at slightly different equilibrium +temperatures under the core-powered mass-loss model. +For the purposes of population-level mass-radius dia- +grams, however, the differences between the two models +are inconsequential. +One can see from Figure 2 that the MESA models are +in excellent agreement with our adopted semi-analytic +photoevaporation models which are also shown in Fig- +ure 2 for small ranges in equilibrium temperatures at +300 ± 10 K, 500 ± 10 K and 800 ± 10 K. Note that +mass-loss is not explicitly included in these MESA mod- +els. +Instead, we adopt the final atmospheric mass- +fractions from the semi-analytic photoevaporation mod- +els (as shown with black triangles in the left-hand pan- +els of Figure 2) and then evolve the planets in MESA +with this atmospheric mass fraction to calculate their +radii after 5 Gyrs. Since the majority of atmospheric +escape under photoevaporation typically occurs in the +first ∼ 100 Myr, this is akin to beginning the MESA sim- +ulations at the end of this period in order to accurately +determine their radii at 5 Gyrs. As Figure 2 demon- +strates, these models robustly confirm the mass-radius +relations found with our semi-analytic approach. +Figure 2 also highlights important points about at- +mospheric evolution. +Firstly, in the left-hand panels, +the final atmospheric mass fractions demonstrate that +planets of different core masses evolve to host very dif- +ferent atmospheric mass fractions. For an initial boil-off +distribution represented with grey points (see Equation +2), low-mass planets are stripped of their atmosphere +(numerically identified with an atmospheric mass frac- +tion ≤ 10−4) whilst the highest-mass planets retain most +of their initial atmospheric mass and therefore match +the initial distribution. Intermediate-mass planets, how- +ever, lose progressively less atmosphere with increas- +ing core mass. A common generalisation is that sub- +Neptunes host an atmospheric mass-fraction of ∼ 1% +(Owen & Wu 2017), since this value maximises the +mass-loss timescale and naturally leads to a population +of planets that retain their H/He atmosphere. Whilst +this is good to an order of magnitude, Figure 2 clearly +demonstrates that this an oversimplification, with larger +planets naturally retaining a greater atmospheric mass +fraction due to their increased gravitational potential- +wells (which is also shown analytically in Appendix A). +Furthermore, this distribution changes as a function of +equilibrium temperature (i.e. different rows in Figure 2), +with planets at lower equilibrium temperatures able to +maintain more atmospheric mass for a given core mass. +Figure 2 also demonstrates the importance of initial con- +ditions for such planets, since high-mass planets main- +tain an atmospheric mass that follows their initial dis- +tribution. The population-level mass-radius diagram (as +shown in Figure 1) is therefore a superposition of differ- +ent planets at different core masses, equilibrium temper- +atures and ages, with their initial conditions playing a +progressively more influential role for higher masses. +3.3. Comparison with Zeng et. al. 2019 models +In the right-hand panel of Figure 2, we compare our +semi-analytic and MESA models with numerical models of +Zeng et al. (2019), which provide mass-radius relations +for rocky cores hosting a H/He atmospheric mass frac- +tion under the assumption of constant specific entropy, +defined with a temperature at a pressure of 100 bar, al- +though we highlight that this temperature is frequently +misinterpreted as the planetary equilibrium tempera- +ture. For reference, for an adiabat with γ = 5/3, set such +that its temperature is 500 K and 100 bar, the temper- +ature at 1 bar is ≲ 80 K, which is far below the typ- +ical planetary equilibrium temperatures currently ob- +served. We stress that such models are not applicable +to evolved sub-Neptunes to perform quantitative analy- +sis. Examples of these mass-radius relations are shown +in black-dashed lines in Figure 2 for atmospheric mass +fractions of 0.1% and 1.0% with specific entropy set with +a temperature of 500 K and 100 bar. +These curves +have a characteristic and dramatic increase in size for +smaller-mass planets, which comes from the assumption +of constant atmospheric mass at constant specific en- +tropy. However, atmospheric evolution naturally allows +planet atmospheres to cool and contract, with smaller- +mass planets cooling more due to their reduced heat +capacity. Combining this with mass-loss, which further +reduces the atmospheric mass retained by smaller mass +planets results in the mass-radius relations found in Fig- +ure 1. We note that if one wishes to analyse an individ- +ual planet in the mass-radius diagram, then the mass- +radius relations of Lopez & Fortney (2014) at constant +age, which are also shown in Figure 2, are more ap- + +8 +1 +2 +3 +4 +10-4 +10-3 +10-2 +10-1 +Figure 2. +The final atmospheric mass fractions (left-hand column) and planet radii (right-hand column) after 5 Gyr of +photoevaporative evolution for populations of planets with equilibrium temperatures of 300 ± 10 K (top row), 500 ± 10 K +(middle row) and 800 ± 10 K (bottom row). Colours represent planet size in the left-hand panels and final atmospheric mass +fraction in the right-hand panels, demonstrating that larger atmospheric mass fractions lead to larger planets and vice versa. All +planets start their evolution with an initial distribution of atmospheric mass fractions (displayed as grey points) that account +for gaseous core accretion and boil-off during protoplanetary disc dispersal (see Equation 2). To corroborate these semi-analytic +results, we also perform numerical models with MESA, which are shown as black triangles. In general, sub-Neptunes only exist +at larger masses for higher equilibrium temperatures. The mass-radius distribution (as seen in Figure 1) is a superposition of +all equilibrium temperatures. In the right-hand column, we compare our mass-radius distributions with the models of Lopez & +Fortney (2014) in blue and Zeng et al. (2019) in black for atmospheric mass fractions of 0.1% and 1.0%. We highlight that the +models of Zeng et al. (2019) assume constant specific entropy defined with a temperature at fixed pressure at 100 bar (not to be +confused with the equilibrium temperature) and therefore suggest a dramatic increase in planet radius for lower-mass planets. +The models of Lopez & Fortney (2014) consider irradiation and cooling for planets with constant atmospheric mass fraction, +meaning they are more appropriate for analysis of planet composition. Our mass-radius models account for loss-induced scaling +of atmospheric mass with planet mass, meaning they are appropriate for comparisons of planet populations in the mass-radius +diagram. + +9 +propriate since these include the essential physical pro- +cesses (cooling and irradiation for a given atmospheric +mass fraction) that shape the radius of small exoplanets +with hydrogen atmospheres. Alternatively, the publicly +available evapmass code from Owen & Campos Estrada +(2020) includes the semi-analytic atmospheric structure +models adopted in this work. In the case of analysing +populations of planets in the mass-radius diagram, we +recommend the relations derived in this work (see Equa- +tion 6). +3.4. Mass-radius relations for sub-Neptunes around +FGK stars +In this letter, we have focused on planets orbiting M +dwarfs, as is the case with the observational work of +Luque & Pall´e (2022). As we have shown, our choice +of physically motivated initial conditions and ranges +in equilibrium temperatures yield mass-radius relations +with an intrinsic spread in planet radii for a given mass +(see Figures 1 and 2). There are, however, additional +factors that can contribute to the mass-radius distribu- +tion spread, that we have not included in our models. +As highlighted in Kubyshkina & Fossati (2022), vari- +ability in high-energy stellar luminosity (e.g. Tu et al. +2015; Johnstone et al. 2021; Ketzer & Poppenhaeger +2022) can increase the range in planet sizes, since stars +of different initial rotation rates will produce different +X-ray/EUV flux and thus different mass-loss rates for +the orbiting planets. In addition, observational uncer- +tainties in planet radii will increase the spread in the +mass-radius distribution due to purely statistical scat- +ter. +It is interesting to note that the underlying mass- +radius distribution does not significantly change when +considering planets orbiting FGK stars. Although such +planets will receive a larger flux at a given orbital pe- +riod, we find from our mass-radius models that this bias +tends to simply produce a larger ratio of super-Earth to +sub-Neptune occurrence rates, since more planets can +be stripped of their H/He atmosphere. Indeed, this re- +sult is consistent with the demographic work of Petigura +et al. (2022), from which one can calculate the ratio +in occurrence rates of super-Earths to sub-Neptunes to +find it increased, with values of 0.29 ± 0.07, 0.34 ± 0.05 +and 0.54 ± 0.10 for a stellar mass bins of [0.5, 0.7]M⊙, +[0.7, 1.1]M⊙ and [1.1, 1.4]M⊙ respectively. We do high- +light, however, that there are other ways in which this +ratio may increase, such as varying the core mass or or- +bital period distributions as a function of stellar mass. +One major difference between low and high-mass +stars, however, is that transit observations (such as those +from Kepler, K2 and TESS) can achieve a higher pho- +tometric precision around M dwarfs due to their smaller +stellar radii and hence larger Rp/R∗. Such surveys are +also biased to observe planets within a smaller range +in equilibrium temperatures since the transit probabil- +ity of planets at large orbital periods (and therefore low +equilibrium temperatures) decreases rapidly. Different +mission targeting strategies also change the observed +population e.g. Kepler was sensitive to planets with or- +bital periods ∼ 100 days, but specifically targeted FGK +stars, whereas TESS currently targets nearby bright +stars (and is therefore biased to M-dwarfs), with sen- +sitivity out to orbital periods ∼ 30 days. These argu- +ments taken together suggest that the observed mass- +radius distribution around M-dwarfs is expected to have +less scatter compared to that for planets around FGK +stars. This is indeed the case when comparing Figures +1 and S19 from Luque & Pall´e (2022) (see also Fig- +ure 12 from Rogers & Owen 2021, for an example of +a synthetic mass-radius distribution for FGK stars in +the presence of bias and measurement uncertainty). We +find that the underlying mass-radius distribution, in the +absence of statistical scatter and bias4 (as summarised +by Equation 6 under different initial conditions), is ap- +proximately the same across FGKM spectral types but +that the relative occurrence of super-Earths with respect +to sub-Neptunes increases for more massive (luminous) +stellar types. +4. CONCLUSIONS +In this letter, we calculate mass-radius relations of +small, close-in exoplanets that host H/He dominated +atmospheres with self-consistent, physically motivated +evolution models, which have the following form: +Rp +R⊕ += a0 + a1 ln +� Mp +M⊕ +� ++ a2 ln +� Mp +M⊕ +�2 ++ a3 ln +� Mp +M⊕ +�3 ++ a4 ln +� Mp +M⊕ +�4 +, +(6) +where coefficients are summarised for photoevaporation +and core-powered mass-loss models in Tables 1 and 2 re- +spectively. We consider two sets of initial conditions; the +first, in which planets undergo a boil-off phase, whereby +a large fraction of atmospheric mass is lost during proto- +planetary disc dispersal (e.g. Ikoma & Hori 2012; Owen +& Wu 2016; Ginzburg et al. 2016), which yields a rel- +atively tight mass-radius relation after mass-loss and +thermal evolution. In the second scenario, we adopt ag- +nostic initial conditions (see Equation 3), which yields a +4 We also highlight that the bias of planet mass measurements is +currently not quantifiable, since such surveys are not based on +homogeneous observations of a well-defined sample of stars. + +10 +larger spread in final radii. These relations (see Figure +1) incorporate thermodynamic cooling, atmospheric es- +cape and stellar irradiation and are therefore suited for +compositional analyses of populations of sub-Neptunes +in the mass-radius diagram (e.g Wu & Lithwick 2013; +Weiss & Marcy 2014; Hadden & Lithwick 2014; Rogers +2015; Dressing et al. 2015; Wolfgang et al. 2016; Chen +& Kipping 2017; Van Eylen et al. 2021). We show that +accounting for atmospheric mass-loss yields left-over at- +mospheric mass fractions that scale with planet mass +i.e. larger planets retain a larger fraction of their total +mass in hydrogen and show that these results give an +excellent match to the mass and radius measurements +of sub-Neptunes in Luque & Pall´e (2022), independently +of assumed initial conditions. In addition, we show that +the boil-off initial conditions yield a mass-radius relation +that is completely degenerate with that corresponding to +a 1:1 silicate-to-ice ratio. We note that our study moves +beyond the H/He mass-radius relations of Zeng et al. +(2019), as demonstrated in Figure 2, which assume a +constant specific entropy for constant atmospheric mass +factions as a function of planet mass. We find that such +models are therefore not applicable to planets undergo- +ing atmospheric evolution. +In Luque & Pall´e (2022), a sample of observed exo- +planets orbiting M-dwarfs is used to argue that plan- +ets with rocky interiors and H/He atmospheres can- +not explain the observed cluster of planets around the +1:1 silicate-to-ice ratio compositional line in the mass- +radius diagram. +In Figure 1 we have presented our +new mass-radius relations for small planets with hydro- +gen atmospheres (see Equation 6) and show that they +are in fact completely consistent with the data, once +thermal evolution and mass-loss are properly accounted +for. A strong degeneracy therefore still exists between +the water-world and silicate/iron-hydrogen models. We +find that planets with different equilibrium tempera- +tures and atmospheric masses for a given core mass yield +a natural spread in the mass-radius relation (see Fig- +ure 1) that does not vary dramatically for different stel- +lar types. We do note that other factors that we have +not taken into account, such as high-energy stellar lu- +minosity variability (Kubyshkina & Fossati 2022) and +observational uncertainty will act to increase the spread +in the sub-Neptune mass-radius relation. Nevertheless, +we note that many high-mass planets ≳ 10M⊕ in the +sample from Luque & Pall´e (2022), including GJ 436 +b, GJ 3470 b and GJ 1214, which have confirmed es- +caping H/He atmospheric detections, sit well above the +mass-radius relations for both water-worlds and hydro- +gen atmosphere models that assume an initial boil-off +scenario. We speculate that such planets were less sus- +ceptible to boil-off (Owen & Wu 2016; Ginzburg et al. +2016) due to their increased mass (potentially due to +gap-opening in their protoplanetary discs) and therefore +entered the XUV photoevaporation/core-powered mass- +loss phase with larger atmospheric mass fractions. We +highlight that further work is needed to understand this +important stage in exoplanet evolution. +In light of the results shown in Figure 1, we corrob- +orate the well-known result that a planet’s mass and +radius alone are often insufficient to break its internal +composition degeneracy (Valencia et al. 2007; Rogers & +Seager 2010b). Probing for hydrogen and helium pres- +ence around low-mass planets with spectroscopic obser- +vations is one promising avenue (e.g. Ehrenreich et al. +2015; Lavie et al. 2017; Bourrier et al. 2018; Yan & Hen- +ning 2018; Spake et al. 2018; dos Santos et al. 2020; +Ninan et al. 2020; Zhang et al. 2022) although we high- +light that a non-detection in hydrogen Ly-α does not +necessarily indicate the lack of a hydrogen-dominated +atmosphere (Owen et al. 2023). Moreover, observations +from JWST may provide insights into the abundance +of H2O in high mean-molecular weight atmospheres of +sub-Neptunes and thus the prevalence of water-worlds. +In this letter, we have also analysed the occurrence rates +from Petigura et al. (2022) to find that the ratio of +super-Earths to sub-Neptunes increases, with values of +0.29 ± 0.07, 0.34 ± 0.05 and 0.54 ± 0.10 for a stellar +mass bins of [0.5, 0.7]M⊙, [0.7, 1.1]M⊙ and [1.1, 1.4]M⊙ +respectively. Since larger mass stars produce larger lu- +minosities, this result tentatively supports the notion +that stellar irradiation is key in evolving sub-Neptunes +into super-Earths via atmospheric escape. +In addition, if one can accurately measure planet age, +then one can determine how planets and the observed +radius gap, separating the super-Earths from the sub- +Neptunes, evolves with time. Under atmospheric evo- +lution models, the radius gap is expected to evolve on +∼ 100 Myr to Gyr timescales (Gupta & Schlichting 2020; +Rogers & Owen 2021) since hydrogen-dominated atmo- +spheres dramatically change in size as they cool due +to their low mean-molecular weight. Water-worlds on +the other hand will not significantly change in size af- +ter formation since their sizes are dominated by their +ice-silicate composition and not H/He dominated at- +mospheres. Indeed this demographic analysis has been +performed in the works of Berger et al. (e.g. 2020b); +Sandoval et al. (e.g. 2021); Chen et al. (e.g. 2022) to +show that the radius gap evolves on ∼ 100 Myr to Gyr +timescales. Moreover, in a recent study from Fernan- +des et al. (2022), occurrence rates were calculated for +a sample of exoplanets around young stars. They find +tentative evidence for the decrease in sub-Neptune size + +11 +with stellar age, which is indicative of cooling and con- +traction of sub-Neptunes with time, although we note +that presently this sample size is small. A larger sam- +ple of planets with accurate ages may shed light on this +issue. Furthermore, mass measurements of young sub- +Neptunes may show that such planets are indeed in- +flated and therefore extremely under-dense H/He-rich +proto-sub-Neptunes, destined to cool and contract to +the evolved population we observe today (Owen 2020). +Whilst water-worlds may exist in conjunction with plan- +ets hosting H/He rich atmospheres, the evidence for +their existence as a population still remains elusive. +ACKNOWLEDGEMENTS +We would like to kindly thank Akash Gupta, Ruth +Murray-Clay, Erik Petigura and Vincent Van Eylen for +discussions that helped improve the paper. +JGR is +supported by the Alfred P. Sloan Foundation under +grant G202114194 as part of the AEThER collaboration. +HES gratefully acknowledges support from NASA under +grant number 80NSSC21K0392 issued through the Exo- +planet Research Program. 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The location of the radiative-convective boundary is Rrcb with density ρrcb. Following +from Owen & Wu (2017), the mass of the convective interior scales as: +Matm ∝ R3 +rcb ρrcb +� RB +Rrcb +� +1 +γ−1 +I2 +� Rc +Rrcb +, γ +� +, +(A1) +where RB = GMc/2c2 +s is the Bondi radius for isothermal sound speed cs = (kBTeq/µmH)1/2. Here, I2 is a dimensionless +integral which accounts for the mass distribution within the atmosphere: +I2 +� Rc +Rrcb +, γ +� += +� 1 +Rc/Rp +x2 +� 1 +x − 1 +� +1 +γ−1 +dr. +(A2) +In the case of hydrodynamic escape of planetary atmospheres, the mass-loss rate +˙M scales as: +˙M ∝ R2 +rcb ρrcb cs Mrcb, +(A3) +where Mrcb is the Mach number of the escaping flow, evaluated at the radiative-convective boundary. For an isothermal +outflow, the Mach number is only a function of RB/Rrcb and given by: +Mrcb = +� +−W0 +� +− +� RB +Rrcb +�4 +exp +� +− C − 4 RB +Rrcb +�� +, +(A4) +where W0 is the real branch of the Lambert W function (see Cranmer 2004) and C is a constant. In the case of XUV +photoevaporation, mass-loss rates are typically higher, meaning C < −3, and for core-powered mass-loss, C = −3 (see +Lamers & Cassinelli 1999). For planets that have maintained a significant mass in H/He i.e. sub-Neptunes, one can +state that their mass-loss timescale tloss = Matm/ ˙M, will be approximately constant. Hence, combining Equations A1 +and A3, one finds that: +tloss = Matm +˙M +∝ +Rrcb +� +RB +Rrcb +� +1 +γ−1 +I2 +cs Mrcb +∝ const. +(A5) +This expression is dominated by the exponential term within Mrcb, and only varies logarithmically with C, Rrcb and +I2. Hence, one can state that: +RB +Rrcb +∝ +Mc +Rrcbc2s +∝ const. +(A6) +Now, by combining Equations 8, 9 and 11 from Owen & Wu (2017), (see also Ginzburg et al. (2016)), which assume +radiative diffusion at the radiative-convective boundary for a cooling/Kelvin-Helmholtz timescale τKH, one finds that +the density at the radiative-convective boundary scales as: +ρrcb ∝ RrcbT 3 +eqτKH +Matmκ +�I2 +I1 +� +, +(A7) + +15 +where κ is the opacity and I1 is another dimensionless integral accounting for the binding energy of the planet: +I1 +� Rc +Rrcb +, γ +� += +� 1 +Rc/Rp +x +� 1 +x − 1 +� +1 +γ−1 +dr. +(A8) +Combining the density at the radiative-convective boundary from Equation A7 with the atmospheric mass from +Equation A1, and noting that RB/Rrcb is approximately constant from Equation A6, one can show that: +Mc +Rrcbc2s +∝ X +1 +2 M +− 1 +2 +c +T +1 +4 +eq κ +1 +4 I +1 +4 +1 I +− 1 +2 +2 +. +(A9) +where the atmospheric mass fraction is defined as X ≡ Matm/Mc and we have assumed that for a set of planets with the +same age, their Kelvin-Helmholtz timescale will be approximately constant. Finally, recalling again that Mc/Rrcbc2 +s is +approximately constant from Equation A6, one finds that: +X ∝ Mc T +− 1 +2 +eq κ− 1 +2 I +− 1 +2 +1 +I2. +(A10) +If one numerically evaluates the dimensionless integrals I1 and I2 (see Figure 11 from Owen & Wu 2017), one can +show that I−0.5 +1 +I2 is approximately constant as a function of Rc/Rrcb. +If one also assumes that the opacity κ is +constant, then one finally finds that the atmospheric mass fraction of planets that have undergone mass-loss scales +approximately linearly with core mass and inversely with the square root of equilibrium temperature. Moreover, this +analytic argument is agnostic with respect to mass-loss models i.e. photoevaporation vs. core-powered mass-loss. The +main takeaway result is that larger mass planets at cooler equilibrium temperatures will retain larger atmospheric mass +fractions if they have undergone mass-loss. This result is key to explain the mass-radius distribution of exoplanets. +Note however that from Figure 2, that whilst final atmospheric mass fraction increases with core mass at a given +equilibrium temperature, it is not a linear relation. This is the case when one fully evaluates the integrals of Equations +A2 and A8 and takes non-constant opacities into account, as is the case in the semi-analytic models of Owen & Wu +(2017); Owen & Campos Estrada (2020) and Ginzburg et al. (2018); Gupta & Schlichting (2019). + diff --git a/edE3T4oBgHgl3EQfHQkd/content/tmp_files/load_file.txt b/edE3T4oBgHgl3EQfHQkd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd76f5c85f5a3fb2722410711cd464c5648961bc --- /dev/null +++ b/edE3T4oBgHgl3EQfHQkd/content/tmp_files/load_file.txt @@ -0,0 +1,1171 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf,len=1170 +page_content='Draft version January 12, 2023 Typeset using LATEX twocolumn style in AASTeX631 Conclusive evidence for a population of water-worlds around M-dwarfs remains elusive James G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Rogers ,1 Hilke E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Schlichting ,1 and James E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Owen 2 1Department of Earth, Planetary, and Space Sciences, The University of California, Los Angeles, 595 Charles E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Young Drive East, Los Angeles, CA 90095, USA 2Astrophysics Group, Department of Physics, Imperial College London, Prince Consort Rd, London, SW7 2AZ, UK ABSTRACT The population of small, close-in exoplanets is bifurcated into super-Earths and sub-Neptunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We calculate physically motivated mass-radius relations for sub-Neptunes, with rocky cores and H/He dominated atmospheres, accounting for their thermal evolution, irradiation and mass-loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' For planets ≲ 10 M⊕, we find that sub-Neptunes retain atmospheric mass fractions that scale with planet mass and show that the resulting mass-radius relations are degenerate with results for ‘water-worlds’ consisting of a 1:1 silicate-to-ice composition ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We further demonstrate that our derived mass-radius relation is in excellent agreement with the observed exoplanet population orbiting M-dwarfs and that planet mass and radii alone are insufficient to determine the composition of some sub-Neptunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Finally, we highlight that current exoplanet demographics show an increase in the ratio of super-Earths to sub-Neptunes with both stellar mass (and therefore luminosity) and age, which are both indicative of thermally driven atmospheric escape processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Therefore, such processes should not be ignored when making compositional inferences in the mass-radius diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Keywords: planets and satellites: atmospheres - planets and satellites: physical evolution - planet star interactions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' INTRODUCTION The observed population of small, close-in exoplanets with radii ≲ 4R⊕ and orbital periods ≲ 100 days (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Borucki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Fressin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Silburt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Zink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2022) provide an intriguing problem in terms of their formation pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' With no analogue in our solar system, such planets have been observed to bifurcate into two separate sub-populations, centred at ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='3R⊕ (referred to as ‘super-Earths’) and ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='4R⊕ (referred to as ‘sub-Neptunes’), with a scarcity of planets in-between at ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='8R⊕ labelled as the ‘radius gap’ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Two categories of evolutionary models have emerged to explain this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' The first relies on at- mospheric evolution, as it is known that many sub- Neptunes require a significant H/He dominated atmo- sphere to explain their observed mass and radius (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Weiss & Marcy 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Jontof-Hutter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ben- neke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Under this class of models, super- Earths are expected to have lost their primordial at- mosphere and are thus observed at their core1 radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Sub-Neptunes, on the other hand, have maintained their atmosphere, bloating their size to the observed peak at ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='4R⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Typically, atmospheric mass-loss is thought to cause this bifurcation, with smaller mass, highly ir- radiated planets losing their hydrogen atmospheres to become super-Earths, whilst larger mass, colder plan- ets remain as sub-Neptunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Two successful mass-loss models are XUV photoevaporation, which relies on high- energy stellar flux (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Owen & Wu 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Lopez & Fort- ney 2013) and core-powered mass-loss, which calls upon remnant thermal energy from formation and bolometric stellar luminosity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Gupta & Schlichting 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Other models may also explain the radius gap via atmospheric escape due to giant impacts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Inamdar & Schlichting 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2020) 1 The term ‘core’, as used for the remainder of this letter, refers to the solid/liquid bulk interior of a planet, as opposed to the geological nomenclature of the iron core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In general, the mass of a sub-Neptune is approximately given by its core mass, as the atmospheric mass makes up less, often much less, than 10% of the planet’s total mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='04321v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='EP] 11 Jan 2023 ID2 or through gaseous accretion of primordial atmospheres (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Lee & Connors 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' The alternative model to atmospheric mass-loss and evolution is the ‘water-world’ hypothesis, in which the radius gap arises due to a difference in planet composi- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' super-Earths consisting of a silicate-iron mixture, sub-Neptunes consisting of an ice-silicate mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Since a planetary core of a given mass increases in size for lower bulk densities, the water-world/sub-Neptune pop- ulation exists at a larger size and hence separated from the rocky super-Earths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Mordasini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ray- mond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Turbet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Mousis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' A corollary of this model is that sub- Neptunes form exterior to the protoplanetary disc water- ice line, in which there is an increased solid mass due to the condensation of volatiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Pebble accretion has been shown to predict such planets will have silicate-to-ice ratios of 1:1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Lodders 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Bitsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ven- turini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Br¨ugger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2020), although no ex- planation has been provided as to why such planets did not accrete voluminous H/He dominated atmospheres, as would be expected from accretion and migration in a protoplanetary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Nonetheless, under this model, water-worlds then migrate inwards to the locations at which we observe them today, thus forming the popula- tion of close-in sub-Neptunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Sub-Neptunes consisting, on a population level, of a 1:1 silicate-to-ice mixtures are in tension with the pre- dictions from atmospheric mass-loss models since both photoevaporation and core-powered mass-loss models find that both super-Earths and sub-Neptunes have core bulk densities, and hence compositions, roughly consis- tent with that of Earth (33% iron, 67% silicate e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Wu 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Gupta & Schlichting 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Rogers & Owen 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In light of the difference between the water-world and the atmospheric-mass loss models, it follows that unlocking the origins of the radius gap is of crucial importance for understanding the underlying formation pathways of small, close-in exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In a recent study, Luque & Pall´e (2022) asserted that there is evidence for a population of water-worlds among planets orbiting M-dwarfs, by comparing ob- served planet masses and radii with various planet- composition models in the mass-radius diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' They find that many planets in their sample are consistent with a 1:1 silicate-to-ice composition ratio, as well as population synthesis modelling from Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' They also use the mass-radius relations for rocky bodies hosting H/He dominated atmospheres from Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2019) to claim that the planet sample was inconsis- tent with rocky cores hosting H/He atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Un- fortunately, these adopted mass-radius models for H/He dominated atmospheres were not appropriate for plan- ets at fixed age i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' analogous to stellar isochrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In order to do this analysis, one must consider the thermal evolution and atmospheric mass-loss with the associated changes in entropy of H/He atmospheres over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In this letter, we calculate physically motivated, self- consistent, population-level mass-radius relations for rocky planets hosting H/He atmospheres, which cru- cially take into account atmospheric evolution and ir- radiation from the host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We show that the sample of observed sub-Neptunes around M-dwarfs from Luque & Pall´e (2022) is, in fact, also consistent with rocky cores hosting H/He dominated atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We con- clude that, currently, there is no conclusive evidence for differentiating, on a population level, between water- worlds and rocky cores hosting H/He atmospheres from sub-Neptune masses and radii alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' METHOD It is commonplace for mass-radius diagrams to be used as a visual guide to the population of observed exoplan- ets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g Wu & Lithwick 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Weiss & Marcy 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Had- den & Lithwick 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Rogers 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Dressing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Wolfgang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Chen & Kipping 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' To interpret the observations, theoretical mass-radius relations are used to plot a planet’s size as a function of mass for a given composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' For solid bod- ies consisting of iron, silicate and ice mass fractions, the models of Fortney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2019) are commonly used, in which the planet radius Rp scales ap- proximately as Rp/R⊕ ∝ (Mp/M⊕)1/4, where Mp is the planet mass (Valencia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Specifically the mod- els of Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2019), which were adopted in Luque & Pall´e (2022), give the following mass-radius relation for water-worlds consisting of a 1:1 silicate-to-ice ratio: Rp R⊕ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='24 � Mp M⊕ �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='27 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (1) Our task is to determine the mass-radius relation for rocky/iron-rich cores with a H/He dominated atmo- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Crucially, we aim to calculate physically- motivated, self-consistent mass-radius relations, which incorporate the physics of atmospheric evolution, includ- ing mass-loss and cooling, which strongly modifies the mass-radius relation from the canonical H/He results of Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We highlight that the H/He mass- radius models from Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2019) assume constant specific entropy in a purely adiabatic atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' The assumption of constant specific entropy (which sets the adiabat) for planets of varying mass is not accurate for planets that have undergone thermodynamic processes such as cooling and mass-loss, which naturally reduce 3 the specific entropy of a planet and depend on many variables such as planet mass and equilibrium temper- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Moreover, in the Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2019) mass-radius relations, each model’s specific entropy is parameterised with a temperature defined at a fixed pressure of 100 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We note that this temperature is often mistaken for the equilibrium temperature of a given planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' The tem- perature and density of a purely adiabatic atmosphere will drop far below the equilibrium temperature within a few scale heights of the planet’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In reality, an outer radiative layer will form as a planet comes into radiative equilibrium with the host star (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Guillot 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Piso & Youdin 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We point the reader to the mass-radius relations of Lopez & Fortney (2014), which account for thermal evolution, including radiative-convective mod- els that provide planet size at a constant age for a given mass and H/He mass fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Method Atmospheric mass-loss sculpts the exoplanet popula- tion such that planets with larger core masses and there- fore deeper gravitational potential wells retain larger atmospheric masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' This is also true for planets at cooler equilibrium temperatures, since they receive a smaller integrated stellar flux, which drives hydrody- namic escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Whilst one can explore these basic de- pendencies analytically (see Appendix A), the easiest way to fully understand these effects, in conjunction with thermodynamic cooling and contraction, is with numerical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Firstly we use the semi-analytic nu- merical models for XUV photoevaporation from Owen & Wu (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Owen & Campos Estrada (2020) and core- powered mass-loss from Gupta & Schlichting (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2022) to numerically model populations of planets undergoing atmospheric-mass loss driven by both mechanisms (see Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2021, for a full dis- cussion of both models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' For both models we assume an atmospheric adiabatic index of γ = 5/3, a core heat capacity of 107 erg g−1 K−1 (Valencia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2010) and an opacity scaling law of κ ∝ P αT β, where α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='68, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='45 and κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='29 × 10−2 cm2 g−1 at 100 bar and 1000 K (Rogers & Seager 2010a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Both models rely on the hydrodynamic escape of hydrogen-dominated mate- rial, hence we expect the predicted mass-radius distri- butions to be very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Chronologically speaking, there are three dominant atmospheric processes that small, close-in exoplanets with H/He dominated atmospheres undergo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Firstly, at- mospheric mass is accrued via core-nucleated accretion whilst immersed in a protoplanetary disc (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Rafikov 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Piso & Youdin 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Then, as the disc disperses, the atmo- spheric mass of some planets is rapidly removed through a “boil-off” process (also referred to as “spontaneous mass-loss”) as the confining pressure from the disc is re- moved on timescales ∼ 105 yrs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ikoma & Hori 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Owen & Wu 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' This is ap- propriate for smaller mass planets ≲ 10M⊕, since larger mass cores may open gaps in the gaseous protoplanetary disc, resulting in different atmospheric evolution during disc dispersal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Finally, once the disc has completely dis- persed, these latter processes transition into XUV pho- toevaporation and core-powered mass-loss (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Lopez & Fortney 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Owen & Wu 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Gupta & Schlichting 2019) combined with thermal cool- ing and contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Since we are not explicitly incorporating gaseous ac- cretion and boil-off, our initial conditions must encom- pass such processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' To account for both scenarios, namely in which boil-off does/does not occur, we adopt two sets of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' The first scenario assumes that planets have undergone a boil-off phase during disc dispersal, for which we assume that planets host an ini- tial atmospheric mass-fraction according to: Xinit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='01 � Mc M⊕ �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='44 � Teq 1000 K �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='25 , (2) which comes from the theoretical predictions of Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2016), which account for core accretion and boil-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In the inference work from Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2022), the authors showed that this relation can be ac- curately recovered from the data by inferring the corre- lation between core mass and atmospheric mass fraction prior to XUV photoevaporation for a sample of Kepler, K2 and TESS planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In the second scenario, we do not enforce a boil-off phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' There is currently uncertainty as to the details of this mechanism, as it is a non-standard escape process, particularly at high masses whereby gaps can be opened in the protoplanetary disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In light of this, we provide an additional agnostic set of initial atmospheric mass fractions, drawn log-uniformly in the range: log Xinit ∼ U(10−3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='3), (3) where U is a uniform distribution, where the lower limit avoids large mass cores hosting negligible atmospheric mass fractions (we assume planets with X ≤ 10−4 to be completely stripped i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' super-Earths), whilst the up- per limit is chosen to avoid self-gravitating atmospheres which are known to be extremely rare and the semi- analytic models do not account for (Wolfgang & Lopez 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In essence, this distribution accounts for all possible initial atmospheric conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 4 GJ 3470 b GJ 436 b K2 18 b GJ 1214 b GJ 3470 b GJ 436 b K2 18 b GJ 1214 b Photoevaporation Boil-Off Initial Conditions Core-Powered Mass-Loss Agnostic Initial Conditions Observations Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Synthetic mass-radius distributions are shown for populations of planets evolved with photoevaporation and core- powered mass-loss in left and right-hand panels, respectively, coloured by their equilibrium temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Super-Earths are stripped of their H/He dominated atmospheres and fall onto a relation consistent with an Earth-like composition (brown-dashed), whilst sub-Neptunes retain a significant atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In the top panels, we assume an initial distribution of atmospheric masses appropriate for a boil-off scenario (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2), in which planets lose a significant amount of H/He mass during disc dispersal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We characterise the resulting narrow mass-radius distribution with a median line (orange dashed, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In the middle panels, we adopt agnostic initial conditions (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 3) and parameterise this mass-radius relation with 2σ limits (orange dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In the bottom panels, we compare our theoretical mass-radius distributions (orange dashed/dotted lines, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 6) with the observed sample of M-dwarf orbiting exoplanets from Luque & Pall´e (2022), together with the mass-radius relation for water-worlds (blue solid line, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We find that boil-off initial conditions provide mass-radius relations that are completely degenerate with that of water-worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Furthermore, even when adopting agnostic initial conditions, the observations are accurately reproduced since the mass-radius distribution is naturally explained due to mass-loss and cooling/contraction of H/He dominated atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We highlight planets with confirmed escaping H/He detections with blue-shaded regions (namely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' K2 18 b, GJ 3470 b, GJ 436 b and, tentatively, GJ 1214 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 5 As we shall show, even this agnostic set of initial condi- tions accurately reproduces the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We assume planetary cores are of Earth-like composi- tion, such that they have a silicate-to-iron ratio of 67:33 (Owen & Wu 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Gupta & Schlichting 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Rogers & Owen 2021), making use of the mass-radius relations from Fortney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' To approximately match the stellar sample from Luque & Pall´e (2022), we adopt a Gaussian stellar mass distribution centred at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='3M⊙ with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='1M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We evolve a popu- lation of 105 planets for each mass-loss model for 5 Gyrs to match the approximate ages of observed planets, al- though this final age makes no difference to the final mass-radius distribution2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We randomly draw orbital periods from a broken power law: dN d logP ∝ � � � P a, P < P0 days P b, P > P0 days, (4) where a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0, b = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='5 and P0 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0 are chosen to approximately match the population of observed plan- ets orbiting M-dwarfs from Kepler (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We also place an upper limit on orbital periods of 30 days, since most M dwarf orbiting planets with mea- sured masses and radii are observed with TESS, which has a baseline capable of observing planets out to this orbital separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We randomly draw the planet core masses in a log-uniform manner, so as to evenly sample the mass-radius diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Finally, we remove planets with an RV semi-amplitude ≤ 30 cm s−1 to approxi- mate current RV sensitivity limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' RESULTS AND DISCUSSION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Mass-radius relation for sub-Neptunes with rocky cores and H/He atmospheres Figure 1 demonstrates the mass-radius relations for a population of rocky cores, initially hosting H/He rich atmospheres, that have undergone thermal evo- lution and atmospheric mass-loss over 5 Gyrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Since both photoevaporation (see left-hand panels) and core- powered mass-loss models (see right-hand panels) de- rive from hydrodynamic escape mechanisms, their pre- dictions are very similar in this plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' The bimodal distribution is clearly seen, with super-Earths typically residing at orbital separations corresponding to higher equilibrium temperatures and having been stripped of their hydrogen-dominated atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' As such, super- Earths fall on an Earth-like composition line in the 2 This is because the trend of entropy with mass is maintained across various ages, meaning that the slope and position of the mass-radius plane are age-insensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' mass-radius diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Sub-Neptunes, on the other hand, maintain, despite some atmospheric mass-loss, a H/He atmosphere, the amount of which scales with core mass among other variables, such that more massive cores retain larger atmospheric mass-fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' These H/He atmospheres increase the radii of sub-Neptunes above that expected for an Earth-composition core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We note that the mass-radius relation for planets in the absence of atmospheric mass-loss is less steep with planet mass for a given atmospheric mass fraction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' see Figure 1 of Lopez & Fortney 2014), as opposed to the models of Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2019) (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Hence, the mass- radius observations for sub-Neptunes can only be fit with H/He atmospheric mass-fractions that scale with planet mass, which is a natural outcome of the hydro- dynamic atmospheric-loss processes discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In the top and middle panels, we show mass-radius distributions for both sets of initial conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' boil-off (see Equation 2) and agnostic (see Equation 3) respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' One can see that the boil-off scenario produces a narrow mass-radius distribution, whilst the agnostic initial conditions produce a wider range in sub-Neptune radii for a given mass, owing to the increased range in initial atmospheric mass fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Note however that even with this set of agnostic initial conditions, the wider spread in sub-Neptune sizes shrinks as the planets cool and contract to smaller radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' As such, the majority of sub-Neptunes sit close to the models which started with boil-off initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' This is because thermal evo- lution and mass-loss of H/He dominated atmospheres naturally produce this relation, independent of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' To provide a useful reference for comparison with future observations, we quantify these mass-radius re- lations, which we highlight are appropriate for sub- Neptunes in the range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0 ≲ Mp/M⊕ ≲ 30, with quartic logarithmic functions: Rp R⊕ = a0 + a1 ln � Mp M⊕ � + a2 ln � Mp M⊕ �2 + a3 ln � Mp M⊕ �3 + a4 ln � Mp M⊕ �4 , (5) where coefficients are summarised for photoevaporation and core-powered mass-loss in Tables 1 and 2 respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Since the boil-off scenario is extremely narrow (top panels of Figure 1), we quantify its median value (orange dashed line) for both models by calculating their median planet size for bins in planet mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We then fit these median values to Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Similarly, for the ag- nostic initial conditions (bottom panels of Figure 1), we quantify this wider mass-radius distribution by finding 2σ limits in planet size for planet mass bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 6 Boil-Off Agnostic (lower) Agnostic (upper) a0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='3104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='2131 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='5776 a1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='2862 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='2326 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='7713 a2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='1329 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='5921 a3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='2325 a4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0301 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Coefficients for mass-radius relations for photo- evaporation, given by a quartic logarithmic equation from Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Boil-off initial atmospheric conditions (see dashed-orange line in Figure 1) are from Equation 2, agnos- tic initial atmospheric conditions (see dotted-orange lines in Figure 1) are from Equation 3, with upper and lower planet size bounds given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Boil-Off Agnostic (lower) Agnostic (upper) a0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='3255 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='5776 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='2131 a1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='4168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='7713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='2326 a2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='1567 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='5921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0139 a3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='07224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='2325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0367 a4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='01092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0065 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Same as Table 1, but for core-powered mass-loss models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In the bottom panels of Figure 1, we show our pre- dicted mass-radius relations from Equation 6 alongside the observed sample from Luque & Pall´e (2022), consist- ing of 48 planets orbiting 26 M-dwarfs systems with stel- lar masses 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='1 ≲ M∗/M⊙ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Figure 1 clearly demon- strates that the mass-radius observations are in excellent agreement with sub-Neptunes which have rocky interi- ors and H/He atmospheres provided that their thermal evolution and mass-loss histories are accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In the case of boil-off initial conditions, (orange- dashed line) the mass-radius relation from our atmo- spheric evolution and mass-loss models is degenerate with bodies of a 1:1 silicate-to-ice ratio (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2019) (blue solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In the case of agnostic initial conditions, (orange dotted-lines) the mass-radius rela- tion encompasses all observed planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Finally, we also highlight planets with blue-shaded circles that have con- firmed escaping hydrogen/helium atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Namely, these are K2 18 b (Benneke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' dos Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2020), GJ 436 b (Bean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Pont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Knutson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2016), GJ 3470 b (Fukui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Nascimbeni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Crossfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Dragomir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Awiphan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Bourrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ninan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2020) and GJ 1214 b (although we highlight that this is a tentative detection from Orell-Miquel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' K2 18 b is an interesting case, since it is close3 to the mass-radius relations for atmospheric evolution (orange dashed line) and water-worlds (blue solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' However, the direct hydrogen detection suggests it is inconsistent with the water-world hypothesis since such planets can- not host significant hydrogen atmospheres whilst still being consistent with observed masses and radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We also note that whilst many observed intermediate mass planets (2 ≲ Mp/M⊕ ≲ 10) are tightly clustered around the mass-radius relations for water-worlds and H/He atmospheres with boil-off initial conditions, there are many high-mass planets, including those with escap- ing H/He detections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' GJ 3470 b, GJ 436 b, and GJ 1214 b, that sit above both of these mass-radius relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' They do however sit within the bounds of H/He mass relations with agnostic initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' As discussed in Section 2, boil-off is likely inefficient for planets with Mc ≳ 10M⊕ since such planets will begin to open gaps in their protoplanetary discs, hence implying the agnos- tic initial conditions (Equation 3) are more appropriate for such planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' The observations appear to support this notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We highlight that more work is needed to understand boil-off and that these planets provide im- portant tests of such processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Verifying mass-radius relations with MESA Whilst the semi-analytic model of atmospheric evo- lution for photoevaporation from Owen & Wu (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Owen & Campos Estrada (2020) and core-powered mass-loss from Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Gupta & Schlicht- ing (2019) are computationally inexpensive and thus al- low large populations of planets to be generated, they lack complex physics such as a detailed model for convec- tion, self-gravity and realistic equations of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' There- fore, as in Owen & Wu (2017), we corroborate our semi- analytical modelling from Figure 1 by comparing our results with numerical models performed with Modules for Experiments in Stellar Astrophysics (MESA) (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2011, 2013, 2015, 2018), which solves and evolves the stellar structure equations with accurate H/He equa- tions of state from Saumon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (1995) and dust-free opacity tables from Freedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2008) for low- mass and irradiated planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' These sophisticated mod- els remove free parameters from the problem, such as choices in adiabatic index and opacities since these are determined self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We follow previous works to model low-mass planets (Owen & Wu 2013, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Chen & Rogers 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Kubyshkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Malsky & 3 In fact, other literature values would place K2 18 b precisely on the mass-radius relations for H/He atmospheres and water-worlds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Sarkis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 7 Rogers 2020), and evolve each model for 5 Gyrs, adopt- ing stellar irradiation performed with the F∗ − Σ rou- tine from MESA (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2013), which injects ir- radiative flux within a column density of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' For these models, we follow Owen & Wu (2013, 2016) and assume Σ = 250 g cm−2, appropriate for opacities to incoming stellar irradiation of κν = 4 × 10−3 cm2 g−1 (Guillot 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In Figure 2, we show populations of planets evolved with MESA at equilibrium temperatures of 300K, 500K and 800K in the top, middle and bottom panels respec- tively, represented with black triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' For simplicity, we only compare these results with the semi-analytic photoevaporation models since planets of different core masses can be stripped at slightly different equilibrium temperatures under the core-powered mass-loss model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' For the purposes of population-level mass-radius dia- grams, however, the differences between the two models are inconsequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' One can see from Figure 2 that the MESA models are in excellent agreement with our adopted semi-analytic photoevaporation models which are also shown in Fig- ure 2 for small ranges in equilibrium temperatures at 300 ± 10 K, 500 ± 10 K and 800 ± 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Note that mass-loss is not explicitly included in these MESA mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Instead, we adopt the final atmospheric mass- fractions from the semi-analytic photoevaporation mod- els (as shown with black triangles in the left-hand pan- els of Figure 2) and then evolve the planets in MESA with this atmospheric mass fraction to calculate their radii after 5 Gyrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Since the majority of atmospheric escape under photoevaporation typically occurs in the first ∼ 100 Myr, this is akin to beginning the MESA sim- ulations at the end of this period in order to accurately determine their radii at 5 Gyrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' As Figure 2 demon- strates, these models robustly confirm the mass-radius relations found with our semi-analytic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Figure 2 also highlights important points about at- mospheric evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Firstly, in the left-hand panels, the final atmospheric mass fractions demonstrate that planets of different core masses evolve to host very dif- ferent atmospheric mass fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' For an initial boil-off distribution represented with grey points (see Equation 2), low-mass planets are stripped of their atmosphere (numerically identified with an atmospheric mass frac- tion ≤ 10−4) whilst the highest-mass planets retain most of their initial atmospheric mass and therefore match the initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Intermediate-mass planets, how- ever, lose progressively less atmosphere with increas- ing core mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' A common generalisation is that sub- Neptunes host an atmospheric mass-fraction of ∼ 1% (Owen & Wu 2017), since this value maximises the mass-loss timescale and naturally leads to a population of planets that retain their H/He atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Whilst this is good to an order of magnitude, Figure 2 clearly demonstrates that this an oversimplification, with larger planets naturally retaining a greater atmospheric mass fraction due to their increased gravitational potential- wells (which is also shown analytically in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Furthermore, this distribution changes as a function of equilibrium temperature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' different rows in Figure 2), with planets at lower equilibrium temperatures able to maintain more atmospheric mass for a given core mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Figure 2 also demonstrates the importance of initial con- ditions for such planets, since high-mass planets main- tain an atmospheric mass that follows their initial dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' The population-level mass-radius diagram (as shown in Figure 1) is therefore a superposition of differ- ent planets at different core masses, equilibrium temper- atures and ages, with their initial conditions playing a progressively more influential role for higher masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Comparison with Zeng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2019 models In the right-hand panel of Figure 2, we compare our semi-analytic and MESA models with numerical models of Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2019), which provide mass-radius relations for rocky cores hosting a H/He atmospheric mass frac- tion under the assumption of constant specific entropy, defined with a temperature at a pressure of 100 bar, al- though we highlight that this temperature is frequently misinterpreted as the planetary equilibrium tempera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' For reference, for an adiabat with γ = 5/3, set such that its temperature is 500 K and 100 bar, the temper- ature at 1 bar is ≲ 80 K, which is far below the typ- ical planetary equilibrium temperatures currently ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We stress that such models are not applicable to evolved sub-Neptunes to perform quantitative analy- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Examples of these mass-radius relations are shown in black-dashed lines in Figure 2 for atmospheric mass fractions of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='1% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0% with specific entropy set with a temperature of 500 K and 100 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' These curves have a characteristic and dramatic increase in size for smaller-mass planets, which comes from the assumption of constant atmospheric mass at constant specific en- tropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' However, atmospheric evolution naturally allows planet atmospheres to cool and contract, with smaller- mass planets cooling more due to their reduced heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Combining this with mass-loss, which further reduces the atmospheric mass retained by smaller mass planets results in the mass-radius relations found in Fig- ure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We note that if one wishes to analyse an individ- ual planet in the mass-radius diagram, then the mass- radius relations of Lopez & Fortney (2014) at constant age, which are also shown in Figure 2, are more ap- 8 1 2 3 4 10-4 10-3 10-2 10-1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' The final atmospheric mass fractions (left-hand column) and planet radii (right-hand column) after 5 Gyr of photoevaporative evolution for populations of planets with equilibrium temperatures of 300 ± 10 K (top row), 500 ± 10 K (middle row) and 800 ± 10 K (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Colours represent planet size in the left-hand panels and final atmospheric mass fraction in the right-hand panels, demonstrating that larger atmospheric mass fractions lead to larger planets and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' All planets start their evolution with an initial distribution of atmospheric mass fractions (displayed as grey points) that account for gaseous core accretion and boil-off during protoplanetary disc dispersal (see Equation 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' To corroborate these semi-analytic results, we also perform numerical models with MESA, which are shown as black triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In general, sub-Neptunes only exist at larger masses for higher equilibrium temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' The mass-radius distribution (as seen in Figure 1) is a superposition of all equilibrium temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In the right-hand column, we compare our mass-radius distributions with the models of Lopez & Fortney (2014) in blue and Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2019) in black for atmospheric mass fractions of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='1% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We highlight that the models of Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2019) assume constant specific entropy defined with a temperature at fixed pressure at 100 bar (not to be confused with the equilibrium temperature) and therefore suggest a dramatic increase in planet radius for lower-mass planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' The models of Lopez & Fortney (2014) consider irradiation and cooling for planets with constant atmospheric mass fraction, meaning they are more appropriate for analysis of planet composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Our mass-radius models account for loss-induced scaling of atmospheric mass with planet mass, meaning they are appropriate for comparisons of planet populations in the mass-radius diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 9 propriate since these include the essential physical pro- cesses (cooling and irradiation for a given atmospheric mass fraction) that shape the radius of small exoplanets with hydrogen atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Alternatively, the publicly available evapmass code from Owen & Campos Estrada (2020) includes the semi-analytic atmospheric structure models adopted in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In the case of analysing populations of planets in the mass-radius diagram, we recommend the relations derived in this work (see Equa- tion 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Mass-radius relations for sub-Neptunes around FGK stars In this letter, we have focused on planets orbiting M dwarfs, as is the case with the observational work of Luque & Pall´e (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' As we have shown, our choice of physically motivated initial conditions and ranges in equilibrium temperatures yield mass-radius relations with an intrinsic spread in planet radii for a given mass (see Figures 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' There are, however, additional factors that can contribute to the mass-radius distribu- tion spread, that we have not included in our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' As highlighted in Kubyshkina & Fossati (2022), vari- ability in high-energy stellar luminosity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Johnstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ketzer & Poppenhaeger 2022) can increase the range in planet sizes, since stars of different initial rotation rates will produce different X-ray/EUV flux and thus different mass-loss rates for the orbiting planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In addition, observational uncer- tainties in planet radii will increase the spread in the mass-radius distribution due to purely statistical scat- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' It is interesting to note that the underlying mass- radius distribution does not significantly change when considering planets orbiting FGK stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Although such planets will receive a larger flux at a given orbital pe- riod, we find from our mass-radius models that this bias tends to simply produce a larger ratio of super-Earth to sub-Neptune occurrence rates, since more planets can be stripped of their H/He atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Indeed, this re- sult is consistent with the demographic work of Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2022), from which one can calculate the ratio in occurrence rates of super-Earths to sub-Neptunes to find it increased, with values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='07, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='10 for a stellar mass bins of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='7]M⊙, [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='1]M⊙ and [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='4]M⊙ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We do high- light, however, that there are other ways in which this ratio may increase, such as varying the core mass or or- bital period distributions as a function of stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' One major difference between low and high-mass stars, however, is that transit observations (such as those from Kepler, K2 and TESS) can achieve a higher pho- tometric precision around M dwarfs due to their smaller stellar radii and hence larger Rp/R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Such surveys are also biased to observe planets within a smaller range in equilibrium temperatures since the transit probabil- ity of planets at large orbital periods (and therefore low equilibrium temperatures) decreases rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Different mission targeting strategies also change the observed population e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Kepler was sensitive to planets with or- bital periods ∼ 100 days, but specifically targeted FGK stars, whereas TESS currently targets nearby bright stars (and is therefore biased to M-dwarfs), with sen- sitivity out to orbital periods ∼ 30 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' These argu- ments taken together suggest that the observed mass- radius distribution around M-dwarfs is expected to have less scatter compared to that for planets around FGK stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' This is indeed the case when comparing Figures 1 and S19 from Luque & Pall´e (2022) (see also Fig- ure 12 from Rogers & Owen 2021, for an example of a synthetic mass-radius distribution for FGK stars in the presence of bias and measurement uncertainty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We find that the underlying mass-radius distribution, in the absence of statistical scatter and bias4 (as summarised by Equation 6 under different initial conditions), is ap- proximately the same across FGKM spectral types but that the relative occurrence of super-Earths with respect to sub-Neptunes increases for more massive (luminous) stellar types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' CONCLUSIONS In this letter, we calculate mass-radius relations of small, close-in exoplanets that host H/He dominated atmospheres with self-consistent, physically motivated evolution models, which have the following form: Rp R⊕ = a0 + a1 ln � Mp M⊕ � + a2 ln � Mp M⊕ �2 + a3 ln � Mp M⊕ �3 + a4 ln � Mp M⊕ �4 , (6) where coefficients are summarised for photoevaporation and core-powered mass-loss models in Tables 1 and 2 re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We consider two sets of initial conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' the first, in which planets undergo a boil-off phase, whereby a large fraction of atmospheric mass is lost during proto- planetary disc dispersal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ikoma & Hori 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Owen & Wu 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2016), which yields a rel- atively tight mass-radius relation after mass-loss and thermal evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In the second scenario, we adopt ag- nostic initial conditions (see Equation 3), which yields a 4 We also highlight that the bias of planet mass measurements is currently not quantifiable, since such surveys are not based on homogeneous observations of a well-defined sample of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 10 larger spread in final radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' These relations (see Figure 1) incorporate thermodynamic cooling, atmospheric es- cape and stellar irradiation and are therefore suited for compositional analyses of populations of sub-Neptunes in the mass-radius diagram (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g Wu & Lithwick 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Weiss & Marcy 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Hadden & Lithwick 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Rogers 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Dressing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Wolfgang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Chen & Kipping 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We show that accounting for atmospheric mass-loss yields left-over at- mospheric mass fractions that scale with planet mass i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' larger planets retain a larger fraction of their total mass in hydrogen and show that these results give an excellent match to the mass and radius measurements of sub-Neptunes in Luque & Pall´e (2022), independently of assumed initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In addition, we show that the boil-off initial conditions yield a mass-radius relation that is completely degenerate with that corresponding to a 1:1 silicate-to-ice ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We note that our study moves beyond the H/He mass-radius relations of Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2019), as demonstrated in Figure 2, which assume a constant specific entropy for constant atmospheric mass factions as a function of planet mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We find that such models are therefore not applicable to planets undergo- ing atmospheric evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In Luque & Pall´e (2022), a sample of observed exo- planets orbiting M-dwarfs is used to argue that plan- ets with rocky interiors and H/He atmospheres can- not explain the observed cluster of planets around the 1:1 silicate-to-ice ratio compositional line in the mass- radius diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In Figure 1 we have presented our new mass-radius relations for small planets with hydro- gen atmospheres (see Equation 6) and show that they are in fact completely consistent with the data, once thermal evolution and mass-loss are properly accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' A strong degeneracy therefore still exists between the water-world and silicate/iron-hydrogen models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We find that planets with different equilibrium tempera- tures and atmospheric masses for a given core mass yield a natural spread in the mass-radius relation (see Fig- ure 1) that does not vary dramatically for different stel- lar types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We do note that other factors that we have not taken into account, such as high-energy stellar lu- minosity variability (Kubyshkina & Fossati 2022) and observational uncertainty will act to increase the spread in the sub-Neptune mass-radius relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Nevertheless, we note that many high-mass planets ≳ 10M⊕ in the sample from Luque & Pall´e (2022), including GJ 436 b, GJ 3470 b and GJ 1214, which have confirmed es- caping H/He atmospheric detections, sit well above the mass-radius relations for both water-worlds and hydro- gen atmosphere models that assume an initial boil-off scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We speculate that such planets were less sus- ceptible to boil-off (Owen & Wu 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2016) due to their increased mass (potentially due to gap-opening in their protoplanetary discs) and therefore entered the XUV photoevaporation/core-powered mass- loss phase with larger atmospheric mass fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' We highlight that further work is needed to understand this important stage in exoplanet evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In light of the results shown in Figure 1, we corrob- orate the well-known result that a planet’s mass and radius alone are often insufficient to break its internal composition degeneracy (Valencia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Rogers & Seager 2010b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Probing for hydrogen and helium pres- ence around low-mass planets with spectroscopic obser- vations is one promising avenue (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ehrenreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Lavie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Bourrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Yan & Hen- ning 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Spake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' dos Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Ninan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2022) although we high- light that a non-detection in hydrogen Ly-α does not necessarily indicate the lack of a hydrogen-dominated atmosphere (Owen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Moreover, observations from JWST may provide insights into the abundance of H2O in high mean-molecular weight atmospheres of sub-Neptunes and thus the prevalence of water-worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In this letter, we have also analysed the occurrence rates from Petigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2022) to find that the ratio of super-Earths to sub-Neptunes increases, with values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='07, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='10 for a stellar mass bins of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='7]M⊙, [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='1]M⊙ and [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='4]M⊙ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Since larger mass stars produce larger lu- minosities, this result tentatively supports the notion that stellar irradiation is key in evolving sub-Neptunes into super-Earths via atmospheric escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In addition, if one can accurately measure planet age, then one can determine how planets and the observed radius gap, separating the super-Earths from the sub- Neptunes, evolves with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Under atmospheric evo- lution models, the radius gap is expected to evolve on ∼ 100 Myr to Gyr timescales (Gupta & Schlichting 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Rogers & Owen 2021) since hydrogen-dominated atmo- spheres dramatically change in size as they cool due to their low mean-molecular weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Water-worlds on the other hand will not significantly change in size af- ter formation since their sizes are dominated by their ice-silicate composition and not H/He dominated at- mospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Indeed this demographic analysis has been performed in the works of Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2020b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Sandoval et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2022) to show that the radius gap evolves on ∼ 100 Myr to Gyr timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Moreover, in a recent study from Fernan- des et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2022), occurrence rates were calculated for a sample of exoplanets around young stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' They find tentative evidence for the decrease in sub-Neptune size 11 with stellar age, which is indicative of cooling and con- traction of sub-Neptunes with time, although we note that presently this sample size is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' A larger sam- ple of planets with accurate ages may shed light on this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Furthermore, mass measurements of young sub- Neptunes may show that such planets are indeed in- flated and therefore extremely under-dense H/He-rich proto-sub-Neptunes, destined to cool and contract to the evolved population we observe today (Owen 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Whilst water-worlds may exist in conjunction with plan- ets hosting H/He rich atmospheres, the evidence for their existence as a population still remains elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We would like to kindly thank Akash Gupta, Ruth Murray-Clay, Erik Petigura and Vincent Van Eylen for discussions that helped improve the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' JGR is supported by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Sloan Foundation under grant G202114194 as part of the AEThER collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' HES gratefully acknowledges support from NASA under grant number 80NSSC21K0392 issued through the Exo- planet Research Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' JEO is supported by a Royal Society University Research Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' This project has received funding from the European Research Coun- cil (ERC) under the European Union’s Horizon 2020 re- search and innovation programme (Grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 853022, PEVAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' For the purpose of open access, the authors have applied a Creative Commons Attribution (CC-BY) licence to any Author Accepted 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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=', Christiansen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=', & Hansen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' 2019, MNRAS, 483, 4479, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='1093/mnras/sty3463 14 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' ANALYTIC ARGUMENTS A natural outcome of atmospheric mass-loss is that larger planetary cores at a cooler equilibrium temperature retain a larger atmospheric mass fraction due to their increased gravitational potential well and reduced irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' As we demonstrate in Figures 1 and 2, if one takes this simple fact into account, the mass-radius relation of planets that have undergone atmospheric escape naturally reproduces the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' To analytically show that larger planetary cores at cooler equilibrium temperatures retain a larger atmospheric mass fraction, consider a planet with core mass Mc and radius Rc, equilibrium temperature Teq, hosting a H/He dominated atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' This atmosphere is split into a convective interior (assumed to be adiabatic with index γ) and radiative exterior (assumed to be isothermal, with a temperature equal to Teq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' The location of the radiative-convective boundary is Rrcb with density ρrcb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Following from Owen & Wu (2017), the mass of the convective interior scales as: Matm ∝ R3 rcb ρrcb � RB Rrcb � 1 γ−1 I2 � Rc Rrcb , γ � , (A1) where RB = GMc/2c2 s is the Bondi radius for isothermal sound speed cs = (kBTeq/µmH)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Here, I2 is a dimensionless integral which accounts for the mass distribution within the atmosphere: I2 � Rc Rrcb , γ � = � 1 Rc/Rp x2 � 1 x − 1 � 1 γ−1 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (A2) In the case of hydrodynamic escape of planetary atmospheres, the mass-loss rate ˙M scales as: ˙M ∝ R2 rcb ρrcb cs Mrcb, (A3) where Mrcb is the Mach number of the escaping flow, evaluated at the radiative-convective boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' For an isothermal outflow, the Mach number is only a function of RB/Rrcb and given by: Mrcb = � −W0 � − � RB Rrcb �4 exp � − C − 4 RB Rrcb �� , (A4) where W0 is the real branch of the Lambert W function (see Cranmer 2004) and C is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' In the case of XUV photoevaporation, mass-loss rates are typically higher, meaning C < −3, and for core-powered mass-loss, C = −3 (see Lamers & Cassinelli 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' For planets that have maintained a significant mass in H/He i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' sub-Neptunes, one can state that their mass-loss timescale tloss = Matm/ ˙M, will be approximately constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Hence, combining Equations A1 and A3, one finds that: tloss = Matm ˙M ∝ Rrcb � RB Rrcb � 1 γ−1 I2 cs Mrcb ∝ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (A5) This expression is dominated by the exponential term within Mrcb, and only varies logarithmically with C, Rrcb and I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Hence, one can state that: RB Rrcb ∝ Mc Rrcbc2s ∝ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (A6) Now, by combining Equations 8, 9 and 11 from Owen & Wu (2017), (see also Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2016)), which assume radiative diffusion at the radiative-convective boundary for a cooling/Kelvin-Helmholtz timescale τKH, one finds that the density at the radiative-convective boundary scales as: ρrcb ∝ RrcbT 3 eqτKH Matmκ �I2 I1 � , (A7) 15 where κ is the opacity and I1 is another dimensionless integral accounting for the binding energy of the planet: I1 � Rc Rrcb , γ � = � 1 Rc/Rp x � 1 x − 1 � 1 γ−1 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (A8) Combining the density at the radiative-convective boundary from Equation A7 with the atmospheric mass from Equation A1, and noting that RB/Rrcb is approximately constant from Equation A6, one can show that: Mc Rrcbc2s ∝ X 1 2 M − 1 2 c T 1 4 eq κ 1 4 I 1 4 1 I − 1 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (A9) where the atmospheric mass fraction is defined as X ≡ Matm/Mc and we have assumed that for a set of planets with the same age, their Kelvin-Helmholtz timescale will be approximately constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Finally, recalling again that Mc/Rrcbc2 s is approximately constant from Equation A6, one finds that: X ∝ Mc T − 1 2 eq κ− 1 2 I − 1 2 1 I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (A10) If one numerically evaluates the dimensionless integrals I1 and I2 (see Figure 11 from Owen & Wu 2017), one can show that I−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='5 1 I2 is approximately constant as a function of Rc/Rrcb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' If one also assumes that the opacity κ is constant, then one finally finds that the atmospheric mass fraction of planets that have undergone mass-loss scales approximately linearly with core mass and inversely with the square root of equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Moreover, this analytic argument is agnostic with respect to mass-loss models i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' photoevaporation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' core-powered mass-loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' The main takeaway result is that larger mass planets at cooler equilibrium temperatures will retain larger atmospheric mass fractions if they have undergone mass-loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' This result is key to explain the mass-radius distribution of exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Note however that from Figure 2, that whilst final atmospheric mass fraction increases with core mass at a given equilibrium temperature, it is not a linear relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' This is the case when one fully evaluates the integrals of Equations A2 and A8 and takes non-constant opacities into account, as is the case in the semi-analytic models of Owen & Wu (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Owen & Campos Estrada (2020) and Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} +page_content=' Gupta & Schlichting (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE3T4oBgHgl3EQfHQkd/content/2301.04321v1.pdf'} diff --git a/etAzT4oBgHgl3EQfaPzl/content/2301.01367v1.pdf 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Murdoch⋆⋆ +University College London +Abstract. In Private Set Intersection protocols (PSIs), a non-empty result always reveals something +about the private input sets of the parties. Moreover, in various variants of PSI, not all parties necessarily +receive or are interested in the result. Nevertheless, to date, the literature has assumed that those parties +who do not receive or are not interested in the result still contribute their private input sets to the PSI +for free, although doing so would cost them their privacy. In this work, for the first time, we propose +a multi-party PSI, called “Anesidora”, that rewards parties who contribute their private input sets to +the protocol. Anesidora is efficient; it mainly relies on symmetric key primitives and its computation +and communication complexities are linear with the number of parties and set cardinality. It remains +secure even if the majority of parties are corrupted by active colluding adversaries. +1 +Introduction +Secure Multi-Party Computation (MPC) allows multiple mutually distrustful parties to jointly compute +a certain functionality on their private inputs without revealing anything beyond the result. Private Set +Intersection (PSI) is a subclass of MPC that aims to efficiently achieve the same security property as MPC +does. PSI has numerous applications. For instance, it has been used in Vertical Federated Learning (VFL) +[35], COVID-19 contact tracing schemes [20], remote diagnostics [11], and finding leaked credentials [43]. +There exist two facts about PSIs: (i) a non-empty result always reveals something about the parties’ +private input sets (i.e., the set elements that are in the intersection), and (ii) various variants of PSIs do not +output the result to all parties, even in those PSIs that do, not all of the parties are necessarily interested +in it. Given these facts, one may ask a natural question: +How can we incentivise the parties that do not receive the result or +are not interested in it to participate in a PSI? +To date, the literature has not answered the above question. The literature has assumed that all parties +will participate in a PSI for free and bear the privacy cost (in addition to computation and computation +overheads imposed by the PSI). In this work, we answer the above question for the first time. We present a +multi-party PSI, called “Anesidora”, that allows a buyer who initiates the PSI computation (and is interested +in the result) to pay other parties proportionate to the number of elements it learns about other parties’ +private inputs.1 Anesidora is efficient and mainly relies on symmetric key primitives. Its computation and +communication complexities are linear with the number of parties and set cardinality. Anesidora remains +secure even if the majority of parties are corrupt by active adversaries which may collude with each other. +We develop Anesidora in a modular fashion. Specifically, we propose the formal notion of “PSI with +Fair Compensation” (PSI +FC) and devise the first construction, called “Justitia”, that realises the notion.2 +PSI +FC ensures that either all parties get the result or if the protocol aborts in an unfair manner (where only +⋆ aydin.abadi@ucl.ac.uk +⋆⋆ s.murdoch@ucl.ac.uk +1 Anesidora is in Greek and Roman mythology an epithet of several goddesses. It means sender of gifts. We call our +protocol which sends rewards (or gifts) to honest parties Anesidora. +2 Justitia is the ancient Roman personification of justice. We call our protocol which ensures that parties are treated +fairly Justitia. +arXiv:2301.03889v1 [cs.CR] 10 Jan 2023 + +dishonest parties learn the result), then honest parties will receive financial compensation, i.e., adversaries are +penalised. Next, we enhance PSI +FC to the notion of “PSI with Fair Compensation and Reward” (PSI +FCR) +and develop Anesidora that realises PSI +FCR. The latter notion ensures that honest parties (a) are rewarded +regardless of whether all parties are honest, or a set of them aborts in an unfair manner and (b) are +compensated in the case of an unfair abort. We formally prove the two PSIs using the simulation-based +model. To devise efficient PSIs, we have developed a primitive, called “unforgeable polynomial” that might +be of independent interest. +A PSI, like Anesidora, that supports more than two parties and rewards set contributors can create +opportunities for much richer analytics and incentivise parties to participate. It can be used (1) by an +advertiser who wants to conduct advertisements targeted at certain customers by first finding their common +shopping patterns distributed across different e-commerce companies’ databases [27], (2) by a malware +detection service that allows a party to send a query to a collection of malware databases held by independent +antivirus companies to find out whether all of them consider a certain application as malware [42], or (3) by +a bank, like “WeBank”, that uses VFL and PSI to gather information about certain customers from various +partners (e.g., national electronic invoice and other financial institutions) to improve its risk management of +loans [13]. In all these cases, the set contributors will be rewarded by such a PSI. +We hope that our work initiates future research on developing reward mechanisms for participants of +generic MPC, as well. Such reward mechanisms have the potential to increase MPC’s real-world adoption. +Our Contributions Summary. In this work, we: (1) devise Anesidora, the first PSI that lets participants +receive a reward for contributing their set elements to the intersection, (2) develop Justitia, the first PSI +that lets either all parties receive the result or if the protocol aborts in an unfair manner, honest parties +receive compensation, and (3) propose formal definitions of the above constructions. +2 +Related Work +Since their introduction in [21], various PSIs have been designed. PSIs can be divided into traditional and +delegated ones. +In traditional PSIs, data owners interactively compute the result using their local data. Very recently, +Raghuraman and Rindal [41] proposed two two-party PSIs, one secure against semi-honest/passive and the +other against malicious/active adversaries. To date, these two protocols are the fastest two-party PSIs. They +mainly rely on Oblivious Key-Value Stores (OKVS) data structure and Vector Oblivious Linear Evaluation +(VOLE). The protocols’ computation cost is O(c), where c is a set’s cardinality. They also impose O(c log c2+ +κ) and O(c · κ) communication costs in the semi-honest and malicious security models respectively, where l +is a set element’s bit-size, and κ is a security parameter. Also, researchers designed PSIs that allow multiple +(i.e., more than two) parties to efficiently compute the intersection. The multi-party PSIs in [26,34] are +secure against passive adversaries while those in [7,23,45,34,38] were designed to remain secure against +active ones. However, Abadi et al. [3] showed that the PSIs in [23] are susceptible to several attacks. To date, +the protocols in [34] and [38] are the most efficient multi-party PSIs designed to be secure against passive +and active adversaries respectively. These protocols are secure even if the majority of parties are corrupt. +The former relies on inexpensive symmetric key primitives such as Programmable Pseudorandom Function +(OPPRF) and Cuckoo Hashing, while the latter mainly uses OPPRF and OKVS. +The overall computation and communication complexities of the PSI in [34] are O(c · m2 + c · m) and +O(c · m2) respectively, as each client needs to interact with the rest (in the “Conditional Zero-Sharing” +phase), where m is the number of clients. Later, to achieve efficiency, Chandran et al. [12] proposed a multi- +party PSI that remains secure only if the minority of the parties is corrupt by a semi-honest adversary; +thus, it offers a weaker security guarantee than the one in [34] does. The PSI in [38] has a parameter t that +determines how many parties can collude with each other and must be set before the protocol’s execution, +where t ∈ {2, m}. The protocol divides the parties into three groups, clients: A1, ..Am−t−1, leader: Am−t, +and servers: Am−t+1, ..Am. Each client needs to send a set of messages to every server and the leader which +2 + +jointly compute the final result. Hence, this protocol’s overall computation and communication complexities +are O(c · κ(m + t2 − t(m + 1))) and O(c · m · κ) respectively. +Dong et al. proposed a “fair” two-party PSIs [17] that ensure either both parties receive the result or +neither does, even if a malicious party aborts prematurely during the protocol’s execution. The protocol relies +on homomorphic encryption, zero-knowledge proofs, and polynomial representation of sets. The protocol’s +computation and communication complexities are O(c2) and O(c) respectively. Since then, various fair two- +party PSIs have been proposed, e.g., in [14,16,15]. To date, the fair PSI in [15] is the most efficient one. +It mainly relies on a combination of ElGamal encryption, verifiable encryption, and zero-knowledge proofs, +which often impose a significant overhead. The protocol’s computation and communication cost is O(c). So +far, there exists no fair multi-party PSI in the literature. Our Justitia is the first fair multi-party PSI. +Delegated PSIs use cloud computing for computation and/or storage, while preserving the privacy of the +computation inputs and outputs from the cloud. They can be divided further into protocols that support +one-off and repeated delegation of PSI computation. The former like [28,30,46] cannot reuse their outsourced +encrypted data and require clients to re-encode their data locally for each computation. The most efficient +such protocol is [28], which has been designed for the two-party setting and its computation and commu- +nication complexity is O(c). In contrast, those protocols that support repeated PSI delegation let clients +outsource the storage of their encrypted data to the cloud only once, and then execute an unlimited number +of computations on the outsourced data. To date, the protocol in [1] is the most efficient PSI that supports re- +peated delegation in the semi-honest model. It relies on the polynomial representation of sets, pseudorandom +function, and hash table. Its overall communication and computation complexities are O(h · d2) and O(h · d) +respectively, where h is the total number of bins in the hash table, d is a bin’s capacity (often d = 100), +and h · d is linear with c. Recently, a multi-party PSI that supports repeated delegation and efficient updates +has been proposed in [2]. It allows a party to efficiently update its outsourced set securely. It is also in the +semi-honest model and uses a pseudorandom function, hash table, and Bloom filter. The protocol imposes +O(h·d2 ·m) and O(h·d·m) computation and communication costs respectively, during the PSI computation. +It also imposes O(d2) computation and communication overheads, during the update phase. +3 +Notations and Preliminaries +3.1 +Notations +Table 1 summarises the main notations used in this paper. +3.2 +Security Model +In this paper, we use the simulation-based paradigm of secure computation [25] to define and prove the +proposed protocols. Since both types of (static) active and passive adversaries are involved in our protocols, +we will provide formal definitions for both types. In this work, we consider a static adversary, we assume +there is an authenticated private (off-chain) channel between the clients and we consider a standard public +blockchain, e.g., Ethereum. +Two-party Computation. A two-party protocol Γ problem is captured by specifying a random process +that maps pairs of inputs to pairs of outputs, one for each party. Such process is referred to as a functionality +denoted by f : {0, 1}∗ ×{0, 1}∗ → {0, 1}∗ ×{0, 1}∗, where f := (f1, f2). For every input pair (x, y), the output +pair is a random variable (f1(x, y), f2(x, y)), such that the party with input x wishes to obtain f1(x, y) while +the party with input y wishes to receive f2(x, y). When f is deterministic, then f1 = f2. In the setting where +f is asymmetric and only one party (say the first one) receives the result, f is defined as f := (f1(x, y), ⊥). +Security in the Presence of Passive Adversaries. In the passive adversarial model, the party corrupted +by such an adversary correctly follows the protocol specification. Nonetheless, the adversary obtains the +3 + +Table 1: Notation Table. +Setting +Symbol +Description +CL +Set of all clients, {A1, ..., Am, D} +D +Dealer client +Am +Buyer client +m +Total number of clients (excluding D) +p +Large prime number +H +Hash function +|S∩| +Intersection size +Smin +Smallest set’s size +Smax +Largest set’s size +| +Divisible +\ +Set subtraction +c +Set’s cardinality +h +Total number of bins in a hash table +d +A bin’s capacity +λ +Security parameter +OLE +Oblivious Linear Evaluation +OLE+ +Advanced OLE +Com +Commitment algorithm of commitment +Ver +Verification algorithm of commitment +MT.genTree +Tree construction algorithm of Merkle tree +MT.prove +Proof generation algorithm of Merkle tree +MT.verify +Verification algorithm of Merkle tree +CT +Coin tossing protocol +VOPR +Verifiable Oblivious Poly. Randomization +ZSPA +Zero-sum Pseudorandom Values Agreement +ZSPA-A +ZSPA with an External Auditor +PSIFC +Multi-party PSI with Fair Compensation +PSIFCR Multi-party PSI with Fair Compensation and Reward +JUS +Protocol that realises PSIFC +ANE +Protocol that realises PSIFCR +PRF +Pseudorandom function +PRP +Pseudorandom permutation +gcd +Greatest common divisor +Generic +ϵ +Negligible function +Setting +Symbol +Description +SCPC +Prisoner’s Contract +SCCC +Colluder’s Contract +SCTC +Traitor’s Contract +¨c +Server’s cost for computing a task +¨ +ch +Auditor’s cost for resolving disputes +¨d +Deposit a server pays to get the job +¨ +w +Amount a server receives for completing the task +Counter +Collusion +Contracts +(pk, sk) +SCJUS’s auditor’s public-private key pair +SCJUS +JUS’s smart contract +ω, ω′, ρ +Random poly. of degree d +γ, δ +Random poly. of degree d + 1 +ν(C) +Blinded poly. sent by each C to SCJUS +φ +Blinded poly. encoding the intersection +χ +Poly. sent to SCJUS to identify misbehaving parties +¯L +List of identified misbehaving parties +A portion of a party’s deposit into SCJUS +¨y +transferred to honest clients if it misbehaves +mk +Master key of PRF +QInit +Initiation predicate +QDel +Delivery predicate +QUF-A +UnFair-Abort predicate +Justitia (JUS) +QF-A +Fair-Abort predicate +SCANE +ANE’s smart contract +¨d′ +Extractor’s deposit +¨y′ +Each client’s deposit into SCJUS +¨l +Reward a client earns for an intersection element +¨r +Extractor’s cost for extracting an intersection element +¨ +f +Shorthand for ¨l(m − 1) +Price a buyer pays for an intersection element +¨v +¨v = m · ¨l + 2¨r +mk′ +Another master key of PRF +ctmk +Encryption of mk under pk +QDel +R +Delivery-with-Reward predicate +Anesidora (ANE) +QUF-A +R +UnFair-Abort-with-Reward predicate +internal state of the corrupted party, including the transcript of all the messages received, and tries to use +this to learn information that should remain private. Loosely speaking, a protocol is secure if whatever can +be computed by a party in the protocol can be computed using its input and output only. In the simulation- +based model, it is required that a party’s view in a protocol’s execution can be simulated given only its input +and output. This implies that the parties learn nothing from the protocol’s execution. More formally, party i’s +view (during the execution of Γ) on input pair (x, y) is denoted by View +Γ +i (x, y) and equals (w, ri, mi +1, ..., mi +t), +where w ∈ {x, y} is the input of ith party, ri is the outcome of this party’s internal random coin tosses, and +mi +j represents the jth message this party receives. The output of the ith party during the execution of Γ on +(x, y) is denoted by Output +Γ +1 (x, y) and can be generated from its own view of the execution. The joint output +of both parties is denoted by Output +Γ(x, y) := (Output +Γ +1 (x, y), Output +Γ +2 (x, y)). +Definition 1. Let f be the deterministic functionality defined above. Protocol Γ security computes f in the +presence of a static passive adversary if there exist polynomial-time algorithms (Sim1, Sim2) such that: +{Sim1(x, f1(x, y))}x,y +c≡ {View +Γ +1 (x, y)}x,y +{Sim2(x, f2(x, y))}x,y +c≡ {View +Γ +2 (x, y)}x,y +Security in the Presence of Active Adversaries. In this adversarial model, the corrupted party may +arbitrarily deviate from the protocol specification, to learn the private inputs of the other parties or to +influence the outcome of the computation. In this case, the adversary may not use the input provided. +Therefore, beyond the possibility that a corrupted party may learn more than it should, correctness is +also required. This means that a corrupted party must not be able to cause the output to be incorrectly +distributed. Moreover, we require independence of inputs meaning that a corrupted party cannot make its +input depend on the other party’s input. To capture the threats, the security of a protocol is analyzed by +comparing what an adversary can do in the real protocol to what it can do in an ideal scenario that is secure +by definition. This is formalized by considering an ideal computation involving an incorruptible Trusted +4 + +Third Party (TTP) to whom the parties send their inputs and receive the output of the ideal functionality. +Below, we describe the executions in the ideal and real models. +First, we describe the execution in the ideal model. Let P1 and P2 be the parties participating in the +protocol, i ∈ {0, 1} be the index of the corrupted party, and A be a non-uniform probabilistic polynomial- +time adversary. Also, let z be an auxiliary input given to A while x and y be the input of party P1 and P2 +respectively. The honest party, Pj, sends its received input to TTP. The corrupted party Pi may either abort +(by replacing the input with a special abort message Λi), send its received input or send some other input +of the same length to TTP. This decision is made by the adversary and may depend on the input value of +Pi and z. If TTP receives Λi, then it sends Λi to the honest party and the ideal execution terminates. Upon +obtaining an input pair (x, y), TTP computes f1(x, y) and f2(x, y). It first sends fi(x, y) to Pi which replies +with “continue” or Λi. In the former case, TTP sends fj(x, y) to Pj and in the latter it sends Λi to Pj. +The honest party always outputs the message that it obtained from TTP. A malicious party may output an +arbitrary function of its initial inputs and the message it has obtained from TTP. The ideal execution of f +on inputs (x, y) and z is denoted by Ideal +f +A(z),i(x, y) and is defined as the output pair of the honest party and +A from the above ideal execution. In the real model, the real two-party protocol Γ is executed without the +involvement of TTP. In this setting, A sends all messages on behalf of the corrupted party and may follow +an arbitrary strategy. The honest party follows the instructions of Γ. The real execution of Γ is denoted by +Real +Γ +A(z),i(x, y), it is defined as the joint output of the parties engaging in the real execution of Γ (on the +inputs), in the presence of A. +Next, we define security. At a high level, the definition states that a secure protocol in the real model +emulates the ideal model. This is formulated by stating that adversaries in the ideal model can simulate +executions of the protocol in the real model. +Definition 2. Let f be the two-party functionality defined above and Γ be a two-party protocol that computes +f. Protocol Γ securely computes f with abort in the presence of static active adversaries if for every non- +uniform probabilistic polynomial time adversary A for the real model, there exists a non-uniform probabilistic +polynomial-time adversary (or simulator) Sim for the ideal model, such that for every i ∈ {0, 1}, it holds +that: +{Ideal +f +Sim(z),i(x, y)}x,y,z +c≡ {Real +Γ +A(z),i(x, y)}x,y,z +3.3 +Smart Contracts +Cryptocurrencies, such as Bitcoin [37] and Ethereum [44], beyond offering a decentralised currency, support +computations on transactions. In this setting, often a certain computation logic is encoded in a computer +program, called a “smart contract”. To date, Ethereum is the most predominant cryptocurrency framework +that enables users to define arbitrary smart contracts. In this framework, contract code is stored on the +blockchain and executed by all parties (i.e., miners) maintaining the cryptocurrency, when the program +inputs are provided by transactions. The program execution’s correctness is guaranteed by the security +of the underlying blockchain components. To prevent a denial-of-service attack, the framework requires a +transaction creator to pay a fee, called “gas”, depending on the complexity of the contract running on it. +3.4 +Counter Collusion Smart Contracts +In order to let a party, e.g., a client, efficiently delegate a computation to a couple of potentially colluding +third parties, e.g., servers, Dong et al. [18] proposed two main smart contracts; namely, “Prisoner’s Contract” +(SCPC) and “Traitor’s Contract” (SCTC). The Prisoner’s contract is signed by the client and the servers. This +contract tries to incentivize correct computation by using the following idea. It requires each server to pay +a deposit before the computation is delegated. It is equipped with an external auditor that is invoked to +detect a misbehaving server only when the servers provide non-equal results. +If a server behaves honestly, then it can withdraw its deposit. Nevertheless, if a cheating server is detected +by the auditor, then (a portion) of its deposit is transferred to the client. If one of the servers is honest and +5 + +the other one cheats, then the honest server receives a reward taken from the cheating server’s deposit. +However, the dilemma, created by SCPC between the two servers, can be addressed if they can make an +enforceable promise, say via a “Colluder’s Contract” (SCCC), in which one party, called “ringleader”, would +pay its counterparty a bribe if both follow the collusion and provide an incorrect computation to SCPC. To +counter SCCC, Dong et al. proposed SCTC, which incentivises a colluding server to betray the other server and +report the collusion without being penalised by SCPC. In this work, we slightly adjust and use these contracts. +We have stated the related parameters of these tree contracts in Table 1. We refer readers to Appendix C +for the full description of the parameters and contracts. +3.5 +Pseudorandom Function and Permutation +Informally, a pseudorandom function is a deterministic function that takes a key of length λ and an input; and +outputs a value indistinguishable from that of a truly random function. In this paper, we use pseudorandom +functions: PRF : {0, 1}λ × {0, 1}∗ → Fp, where |p| = λ is the security parameter. In practice, a pseudorandom +function can be obtained from an efficient block cipher [29]. +The definition of a pseudorandom permutation, PRP : {0, 1}λ × {0, 1}∗ → Fp, is very similar to that +of a pseudorandom function, with a difference; namely, it is required the keyed function PRP(k, .) to be +indistinguishable from a uniform permutation, instead of a uniform function. In cryptographic schemes that +involve PRP, sometimes honest parties may require to compute the inverse of pseudorandom permutation, +i.e., PRP−1(k, .), as well. In this case, it would require that PRP(k, .) be indistinguishable from a uniform +permutation even if the distinguisher is additionally given oracle access to the inverse of the permutation. +3.6 +Commitment Scheme +A commitment scheme involves a sender and a receiver. It also involves two phases; namely, commit and +open. In the commit phase, the sender commits to a message: x as Com(x, r) = com, that involves a secret +value: r +$← {0, 1}λ. At the end of the commit phase, the commitment com is sent to the receiver. In the open +phase, the sender sends the opening ˆx := (x, r) to the receiver who verifies its correctness: Ver(com, ˆx) +?= 1 +and accepts if the output is 1. A commitment scheme must satisfy two properties: (a) hiding: it is infeasible +for an adversary (i.e., the receiver) to learn any information about the committed message x, until the +commitment com is opened, and (b) binding: it is infeasible for an adversary (i.e., the sender) to open a +commitment com to different values ˆx′ := (x′, r′) than that was used in the commit phase, i.e., infeasible +to find ˆx′, s.t. Ver(com, ˆx) = Ver(com, ˆx′) = 1, where ˆx ̸= ˆx′. There exist efficient commitment schemes +both in (a) the standard model, e.g., Pedersen scheme [40], and (b) the random oracle model using the +well-known hash-based scheme such that committing is : H(x||r) = com and Ver(com, ˆx) requires checking: +H(x||r) +?= com, where H : {0, 1}∗ → {0, 1}λ is a collision-resistant hash function, i.e., the probability to find +x and x′ such that H(x) = H(x′) is negligible in the security parameter λ. +3.7 +Hash Tables +A hash table is an array of bins each of which can hold a set of elements. It is accompanied by a hash function. +To insert an element, we first compute the element’s hash, and then store the element in the bin whose index +is the element’s hash. In this paper, we set the table’s parameters appropriately to ensure the number of +elements in each bin does not exceed a predefined capacity. Given the maximum number of elements c and +the bin’s maximum size d, we can determine the number of bins, h, by analysing hash tables under the balls +into the bins model [8]. In Appendix A, we explain how the hash table parameters are set. +3.8 +Merkle Tree +A Merkle tree is a data structure that supports a compact commitment of a set of values/blocks. As a +result, it includes two parties, prover P and verifier V. The Merkle tree scheme includes three algorithms +(MT.genTree, MT.prove, MT.verify), defined as follows: +6 + +• The algorithm that constructs a Merkle tree, MT.genTree, is run by V. It takes blocks, u := u1, ..., un, +as input. Then, it groups the blocks in pairs. Next, a collision-resistant hash function, H(.), is used to +hash each pair. After that, the hash values are grouped in pairs and each pair is further hashed, and +this process is repeated until only a single hash value, called “root”, remains. This yields a tree with the +leaves corresponding to the input blocks and the root corresponding to the last remaining hash value. V +sends the root to P. +• The proving algorithm, MT.prove, is run by P. It takes a block index, i, and a tree as inputs. It outputs a +vector proof, of log2(n) elements. The proof asserts the membership of i-th block in the tree, and consists +of all the sibling nodes on a path from the i-th block to the root of the Merkle tree (including i-th block). +The proof is given to V. +• The verification algorithm, MT.verify, is run by V. It takes as an input i-th block, a proof, and the tree’s +root. It checks if the i-th block corresponds to the root. If the verification passes, it outputs 1; otherwise, +it outputs 0. +The Merkle tree-based scheme has two properties: correctness and security. Informally, the correctness +requires that if both parties run the algorithms correctly, then a proof is always accepted by V. The security +requires that a computationally bounded malicious P cannot convince V into accepting an incorrect proof, +e.g., proof for a non-member block. The security relies on the assumption that it is computationally infeasible +to find the hash function’s collision. Usually, for the sake of simplicity, it is assumed that the number of +blocks, n, is a power of 2. The height of the tree, constructed on m blocks, is log2(n). +3.9 +Polynomial Representation of Sets +The idea of using a polynomial to represent a set’s elements was proposed by Freedman et al. in [21]. Since +then, the idea has been widely used, e.g., in [4,5,24,33]. In this representation, set elements S = {s1, ..., sd} +are defined over Fp and set S is represented as a polynomial of form: p(x) = +d� +i=1 +(x − si), where p(x) ∈ Fp[X] +and Fp[X] is a polynomial ring. Often a polynomial, p(x), of degree d is represented in the “coefficient form” +as follows: p(x) = a0+a1·x+...+ad·xd. The form +d� +i=1 +(x−si) is a special case of the coefficient form. As shown +in [10,33], for two sets S(A) and S(B) represented by polynomials pA and pB respectively, their product, which +is polynomial pA · pB, represents the set union, while their greatest common divisor, gcd(pA, pB), represents +the set intersection. For two polynomials pA and pB of degree d, and two random polynomials γA and γB of +degree d, it is proven in [10,33] that: θ = γA · pA + γB · pB = µ · gcd(pA, pB), where µ is a uniformly random +polynomial, and polynomial θ contains only information about the elements in S(A) ∩ S(B), and contains no +information about other elements in S(A) or S(B). +Given a polynomial θ that encodes sets intersection, one can find the set elements in the intersection +via one of the following approaches. First, via polynomial evaluation: the party who already has one of the +original input sets, say pA, evaluates θ at every element si of pA and considers si in the intersection if +pA(si) = 0. Second, via polynomial root extraction: the party who does not have one of the original input +sets, extracts the roots of θ, which contain the roots of (i) random polynomial µ and (ii) the polynomial +that represents the intersection, i.e., gcd(pA, pB). In this approach, to distinguish errors (i.e., roots of µ) +from the intersection, PSIs in [1,33] use the “hash-based padding technique”. In this technique, every element +ui in the set universe U, becomes si = ui||H(ui), where H is a cryptographic hash function with a sufficiently +large output size. Given a field’s arbitrary element, s ∈ Fp and H’s output size |H(.)|, we can parse s into +x1 and x2, such that s = x1||x2 and |x2| = |H(.)|. In a PSI that uses polynomial representation and this +padding technique, after we extract each root of θ, say s, we parse it into (x1, x2) and check x2 +?= H(x1). If +the equation holds, then we consider s as an element of the intersection. +3.10 +Horner’s Method +Horner’s method [19] allows for efficiently evaluating polynomials at a given point, e.g., x0. Specifically, given +a polynomial of the form: τ(x) = a0+a1·x+a2·x2+...+an·xn and a point: x0, one can efficiently evaluate the +7 + +polynomial at x0 iteratively, in the following fashion: τ(x0) = a0 + x0(a1 + x0(a2 + ... + x0(an−1 + x0 · an)...))). +Evaluating a polynomial of degree n naively requires n additions and +(n2+n) +2 +multiplications. However, +using Horner’s method the evaluation requires only n additions and n multiplications. We use this method +throughout the paper. +3.11 +Oblivious Linear Function Evaluation +Oblivious Linear function Evaluation (OLE) is a two-party protocol that involves a sender and receiver. In +OLE, the sender has two inputs a, b ∈ Fp and the receiver has a single input, c ∈ Fp. The protocol allows +the receiver to learn only s = a · c + b ∈ Fp, while the sender learns nothing. Ghosh et al. [22] proposed +an efficient OLE that has O(1) overhead and involves mainly symmetric key operations. Later, in [23] an +enhanced OLE, called OLE+ was proposed. The latter ensures that the receiver cannot learn anything about +the sender’s inputs, even if it sets its input to 0. In this paper, we use OLE+. We refer readers to Appendix +B, for its construction. +3.12 +Coin-Tossing Protocol +A Coin-Tossing protocol, CT, allows two mutually distrustful parties, say A and B, to jointly generate a single +random bit. Formally, CT computes the functionality fCT(inA, inB) → (outA, outB), which takes inA and inB +as inputs of A and B respectively and outputs outA to A and outB to B, where outA = outB. A basic security +requirement of a CT is that the resulting bit is (computationally) indistinguishable from a truly random bit. +Blum proposed a simple CT in [9] that works as follows. Party A picks a random bit inA +$← {0, 1}, commits +to it and sends the commitment to B which sends its choice of random input, inB +$← {0, 1}, to A. Then, A +sends the opening of the commitment (including inA) to B, which checks whether the commitment matches +its opening. If so, each party computes the final random bit as inA ⊕ inB. +There have also been fair coin-tossing protocols, e.g., in [36], that ensure either both parties learn the +result or nobody does. These protocols can be generalised to multi-party coin-tossing protocols to generate +a random string (rather than a single bit), e.g., see [6,31]. The overall computation and communication +complexities of (fair) multi-party coin-tossing protocols are often linear with the number of participants. In +this paper, any secure multi-party CT that generates a random string can be used. For the sake of simplicity, +we let a multi-party fCT take m inputs and output a single value, i.e., fCT(in1, ..., inm) → out. +4 +Definition of Multi-party PSI with Fair Compensation +In this section, we present the notion of multi-party PSI with Fair Compensation (PSI +FC) which allows +either all clients to get the result or the honest parties to be financially compensated if the protocol aborts +in an unfair manner, where only dishonest parties learn the result. +In a PSI +MFC, three types of parties are involved; namely, (1) a set of clients {A1, ..., Am} potentially +malicious (i.e., active adversaries) and all but one may collude with each other, (2) a non-colluding dealer, D, +potentially semi-honest (i.e., a passive adversary) and has an input set, and (3) an auditor Aud potentially +semi-honest, where all parties except Aud have input set. For simplicity, we assume that given an address +one can determine whether it belongs to Aud. +The basic functionality that any multi-party PSI computes can be defined as f PSI(S1, ..., Sm+1) → +(S∩, ..., S∩) +� +�� +� +m+1 +, where S∩ = S1 ∩ S2, ..., ∩ Sm+1. To formally define a PSI +FC, we equip the above PSI func- +tionality with four predicates, Q := (QInit, QDel, QUF-A, QF-A), which ensure that certain financial conditions +are met. We borrow three of these predicates (i.e., QInit, QDel, QUF-A) from the “fair and robust multi-party +computation” initially proposed in [32]; nevertheless, we will (i) introduce an additional predicate QF-A and +(ii) provide more formal accurate definitions of these predicates. +Predicate QInit specifies under which condition a protocol that realises PSI +FC should start executing, i.e., +when all set owners have enough deposit. Predicate QDel determines in which situation parties receive their +8 + +output, i.e., when honest parties receive their deposit back. Predicate QUF-A specifies under which condition +the simulator can force parties to abort if the adversary learns the output, i.e., when an honest party receives +its deposit back plus a predefined amount of compensation. Predicate QF-A specifies under which condition +the simulator can force parties to abort if the adversary receives no output, i.e., when honest parties receive +their deposits back. We observed that the latter predicate should have been defined in the generic framework +in [32] too; as the framework should have also captured the cases where an adversary may abort without +learning any output after the onset of the protocol. Intuitively, by requiring any protocol that realises PSI +FC +to implement a wrapped version of f PSI that includes Q, we will ensure that an honest set owner only aborts +in an unfair manner if QUF-A returns 1, it only aborts in a fair manner if QF-A returns 1, and outputs a valid +value if QDel returns 1. Now, we formally define each of these predicates. +Definition 3 (QInit: Initiation predicate). Let G be a stable ledger, adrsc be smart contract sc’s address, +Adr be a set of m+1 distinct addresses, and ¨x be a fixed amount of coins. Then, predicate QInit(G, adrsc, m+ +1, Adr, ¨x) returns 1 if every address in Adr has at least ¨x coins in sc; otherwise, it returns 0. +Definition 4 (QDel: Delivery predicate). Let pram := (G, adrsc, ¨x) be the parameters defined above, and +adri ∈ Adr be the address of an honest party. Then, predicate QDel(pram, adri) returns 1 if adri has sent ¨x +amount to sc and received ¨x amount from it; thus, its balance in sc is 0. Otherwise, it returns 0. +Definition 5 (QUF-A: UnFair-Abort predicate). Let pram := (G, adrsc, ¨x) be the parameters defined +above, and Adr′ ⊂ Adr be a set containing honest parties’ addresses, m′ = |Adr′|, and adri ∈ Adr′. Let also +G be a compensation function that takes as input three parameters ( ¨ +deps, adri, m′), where +¨ +deps is the amount +of coins that all m+1 parties deposit. It returns the amount of compensation each honest party must receive, +i.e., G( ¨ +deps, ardi, m′) → ¨xi. Then, predicate QUnF-Abt is defined as QUF-A(pram, G, +¨ +deps, m′, adri) → (a, b), +where a = 1 if adri is an honest party’s address and adri has sent ¨x amount to sc and received ¨x + ¨xi from +it, and b = 1 if adri is Aud’s address and adri received ¨xi from sc. Otherwise, a = b = 0. +Definition 6 (QF-A: Fair-Abort predicate). Let pram := (G, adrsc, ¨x) be the parameters defined above, +and Adr′ ⊂ Adr be a set containing honest parties’ addresses, m′ = |Adr′|, adri ∈ Adr′, and adrj be +Aud’s address. Let G be the compensation function, defined above and let G(deps, ardj, m′) → ¨xj be the +compensation that the auditor must receive. Then, predicate QF-A(pram, G, +¨ +deps, m′, adri, adrj) returns 1, +if adri (s.t. adri ̸= adrj) has sent ¨x amount to sc and received ¨x from it, and adrj received ¨xj from sc. +Otherwise, it returns 0. +Next, we present a formal definition of PSI +FC. +Definition 7 (PSI +FC). Let f PSI be the multi-party PSI functionality defined above. We say protocol Γ +realises f PSI with Q-fairness in the presence of m − 1 static active-adversary clients (i.e., Ajs) or a static +passive dealer D or passive auditor Aud, if for every non-uniform probabilistic polynomial time adversary A +for the real model, there exists a non-uniform probabilistic polynomial-time adversary (or simulator) Sim for +the ideal model, such that for every I ∈ {A1, ..., Am, D, Aud}, it holds that: +{Ideal +W(fPSI,Q) +Sim(z),I +(S1, ..., Sm+1)}S1,...,Sm+1,z +c≡ {Real +Γ +A(z),I(S1, ..., Sm+1)}S1,...,Sm+1,z +where z is an auxiliary input given to A and W(f PSI, Q) is a functionality that wraps f PSI with predicates +Q := (QInit, QDel, QUF-A, QF-A). +5 +Other Subroutines Used in Justitia +In this section, we present three subroutines and a primitive that we developed and are used in the instan- +tiation of PSI +FC, i.e., Justitia. +9 + +5.1 +Verifiable Oblivious Polynomial Randomisation (VOPR) +In the VOPR, two parties are involved, (i) a sender which is potentially a passive adversary and (ii) a receiver +that is potentially an active adversary. The protocol allows the receiver with input polynomial β (of degree +e′) and the sender with input random polynomials ψ (of degree e) and α (of degree e + e′) to compute: +θ = ψ · β + α, such that (a) the receiver learns only θ and nothing about the sender’s input even if it sets +β = 0, (b) the sender learns nothing, and (c) the receiver’s misbehaviour is detected in the protocol. Thus, +the functionality that VOPR computes is defined as f VOPR((ψ, α), β) → (⊥, ψ · β + α). We will use VOPR in +Justitia for two main reasons: (a) to let a party re-randomise its counterparty’s polynomial (representing its +set) and (b) to impose a MAC-like structure to the randomised polynomial; such a structure will allow a +verifier to detect if VOPR’s output has been modified. +Now, we outline how we design VOPR without using any (expensive) zero-knowledge proofs.3 In the +setup phase, both parties represent their input polynomials in the regular coefficient form; therefore, the +sender’s polynomials are defined as ψ = +e� +i=0 +gi · xi and α = +e+e′ +� +j=0 +aj · xj and the receiver’s polynomial is +defined as β = +e′� +i=0 +bi · xi. However, the sender computes each coefficient aj (of polynomial α) as follows, +aj = +k=e′ +t=e +� +t,k=0 +at,k, where t + k = j and each at,k is a random value. For instance, if e = 4 and e′ = 3, then +a3 = a0,3 + a3,0 + a1,2 + a2,1. Shortly, we explain why polynomial α is constructed this way. +In the computation phase, to compute polynomial θ, the two parties interactively multiply and add the +related coefficients in a secure way using OLE+ (presented in Section 3.11). Specifically, for every j (where +0 ≤ j ≤ e′) the sender sends gi and ai,j to an instance of OLE+, while the receiver sends bj to the same instance, +which returns ci,j = gi · bj + ai,j to the receiver. This process is repeated for every i, where 0 ≤ i ≤ e. Then, +the receiver uses ci,j values to construct the resulting polynomial, θ. +The reason that the sender imposes the above structure to (the coefficients of) α in the setup, is to let +the parties securely compute θ via OLE+. Specifically, by imposing this structure (1) the sender can blind +each product gi · bj with random value ai,j which is a component of α’s coefficient and (2) the receiver can +construct a result polynomial of the form θ = ψ · β + α. +Now, we outline how the verification works. To check the result’s correctness, the sender picks and sends +a random value z to the receiver which computes θ(z) and β(z) and sends these two values to the sender. +The sender computes ψ(z) and α(z) and then checks if equation θ(z) = ψ(z) · β(z) + α(z) holds. It accepts +the result if the check passes. Figure 1 describes VOPR in detail. +Theorem 1. Let f VOPR be the functionality defined above. If the enhanced OLE (i.e., OLE+) is secure against +malicious (or active) adversaries, then the Verifiable Oblivious Polynomial Randomisation (VOPR), presented +in Figure 1, securely computes f VOPR in the presence of (i) a semi-honest sender and honest receiver or (ii) a +malicious receiver and honest sender. +Proof. Before proving Theorem 1, we present Lemma 1 and Theorem 2 that will be used in the proof of +Theorem 1. Informally, Lemma 1 states that the evaluation of a random polynomial at a fixed value results +in a uniformly random value. +Lemma 1. Let xi be an element of a finite field Fp, picked uniformly at random and µ(x) be a random +polynomial of constant degree d and defined over Fp[X]. Then, the evaluation of µ(x) at xi is distributed +uniformly at random over the non-zero elements of the field, i.e., Pr[µ(xi) = y] = +1 +p−1, where y is arbitrary +elements of F∗ +p. +Proof. Let µ(x) = a0 + +d� +j=1 +ajxj, where the coefficients are distributed uniformly at random over the field. +We know that if xi is a root of µ(x), then because the polynomial can have at most d roots, we have +3 Previously, Ghosh et al. [23] designed a protocol called Oblivious Polynomial Addition (OPA) to meet similar +security requirements that we laid out above. But, as shown in [3], OPA is susceptible to several serious attacks. +10 + +• Input. +• Public Parameters: upper bound on input polynomials’ degree: e and e′. +• Sender Input: random polynomials: ψ = +e� +i=0 +gi · xi and α = +e+e′ +� +j=0 +aj · xj, where +gi +$← Fp. Each aj has the form: aj = +k=e′ +t=e +� +t,k=0 +at,k, such that t + k = j and at,k +$← Fp. +• Receiver Input: polynomial β = β1·β2 = +e′ +� +i=0 +bi·xi, where β1 is a random polynomial +of degree 1 and β2 is an arbitrary polynomial of degree e′ − 1. +• Output. The receiver gets θ = ψ · β + α. +1. Computation: +(a) Sender and receiver together for every j, 0 ≤ j ≤ e′, invoke e+1 instances of OLE+. +In particular, ∀j, 0 ≤ j ≤ e′: sender sends gi and ai,j while the receiver sends bj to +OLE+ that returns: ci,j = gi · bj + ai,j to the receiver (∀i, 0 ≤ i ≤ e). +(b) The receiver sums component-wise values ci,j that results in polynomial: +θ = ψ · β + α = +i=e +j=e′ +� +i,j=0 +ci,j · x +i+j +2. Verification: +(a) Sender: picks a random value z and sends it to the receiver. +(b) Receiver: sends θz = θ(z) and βz = β(z) to the sender. +(c) Sender: computes ψz = ψ(z) and αz = α(z) and checks if equation θz = ψz·βz +αz +holds. If the equation holds, it concludes that the computation was performed +correctly. Otherwise, it aborts. +Fig. 1: Verifiable Oblivious Polynomial Randomization (VOPR) +Pr[µ(xi) = 0] = +d +p. Next, we focus on the case where xi is not a root of the polynomial (thus y ̸= 0). +For any choice of xi, a1, ..., ad, there exists exactly one value of a0 that makes µ(xi) = y, i.e., µ(xi) = y iff +a0 = y − +d� +j=1 +ajxj +i. As a0 is picked uniformly at random, the probability that it equals a certain value that +makes µ(xi) = y is +1 +p−1. Thus, Pr[µ(xi) = y] = +1 +p−1, ∀y ∈ F∗ +p. +2 +Informally, Theorem 2 states that the product of two arbitrary polynomials (in coefficient form) is a +polynomial whose roots are the union of the two original polynomials. Below, we formally state it. The +theorem has been taken from [3]. +Theorem 2. Let p and q be two arbitrary non-constant polynomials of degree d and d′ respectively, such +that p, q ∈ Fp[X] and they are in coefficient form. Then, the product of the two polynomials is a polynomial +whose roots include precisely the two polynomials’ roots. +We refer readers to Appendix D for the proof of Theorem 2. Next, we prove the main theorem, i.e., +Theorem 1, by considering the case where each party is corrupt, in turn. +Case 1: Corrupt sender. In the real execution, the sender’s view is defined as follows: +View +VOPR +S +� +(ψ, α), β +� += {ψ, α, rS, β(z), θ(z), View +OLE+ +S +, ⊥} +where rS is the outcome of internal random coins of the sender and View +OLE+ +S +refers to the sender’s real-model +view during the execution of OLE+. The simulator Sim +VOPR +S , which receives ψ and α, works as follows. +11 + +1. generates an empty view. It appends to the view polynomials (ψ, α) and coins r′ +S chosen uniformly at +random. +2. computes polynomial β = β1 · β2, where β1 is a random polynomial of degree 1 and β2 is an arbitrary +polynomial of degree e′ − 1. Next, it constructs polynomial θ as follows: θ = ψ · β + α. +3. picks value z +$← Fp. Then, it evaluates polynomials β and θ at point z. This results in values βz and θz +respectively. It appends these two values to the view. +4. extracts the sender-side simulation of OLE+ from OLE+’s simulator. Let Sim +OLE+ +S +be this simulation. Note, +the latter simulation is guaranteed to exist, as OLE+ has been proven secure (in [23]). It appends Sim +OLE+ +S +and ⊥ to its view. +Now, we are ready to show that the two views are computationally indistinguishable. The sender’s inputs +are identical in both models, so they have identical distributions. Since the real-model semi-honest adversary +samples its randomness according to the protocol’s description, the random coins in both models have +identical distributions. Next, we explain why values β(z) in the real model and βz in the ideal model are +(computationally) indistinguishable. In the real model, β(z) is the evaluation of polynomial β = β1 · β2 at +random point z, where β1 is a random polynomial. We know that β(z) = β1(z) · β2(z), for any (non-zero) +z. Moreover, by Lemma 1, we know that β1(z) is a uniformly random value. Therefore, β(z) = β1(z) · β2(z) +is a uniformly random value as well. In the ideal world, polynomial β has the same structure as β has (i.e., +β = β1 · β2, where β1 is a random polynomial). That means βz is a uniformly random value too. Thus, +β(z) and βz are computationally indistinguishable. Next, we turn our attention to values θ(z) in the real +model and θz in the ideal model. We know that θ(z) is a function of β1(z), as polynomial θ has been defined +as θ = ψ · (β1 · β2) + α. Similarly, θz is a function of βz. As we have already discussed, β(z) and βz are +computationally indistinguishable, so are their functions θ(z) and θz. Moreover, as OLE+ has been proven +secure, View +OLE+ +S +and Sim +OLE+ +S +are computationally indistinguishable. It is also clear that ⊥ is identical in both +models. We conclude that the two views are computationally indistinguishable. +Case 2: Corrupt receiver. Let Sim +VOPR +R +be the simulator, in this case, which uses a subroutine adversary, +AR. Sim +VOPR +R +works as follows. +1. simulates OLE+ and receives AR’s input coefficients bj for all j, 0 ≤ j ≤ e′, as we are in fOLE+-hybrid +model. +2. reconstructs polynomial β, given the above coefficients. +3. simulates the honest sender’s inputs as follows. It picks two random polynomials: ψ = +e� +i=0 +gi · xi and +α = +e+e′ +� +j=0 +aj · xj, such that gi +$← Fp and every aj has the form: aj = +k=e′ +t=e +� +t,k=0 +at,k, where t + k = j and +at,k +$← Fp. +4. sends to OLE+’s functionality values gi and ai,j and receives ci,j from this functionality (for all i, j). +5. sends all ci,j to TTP and receives polynomial θ. +6. picks a random value z from Fp. Then, it computes ψz = ψ(z) and αz = α(z). +7. sends z and all ci,j to AR which sends back θz and βz to the simulator. +8. sends ψz and αz to AR. +9. checks if the following relation hold: +β¯z = β(z) +∧ +θz = θ(z) +∧ +θ(z) = ψz · βz + αz +(1) +If Relation 1 does not hold, it aborts (i.e., sends abort signal Λ to the sender) and still proceeds to the +next step. +10. outputs whatever AR outputs. +We first focus on the adversary’s output. Both values of z in the real and ideal models have been picked +uniformly at random from Fp; therefore, they have identical distributions. In the real model, values ψz and +αz are the result of the evaluations of two random polynomials at (random) point z. In the ideal model, +values ψz and αz are also the result of the evaluations of two random polynomials (i.e., ψ and α) at point z. +12 + +By Lemma 1, we know that the evaluation of a random polynomial at an arbitrary value yields a uniformly +random value in Fp. Therefore, the distribution of pair (ψz, αz) in the real model is identical to that of +pair (ψz, αz) in the ideal model. Moreover, the final result (i.e., values ci,j) in the real model has the same +distribution as the final result (i.e., values ci,j) in the ideal model, as they are the outputs of the ideal calls +to fOLE+, as we are in the fOLE+-hybrid model. +Next, we turn our attention to the sender’s output. We will show that the output distributions of the +honest sender in the ideal and real models are statistically close. Our focus will be on the probability that +it aborts in each model, as it does not receive any other output. In the ideal model, Sim +VOPR +R +is given the +honestly generated result polynomial θ (computed by TTP) and the adversary’s input polynomial β. Sim +VOPR +R +aborts with a probability of 1 if Relation 1 does not hold. However, in the real model, the honest sender +(in addition to its inputs) is given only βz and θz and is not given polynomials β and θ; it wants to check +if the following equation holds, θz = ψz · βz + αz. Note, polynomial θ = ψ · β + α (resulted from ci,j) +is well-structured, as it satisfies the following three conditions, regardless of the adversary’s input β to +OLE+, (i) deg(θ) = Max +� +deg(β) + deg(ψ), deg(α) +� +, as Fp[X] is an integral domain and (ψ, α) are random +polynomials, (ii) the roots of the product polynomial ν = ψ · β contains exactly both polynomials’ roots, +by Theorem 2, and (iii) the roots of ν + α is the intersection of the roots of ν and α, as shown in [33]. +Furthermore, polynomial θ reveals no information (about ψ and α except their degrees) to the adversary +and the pair (ψz, αz) is given to the adversary after it sends the pair (θz, βz) to the sender. There are exactly +four cases where pair (θz, βz) can be constructed by the real-model adversary. Below, we state each case and +the probability that the adversary is detected in that case during the verification, i.e., θz +?= ψz · βz + αz. +1. θz = θ(z) ∧ βz = β(z). This is a trivial non-interesting case, as the adversary has behaved honestly, so +it can always pass the verification. +2. θz ̸= θ(z) ∧ βz = β(z). In this case, the adversary is detected with a probability of 1. +3. θz = θ(z) ∧ βz ̸= β(z). In this case, the adversary is also detected with a probability of 1. +4. θz ̸= θ(z) ∧ βz ̸= β(z). In this case, the adversary is detected with an overwhelming probability, i.e., +1 − +1 +22λ . +As we illustrated above, in the real model, the lowest probability that the honest sender would abort in +case of adversarial behaviour is 1 − +1 +22λ . Thus, the honest sender’s output distributions in the ideal and real +models are statistically close, i.e., 1 vs 1 − +1 +22λ . +We conclude that the distribution of the joint outputs of the honest sender and adversary in the real and +ideal models are computationally indistinguishable. +2 +5.2 +Zero-sum Pseudorandom Values Agreement Protocol (ZSPA) +The ZSPA allows m parties (the majority of which is potentially malicious) to efficiently agree on (a set of +vectors, where each vector has) m pseudorandom values such that their sum equals zero. At a high level, +the parties first sign a smart contract and then run a coin-tossing protocol CT to agree on a key: k. Next, +one of the parties generates m − 1 pseudorandom values zj (where 1 ≤ j ≤ m − 1) using key k and PRF. +It sets the last value as the additive inverse of the sum of the values generated, i.e. zm = − +m−1 +� +j=1 +zj. Then, it +constructs a Merkel tree on top of the pseudorandom values and stores only the tree’s root g and the key’s +hash value q in the smart contract. Then, each party (using the key) locally checks if the values (on the +contract) have been constructed correctly; if so, then it sends a signed “approved” message to the contract. +Hence, the functionality that ZSPA computes is defined as f ZSPA (⊥, ..., ⊥) +� +�� +� +m +→ ((k, g, q), ..., (k, g, q)) +� +�� +� +m +, where g is +the Markle tree’s root built on the pseudorandom values zi,j, q is the hash value of the key used to generate +the pseudorandom values, and m ≥ 2. Figure 2 presents ZSPA in detail. +Briefly, ZSPA will be used in Justitia to allow clients {A1, ..., Am} to provably agree on a set of pseudo- +random polynomials whose sum is zero. Each of these polynomials will be used by a client to blind/encrypt +the messages it sends to the smart contract, to protect the privacy of the plaintext message (from Aud, D, +13 + +and the public). To compute the sum of the plaintext messages, one can easily sum the blinded messages, +which removes the blinding polynomials. +• Parties. A set of clients {A1, ..., Am}. +• Input. m: the total number of participants, adr: a deployed smart contract’s address, +and b: the total number of vectors. Let b′ = b − 1. +• Output. k: a secret key that generates b vectors [z0,1, ..., z0,m], ..., [zb′,1, ..., zb′,m] of pseu- +dorandom values, h: hash of the key, g: a Merkle tree’s root, and a vector of signed +messages. +1. Coin-tossing. CT(in1, ..., inm) → k. +All participants run a coin-tossing protocol to agree on PRF’s key, k. +2. Encoding. Encode(k, m) → (g, q). +One of the parties takes the following steps: +(a) for every i (where 0 ≤ i ≤ b′), generates m pseudorandom values as follows. +∀j, 1 ≤ j ≤ m − 1 : zi,j = PRF(k, i||j), +zi,m = − +m−1 +� +j=1 +zi,j +(b) constructs +a +Merkel +tree +on +top +of +all +pseudorandom +values, +MT.genTree(z0,1, ..., zb′,m) → g. +(c) sends the Merkel tree’s root: g, and the key’s hash: q = H(k) to adr. +3. Verification. Verify(k, g, q, m) → (a, s). +Each party checks if, all zi,j values, the root g, and key’s hash q have been correctly +generated, by retaking step 2. If the checks pass, it sets a = 1, sets s to a singed +“approved” message, and sends s to adr. Otherwise, it aborts by returning a = 0 and +s = ⊥. +Fig. 2: Zero-sum Pseudorandom Values Agreement (ZSPA) +Theorem 3. Let f ZSPA be the functionality defined above. If CT is secure against a malicious adversary and the +correctness of PRF, H, and Merkle tree holds, then ZSPA, in Figure 2, securely computes f ZSPA in the presence +of m − 1 malicious adversaries. +Proof. For the sake of simplicity, we will assume the sender, which generates the result, sends the result +directly to the rest of the parties, i.e., receivers, instead of sending it to a smart contract. We first consider +the case in which the sender is corrupt. +Case 1: Corrupt sender. Let Sim +ZSPA +S +be the simulator using a subroutine adversary, AS. Sim +ZSPA +S +works as +follows. +1. simulates CT and receives the output value k from fCT, as we are in fCT-hybrid model. +2. sends k to TTP and receives back from it m pairs, where each pair is of the form (g, q). +3. sends k to AS and receives back from it m pairs where each pair is of the form (g′, q′). +4. checks whether the following equations hold (for each pair): g = g′ +∧ +q = q′. If the two equations do +not hold, then it aborts (i.e., sends abort signal Λ to the receiver) and proceeds to the next step. +5. outputs whatever AS outputs. +We first focus on the adversary’s output. In the real model, the only messages that the adversary receives +are those messages it receives as the result of the ideal call to fCT. These messages have identical distribution +to the distribution of the messages in the ideal model, as the CT is secure. Now, we move on to the receiver’s +14 + +output. We will show that the output distributions of the honest receiver in the ideal and real models +are computationally indistinguishable. In the real model, each element of pair (g, p) is the output of a +deterministic function on the output of fCT. We know the output of fCT in the real and ideal models have an +identical distribution, and so do the evaluations of deterministic functions (i.e., Merkle tree, H, and PRF) on +them, as long as these three functions’ correctness holds. Therefore, each pair (g, q) in the real model has an +identical distribution to pair (g, q) in the ideal model. For the same reasons, the honest receiver in the real +model aborts with the same probability as Sim +ZSPA +S +does in the ideal model. We conclude that the distributions +of the joint outputs of the adversary and honest receiver in the real and ideal models are (computationally) +indistinguishable. +Case 2: Corrupt receiver. Let Sim +ZSPA +R +be the simulator that uses subroutine adversary AR. Sim +ZSPA +R +works +as follows. +1. simulates CT and receives the output value k from fCT. +2. sends k to TTP and receives back m pairs of the form (g, q) from TTP. +3. sends (k, g, q) to AR and outputs whatever AR outputs. +In the real model, the adversary receives two sets of messages, the first set includes the transcripts +(including k) it receives when it makes an ideal call to fCT and the second set includes pair (g, q). As we +already discussed above (because we are in the fCT-hybrid model) the distributions of the messages it receives +from fCT in the real and ideal models are identical. Moreover, the distribution of fCT’s output (i.e., ¯k and k) in +both models is identical; therefore, the honest sender’s output distribution in both models is identical. As we +already discussed, the evaluations of deterministic functions (i.e., Merkle tree, H, and PRF) on fCT’s outputs +have an identical distribution. Therefore, each pair (g, q) in the real model has an identical distribution to +the pair (g, q) in the ideal model. Hence, the distribution of the joint outputs of the adversary and honest +receiver in the real and ideal models is indistinguishable. +2 +In addition to the security guarantee (i.e., computation’s correctness against malicious sender or receiver) +stated by Theorem 3, ZSPA offers (a) privacy against the public, and (b) non-refutability. Informally, privacy +here means that given the state of the contract (i.e., g and q), an external party cannot learn any information +about any of the pseudorandom values, zj; while non-refutability means that if a party sends “approved” +then in future cannot deny the knowledge of the values whose representation is stored in the contract. +Theorem 4. If H is preimage resistance, PRF is secure, the signature scheme used in the smart contract is +secure (i.e., existentially unforgeable under chosen message attacks), and the blockchain is secure (i.e., offers +liveness property and the hash power of the adversary is lower than those of honest miners) then ZSPA offers +(i) privacy against the public and (ii) non-refutability. +Proof. First, we focus on privacy. Since key k, for PRF, has been picked uniformly at random and H is preimage +resistance, the probability that given g the adversary can find k is negligible in the security parameter, i.e., +ϵ(λ). Furthermore, because PRF is secure (i.e., its outputs are indistinguishable from random values) and H is +preimage resistance, given the Merkle tree’s root g, the probability that the adversary can find a leaf node, +which is the output of PRF, is ϵ(λ) too. +2 +5.3 +ZSPA’s Extension: ZSPA with an External Auditor (ZSPA-A) +In this section, we present an extension of ZSPA, called ZSPA-A which lets a (trusted) third-party auditor, +Aud, help identify misbehaving clients in the ZSPA and generate a vector of random polynomials. Informally, +ZSPA-A requires that misbehaving parties are always detected, except with a negligible probability. Aud of +this protocol will be invoked by Justitia when Justitia’s smart contract detects that a combination of the +messages sent by the clients is not well-formed. Later, in Justitia’s proof, we will show that even a semi- +honest Aud who observes all messages that clients send to Justitia’s smart contracts, cannot learn anything +about their set elements. We present ZSPA-A in Figure 3. +15 + +• Parties. A set of clients {A1, ..., Am} and an external auditor, Aud. +• Input. m: the total number of participants (excluding the auditor), ζ: a random poly- +nomial of degree 1, b: the total number of vectors, and adr: a deployed smart contract’s +address. Let b′ = b − 1. +• Output +of +each +Aj. +k: +a +secret +key +that +generates +b +vectors +[z0,1, ..., z0,m], ..., [zb′,1, ..., zb′,m] of pseudorandom values, h: hash of the key, g: a +Merkle tree’s root, and a vector of signed messages. +• Output of Aud. L: a list of misbehaving parties’ indices, and #»µ: a vector of random +polynomials. +1. ZSPA invocation. ZSPA(⊥, ..., ⊥) → +� +(k, g, q), ..., (k, g, q) +� +. +All +parties +in {A1, ..., Am} +call +the +same instance +of ZSPA, +which +results +in +(k, g, q), ..., (k, g, q). +2. Auditor computation. Audit(#»k , q, ζ, b, g) → (L, #»µ). +Aud takes the below steps. Note, each kj ∈ #»k is given by Aj. An honest party’s input, +kj, equals k, where 1 ≤ j ≤ m. +(a) runs the checks in the verification phase (i.e., Phase 3) of ZSPA for every j, i.e., +Verify(kj, g, q, m) → (aj, s). +(b) appends j to L, if any checks fails, i.e., if aj = 0. In this case, it skips the next two +steps for the current j. +(c) For every i (where 0 ≤ i ≤ b′), it recomputes m pseudorandom values: ∀j, 1 ≤ j ≤ +m − 1 : zi,j = PRF(k, i||j), +zi,m = − +m−1 +� +j=1 +zi,j. +(d) generates polynomial µ(j) as follows: µ(j) = ζ · ξ(j) − τ (j), where ξ(j) is a random +polynomial of degree b′ − 1 and τ (j) = +b′ +� +i=0 +zi,j · xi. By the end of this step, a vector +#»µ containing at most m polynomials is generated. +(e) returns list L and #»µ. +Fig. 3: ZSPA with an external auditor (ZSPA-A) +Theorem 5. If ZSPA is secure, H is second-preimage resistant, and the correctness of PRF, H, and Merkle +tree holds, then ZSPA-A securely computes f ZSPA-A in the presence of m − 1 malicious adversaries. +Proof. First, we consider the case where a sender, who (may collude with m−2 senders and) generates pairs +(g, q), is corrupt. +Case 1: Corrupt sender. Let Sim +ZSPA-A +S +be the simulator using a subroutine adversary, AS. Below, we explain +how Sim +ZSPA-A +S +works. +1. simulates CT and receives the output value k from fCT. +2. sends k to TTP and receives back from it m pairs, where each pair is of the form (g, q). +3. sends k to AS and receives back from it m pairs where each pair is of the form (g′, q′). +4. constructs an empty vector L. Sim +ZSPA-A +S +checks whether the following equations hold for each j-th pair: +g = g′ +∧ +q = q′. If these two equations do not hold, it sends an abort message Λ to other receiver +clients, appends the index of the pair (i.e., j) to L, and proceeds to the next step for the valid pairs. In +the case where there are no valid pairs, it moves on to step 9. +5. picks a random polynomial ζ of degree 1. Moreover, for every j /∈ L, Sim +ZSPA-A +S +picks a random polynomial +ξ(j) of degree b′ − 1, where 1 ≤ j ≤ m. +6. computes m pseudorandom values for every i, j′, where 0 ≤ i ≤ b′ and j′ /∈ L as follows. +∀j′, 1 ≤ j′ ≤ m − 1 : zi,j = PRF(k, i||j′) +and +zi,m = − +m−1 +� +j′=1 +zi,j +16 + +7. generates polynomial µ(j), for every j /∈ L, as follows: µ(j) = ζ · ξ(j) − τ (j), where τ (j) = +b′� +i=0 +zi,j · xi. +8. sends the above ζ, ξ(j), and µ(j) to all parties (i.e., AS and the receivers), for every j /∈ L. +9. outputs whatever AS outputs. +Now, we focus on the adversary’s output. In the real model, the messages that the adversary receives +include those messages it receives as the result of the ideal call to fCT and (ζ, ξ(j), µ(j)), where j /∈ L and +1 ≤ j ≤ m. Those messages yielded from the ideal calls have identical distribution to the distribution of the +messages in the ideal model, as CT is secure. The distribution of each µ(j) depends on the distribution of its +components; namely, ζ, ξ(j), and τ j. As we are in the fCT-hybrid model, the distributions of τ (j) in the real +model and τ (j) in the ideal model are identical, as they were derived from the output of fCT. Furthermore, in +the real model, each polynomial ζ and ξ(j) has been picked uniformly at random and they are independent +of the clients’ and the adversary’s inputs. The same arguments hold for (ζ, ξ(j), µ(j)) in the ideal model. +Therefore, (ζ, ξ(j), µ(j)) in the real model and (ζ, ξ(j), µ(j)) in the ideal model have identical distributions. +Next, we turn our attention to the receiver’s output. We will show that the output distributions of an +honest receiver and the auditor in the ideal and real models are computationally indistinguishable. In the +real model, each element of the pair (g, p) is the output of a deterministic function on the output of fCT. We +know the outputs of fCT in the real and ideal models have an identical distribution, and so do the evaluations +of deterministic functions (namely Merkle tree, H, and PRF) on them. Therefore, each pair (g, q) in the real +model has an identical distribution to the pair (g, q) in the ideal model. For the same reasons, the honest +receiver in the real model aborts with the same probability as Sim +ZSPA-A +S +does in the ideal model. The same +argument holds for the arbiter’s output, as it performs the same checks that an honest receiver does. Thus, +the distribution of the joint outputs of the adversary, honest receiver, and honest in the real and ideal models +is computationally indistinguishable. +Case 2: Corrupt receiver. Let Sim +ZSPA-A +R +be the simulator that uses subroutine adversary AR. Below, we +explain how Sim +ZSPA-A +R +works. +1. simulates ZSPA and receives the m output pairs of the form (k, g, q) from f ZSPA. +2. sends (k, g, q) to AR and receives m keys, k′ +j, where 1 ≤ j ≤ m. +3. generates an empty vector L. Next, for every j, Sim +ZSPA-A +R +computes q′ +j as H(k′ +j) = qj. It generates gj as +follows. +(a) for every i (where 0 ≤ i ≤ b′), generates m pseudorandom values as below. +∀j, 1 ≤ j′ ≤ m − 1 : zi,j = PRF(k′ +j, i||j′), +zi,m = − +m−1 +� +j=1 +zi,j +(b) constructs a Merkel tree on top of all pseudorandom values, MT.genTree(z0,1, ..., zb′,m) → g′ +j. +4. checks if the following equations hold for each j-th pair: (k = k′ +j) ∧ (g = g′ +j) ∧ (q = q′ +j). +5. If these equations do not hold for j-th value, it appends j to L and proceeds to the next step for the +valid value. In the case where there is no valid value, it moves on to step 9. +6. picks a random polynomial ζ of degree 1. Also, for every j /∈ L, it picks a random polynomial ξ(j) of +degree b′ − 1, where 1 ≤ j ≤ m. +7. generates polynomial µ(j), for every j /∈ L, as follows: µ(j) = ζ · ξ(j) − τ (j), where τ (j) = +b′� +i=0 +zi,j · xi, and +values zi,j were generated in step 3a. +8. sends the above ζ, ξ(j), and µ(j) to AR, for every j /∈ L and 1 ≤ j ≤ m. +9. outputs whatever AR outputs. +In the real model, the adversary receives two sets of messages, the first set includes the transcripts +(including k, g, q) it receives when it makes an ideal call to f ZSPA and the second set includes pairs (ζ, ξ(j), µ(j)), +for every j /∈ L and 1 ≤ j ≤ m. Since we are in the f ZSPA-hybrid model and (based on our assumption) there is +at least one honest party participated in ZSPA (i.e., there are at most m − 1 malicious participants of ZSPA), +the distribution of the messages it receives from f ZSPA in the real and ideal models is identical. Furthermore, +17 + +as we discussed in Case 1, (ζ, ξ(j), µ(j)) in the real model and (ζ, ξ(j), µ(j)) in the ideal model have identical +distribution. The honest sender’s output distribution in both models is identical, as the distribution of fCT’s +output (i.e., k) in both models is identical. +Now we show that the probability that the auditor aborts in the ideal and real models are statistically +close. In the ideal model, Sim +ZSPA-A +R +is given the ideal functionality’s output that includes key k. Therefore, it +can check whether the key that AR sends to it equals k, i.e., k +?= k′ +j. Thus, it aborts with the probability 1. +However, in the real model, an honest auditor is not given the output of CT (say key k) and it can only check +whether the key is consistent with the hash value q and the Merkle tree’s root g stored on the blockchain. +This means the adversary can distinguish the two models if in the real model it sends a key ¨k, such that +¨k ̸= k and still passes the checks. Specifically, it sends the invalid key ¨k that can generate valid pair (g, q), +as follows: H(¨k) = q and MT.genTree(z′ +0,1, ..., z′ +b′,m) → g, where each z′ +i,j is derived from ¨k using the same +technique described in step 3 above. Nevertheless, this means that the adversary breaks the second preimage +resistance property of H; however, H is the second-preimage resistance and the probability that the adversary +succeeds in finding the second preimage is negligible in the security parameter, i.e., ϵ(λ) where λ is the hash +function’s security parameter. Therefore, in the real model, the auditor aborts if an invalid key is provided +with a probability 1−ϵ(λ) which is statically close to the probability that Sim +ZSPA-A +R +aborts in the same situation +in the ideal model, i.e., 1 − ϵ(λ) vs 1. Hence, the distribution of the joint outputs of the adversary, honest +sender, and honest auditor in the real and ideal models is indistinguishable. +2 +5.4 +Unforgeable Polynomials +In this section, we introduce the notion of “unforgeable polynomials”. Informally, an unforgeable polynomial +has a secret factor. To ensure that an unforgeable polynomial has not been tampered with, a verifier can +check whether the polynomial is divisible by the secret factor. +To turn an arbitrary polynomial π of degree d into an unforgeable polynomial θ, one can (i) pick three +secret random polynomials (ζ, ω, γ) and (ii) compute θ = ζ ·ω ·π +γ mod p, where deg(ζ) = 1, deg(ω) = d, +and deg(γ) = 2d + 1. +To verify whether θ has been tampered with, a verifier (given θ, γ, and ζ) can check if θ − γ is divisible +by ζ. Informally, the security of an unforgeable polynomial states that an adversary (who does not know the +three secret random polynomials) cannot tamper with an unforgeable polynomial without being detected, +except with a negligible probability, in the security parameter. Below, we formally state it. +Theorem 6 (Unforgeable Polynomial). Let polynomials ζ, ω, and γ be three secret uniformly random +polynomials (i.e., ζ, ω, γ +$← Fp[x]), GCD(ζ, γ) = 1, polynomial π be an arbitrary polynomial, deg(ζ) = +1, deg(ω) = d, deg(γ) = 2d+1, deg(π) = d, and p be a λ-bit prime number. Also, let polynomial θ be defined +as θ = ζ · ω · π + γ mod p. Given (θ, π), the probability that an adversary (which does not know ζ, ω, and +γ) can forge θ to an arbitrary polynomial δ such that δ ̸= θ, deg(δ) = const(λ), and ζ divides δ − γ is +negligible in the security parameter λ, i.e., +Pr[ζ | (δ − γ)] ≤ ϵ(λ) +Proof. Let τ = δ − γ and ζ = a · x + b. Since γ is a random polynomial of degree 2d + 1 and unknown +to the adversary, given (θ, π), the adversary cannot learn anything about the factor ζ; as from its point of +view every polynomial of degree 1 in Fp[X] is equally likely to be ζ. Moreover, polynomial τ has at most +Max +� +deg(δ), 2d + 1 +� +irreducible non-constant factors. For ζ to divide τ, one of the factors of τ must be +equal to ζ. We also know that ζ has been picked uniformly at random (i.e., a, b +$← Fp) and by definition +GCD(ζ, γ) = 1. Thus, the probability that ζ divides τ is negligible in the security parameter, λ. Specifically, +Pr[ζ | (δ − γ)] ≤ Max +� +deg(δ), 2d + 1 +� +22λ += ϵ(λ) +2 +18 + +An interesting feature of an unforgeable polynomial is that the verifier can perform the check without +needing to know the original polynomial π. Another appealing feature of the unforgeable polynomial is that it +supports linear combination and accordingly batch verification. Specifically, to turn n arbitrary polynomials +[π1, ..., πn] into unforgeable polynomials, one can construct θi = ζ · ωi · πi + γi mod p, where ∀i, 1 ≤ i ≤ n. +To check whether all polynomials [θ1, ..., θn] are intact, a verifier can (i) compute their sum χ = +n� +i=1 +θi +and (ii) check whether χ− +n� +i=1 +γi is divisible by ζ. Informally, the security of an unforgeable polynomial states +that an adversary (who does not know the three secret random polynomials for each θi) cannot tamper with +any subset of the unforgeable polynomials without being detected, except with a negligible probability. We +formally state it, below. +Theorem 7 (Unforgeable Polynomials’ Linear Combination). Let polynomial ζ be a secret polyno- +mial picked uniformly at random; also, let #»ω = [ω1, ..., ωn] and #»γ = [γ1, ..., γn] be two vectors of secret +uniformly random polynomials (i.e., ζ, ωi, γi +$← Fp[x]), GCD(ζ, γi) = 1, #»π = [π1, ..., πn] be a vector of +arbitrary polynomials, deg(ζ) = 1, deg(ωi) = d, deg(γi) = 2d + 1, deg(πi) = d, p be a λ-bit prime number, +and 1 ≤ i ≤ n. Moreover, let polynomial θi be defined as θi = ζ · ωi · πi + γi mod p, and #»θ = [θ1, ..., θn]. +Given (#»θ , #»π), the probability that an adversary (which does not know ζ, #»ω, and #»γ ) can forge t polynomi- +als, without loss of generality, say θ1, ..., θt ∈ #»θ to arbitrary polynomials δ1, ..., δt such that +t� +j=1 +δj ̸= +t� +j=1 +θj, +deg(δj) = const(λ), and ζ divides ( +t� +j=1 +δj + +n� +j=t+1 +θj − +n� +j=1 +γj) is negligible in the security parameter λ, i.e., +Pr[ζ | ( +t +� +j=1 +δj + +n +� +j=t+1 +θj − +n +� +j=1 +γj)] ≤ ϵ(λ) +Proof. This proof is a generalisation of that of Theorem 6. Let τj = δj − γj and ζ = a · x + b. Since every +γj is a random polynomial of degree 2d + 1 and unknown to the adversary, given (#»θ , #»π), the adversary +cannot learn anything about the factor ζ. Each polynomial τj has at most Max +� +deg(δj), 2d + 1 +� +irreducible +non-constant factors. We know that ζ has been picked uniformly at random (i.e., a, b +$← Fp), by definition +GCD(ζ, γj) = 1, and ζ does divide every θj. Therefore, the probability that ζ divides +t� +j=1 +δj + +n� +j=t+1 +θj − +n� +j=1 +γj +equals the probability that ζ equals to one of the factors of every τj, that is negligible in the security +parameter. Concretely, +Pr[ζ | ( +t +� +j=1 +δj + +n +� +j=t+1 +θj − +n +� +j=1 +γj)] ≤ +t� +j=1 +Max +� +deg(δj), 2d + 1 +� +22λt += ϵ(λ) +2 +It is not hard to see that, Theorem 7 is a generalisation of Theorem 6. Briefly, in Justitia, we will use +unforgeable polynomials (and their linear combinations) to allow a smart contract to efficiently check whether +the polynomials that the clients send to it are intact, i.e., they are VOPR’s outputs. +6 +Justitia: A Concrete Construction of PSI +FC +6.1 +Main Challenges to Overcome +We need to address several key challenges, to design an efficient scheme that realises PSI +FC. Below, we +outline these challenges. +19 + +Keeping Overall Complexities Low. In general, in multi-party PSIs, each client needs to send messages +to the rest of the clients and/or engage in secure computation with them, e.g., in [26,34], which would result +in communication and/or computation quadratic with the number of clients. To address this challenge, we +(a) allow one of the clients as a dealer to interact with the rest of the clients,4 and (b) we use a smart contract, +which acts as a bulletin board to which most messages are sent and also performs lightweight computation +on the clients’ messages. The combination of these approaches will keep the overall communication and +computation linear with the number of clients (and sets’ cardinality). +Securely Randomising Input Polynomials. In multi-party PSIs that rely on the polynomial represen- +tation, it is essential that an input polynomial of a client be randomised by another client [3]. To do that +securely and efficiently, we require the dealer and each client together to engage in an instance of VOPR, which +we developed in Section 5. +Preserving the Privacy of Outgoing Messages. Although the use of regular public smart contracts +(e.g., Ethereum) will help keep overall complexity low, it introduces another challenge; namely, if clients do +not protect the privacy of the messages they send to the smart contracts, then other clients (e.g., dealer) and +non-participants of PSI (i.e., the public) can learn the clients’ set elements and/or the intersection. Because +standard smart contracts do not automatically preserve messages’ privacy. To efficiently protect the privacy +of each client’s messages (sent to the contracts) from the dealer, we require the clients (except the dealer) to +engage in ZSPA-A which lets each of them generate a pseudorandom polynomial with which it can blind its +message. To protect the privacy of the intersection from the public, we require all clients to run a coin-tossing +protocol to agree on a blinding polynomial, with which the final result that encodes the intersection on the +smart contract will be blinded. +Ensuring the Correctness of Subroutine Protocols’ Outputs. In general, any MPC that must remain +secure against an active adversary is equipped with a verification mechanism that ensures an adversary is +detected if it deviates from the protocol and affects messages’ integrity, during the protocol’s execution. +This is the case for the subroutine protocols that we use, i.e., VOPR and ZSPA-A. Nevertheless, this type of +verification itself is not always sufficient. Because in certain cases, the output of an MPC protocol may be fed +as input to another MPC and we need to ensure that the actual/intact output of the first MPC is fed to the +second one. This is the case in our PSI’s subroutines as well. To address this challenge, we use unforgeable +polynomials; specifically, the output of VOPR is an unforgeable polynomial (that encodes the actual output); +if the adversary tampers with the VOPR’s output and uses it later, then a verifier can detect it. We will have +the same integrity guarantee for the output of ZSPA-A for free. Because (i) VOPR is called before ZSPA-A, and +(ii) if clients use intact outputs of ZSPA-A, then the final result (i.e., the sum of all clients’ messages) will not +contain any output of ZSPA-A, as they would cancel out each other. Thus, by checking the correctness of the +final result, one can ensure the correctness of the outputs of VOPR and ZSPA-A, in one go. +6.2 +Description of Justitia (JUS) +An overview. At a high level, Justitia (JUS) works as follows. First, each client encodes its set elements into +a polynomial. All clients sign a smart contract and deposit a predefined amount of coins into it. Next, one of +the clients as a dealer, D, randomises the rest of the clients’ polynomials and imposes a certain structure to +their polynomials. The clients also randomise D’s polynomials. The randomised polynomials reveal nothing +about the clients’ original polynomials representing their set elements. Then, all clients send their randomised +polynomials to the smart contract. The contract combines all polynomials and checks whether the resulting +polynomial still has the structure imposed by D. If the contract concludes that the resulting polynomial does +not have the structure, then it invokes an auditor, Aud, to identify misbehaving clients and penalise them. +4 This approach has similarity with the non-secure PSIs in [23]. +20 + +Nevertheless, if the resulting polynomial has the structure, then the contract outputs an encoded polynomial +and refunds the clients’ deposits. In this case, all clients can use the resulting polynomial (output by the +contract) to locally find the intersection. Figure 4 outlines the interaction between parties. +VOPR +ADCXicjVJLj9MwEHbCazGP7cIFiYtFt9IioSqpxOPGinJAnBYt3V2pCZHjuq21fkTxZKVi5Rfwa7ghrvy +K/Tc4aQ602wMjO/o830znonzQgoLUXQdhLdu37l7b+8+fvDw0eP93sGTM2uqkvEJM9KUFzm1XArNJyBA8oui5FTlkp/nl+Mmfn7FSyuM/gqrgqeKLrSYC0bBu7LeNcaDwyRXyXcOlCRsZoA0R6P4gn5ziWlKD7wkpyQpxfRx9fjTP1sq7rQ4IxJuscC6rU/4m8BrcSWPq6OyWeWme7AqO1vL24orBkVLrTcNtTgDu8+TUl8h6/WgYtUZugrgDfdTZSXYQPEtmh +lWKa2CSWjuNowJSR0sQTPIaD7bCphRXnKVOSs1sjZPK8oKyS7rgUw81Vdymrv1FNRl4z4zMTem3BtJ6/1U4qxdqdwzmy7sRqztyxi5WcTlqt5O0TB3pZhWMH+XOqGLCrhm6/vMK0nAkOZkJkoOQO58oD6WfuOCVvSkjLwjwf7Wcbk7sJzkbD+M3w9ZdR/hDN9U9By9QEcoRm/RMfqETtAEseB9wAMdmPBH+DP8Ff5eU8Og0zxFGxb+QtM/vST✓Cm +2 +A +Cp3icbVFba9swFJbdXVrvlnaPfRFLAi2MYBe29rG0e9hbW2jSQOwFWTlJRHUx0nEhGP+G/r79kL1Pc1Y0xyQ+PguHPQpL6RwGMe/g3Dn1es3b3f3onfvP3z81Nk/GDlTWg5DbqSx45 +w5kELDEAVKGBcWmMol3OX3l2v97gGsE0bf4qATLGFnPBGXpq2nmMol6aq9Quza8qdyKAtsbVxJoRWl1dDlVX38c13Xdi6J+YzcKFmxrYCOR8plBuo4UYrv/X6BZMO1040HcDH0JkhZ0S 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of the novelties of JUS is a lightweight verification mechanism which allows a smart contract to +efficiently verify the correctness of the clients’ messages without being able to learn/reveal the clients’ set +elements. To achieve this, D randomises each client’s polynomials and constructs unforgeable polynomials on +the randomised polynomials (in one go). If any client modifies an unforgeable polynomial that it receives and +sends the modified polynomial to the smart contract, then the smart contract would detect it, by checking +whether the sum of all clients’ (unforgeable) polynomials is divisible by a certain polynomial of degree 1. The +verification is lightweight because: (i) it does not use any public key cryptography (often computationally +expensive), (ii) it needs to perform only polynomial division, and (iii) it can perform batch verification, i.e., +it sums all clients randomised polynomials and then checks the result’s correctness. +Now, we describe JUS in more detail. First, all clients sign and deploy a smart contract, SCJUS. Each +of them put a certain amount of deposit into it. Then, they together run CT to agree on a key, mk, that +will be used to generate a set of blinding polynomials to hide the final result from the public. Next, each +client locally maps its set elements to a hash table and represents the content of each hash table’s bin as a +polynomial, π. After that, for each bin, the following steps are taken. All clients, except D, engage in ZSPA-A +to agree on a set of pseudorandom blinding factors such that their sum is zero. +Then, D randomises and constructs an unforgeable polynomial on each client’s polynomial, π. To do that, +D and every other client engage in VOPR that returns to the client a polynomial. D and every other client +invoke VOPR again to randomise D’s polynomial. VOPR returns to the client another unforgeable polynomial. +Note that the output of VOPR reveals nothing about any client’s original polynomial π, as this polynomial +has been blinded/encrypted with another secret random polynomial by D, during the execution of VOPR. +Each client sums the two polynomials, blinds the result (using the output of ZSPA-A), and sends it to SCJUS. +After all of the clients send their input polynomials to SCJUS, D sends to SCJUS a switching polynomial +that will allow SCJUS to obliviously switch the secret blinding polynomials used by D (during the execution +of VOPR) to blind each client’s original polynomial π to another blinding polynomial that all clients can +21 + +generate themselves, by using key mk. The switching polynomial is constructed in a way that does not affect +the verification of unforgeable polynomials. +Next, D sends to SCJUS a secret polynomial, ζ, that will allow SCJUS to check unforgeable polynomials’ +correctness. Then, SCJUS sums all clients’ polynomials and checks if ζ can divide the sum. SCJUS accepts +the clients’ inputs if the polynomial divides the sum; otherwise, it invokes Aud to identify misbehaving +parties. In this case, all honest parties’ deposit is refunded to them and the deposit of misbehaving parties +is distributed among the honest ones as well. If all clients behave honestly, then each client can locally find +the intersection. To do that, it uses mk to locally remove the blinding polynomial from the sum (that the +contract generated), then evaluates the unblinded polynomial at each of its set elements and considers an +element in the intersection when the evaluation equals zero. +Detailed Description of JUS. Below, we elaborate on how JUS exactly works (see Table 1 for description +of the main notations used). +1. All clients in CL = {A1, ..., Am, D} sign a smart contract: SCJUS and deploy it to a blockchain. All clients +get the deployed contract’s address. Also, all clients engage in CT to agree on a secret master key, mk. +2. Each client in CL builds a hash table, HT, and inserts the set elements into it, i.e., ∀i : H(si) = indx, then +si → HTindx. It pads every bin with random dummy elements to d elements (if needed). Then, for every +bin, it constructs a polynomial whose roots are the bin’s content: π = +d� +i=1 +(x − s′ +i), where s′ +i is either si +or a random value. +3. Every client C in CL \ D, for every bin, agree on b = 3d + 3 vectors of pseudorandom blinding factors: +zi,j, such that the sum of each vector elements is zero, i.e., +m +� +j=1 +zi,j = 0, where 0 ≤ i ≤ b − 1. To do that, +they participate in step 1 of ZSPA-A. By the end of this step, for each bin, they agree on a secret key k +(that will be used to generate the zero-sum values) as well as two values stored in SCJUS, i.e., q: the key’s +hash value and g: a Merkle tree’s root. After time t1, D ensures that all other clients have agreed on the +vectors (i.e., all provided “approved” to the contract). If the check fails, it halts. +4. Each client in CL deposits ¨y + ¨ch amount to SCJUS. After time t2, every client ensures that in total +(¨y + ¨ch) · (m + 1) amount has been deposited. Otherwise, it halts and the clients’ deposit is refunded. +5. D picks a random polynomial ζ +$← Fp[X] of degree 1, for each bin. It, for each client C, allocates to +each bin two random polynomials: ω(D,C), ρ(D,C) +$← Fp[X] of degree d, and two random polynomials: +γ(D,C), δ(D,C) ← Fp[X] of degree 3d + 1. Also, each client C, for each bin, picks two random polynomials: +ω(C,D), ρ(C,D) +$← Fp[X] of degree d. +6. D randomises other clients’ polynomials. To do so, for every bin, it invokes an instance of VOPR (presented +in Fig. 1) with each client C; where D sends ζ · ω(D,C) and γ(D,C), while client C sends ω(C,D) · π(C) to +VOPR. Each client C, for every bin, receives a blind polynomial of the following form: +θ(C) +1 += ζ · ω +(D,C) · ω +(C,D) · π +(C) + γ +(D,C) +from VOPR. If any party aborts, the deposit would be refunded to all parties. +7. Each client C randomises D’s polynomial. To do that, each client C, for each bin, invokes an instance of +VOPR with D, where each client C sends ρ(C,D), while D sends ζ · ρ(D,C) · π(D) and δ(D,C) to VOPR. Every +client C, for each bin, receives a blind polynomial of the following form: +θ(C) +2 += ζ · ρ +(D,C) · ρ +(C,D) · π +(D) + δ +(D,C) +from VOPR. If any party aborts, the deposit would be refunded to all parties. +8. Each client C, for every bin, masks the sum of polynomials θ(C) +1 +and θ(C) +2 +using the blinding factors: zi,c, +generated in step 3. Specifically, it computes the following blind polynomial (for every bin): +ν +(C) = θ +(C) +1 ++ θ +(C) +2 ++ τ +(C) +22 + +where τ (C) = +3d+2 +� +i=0 +zi,c · xi. Next, it sends all ν(C) to SCJUS. If any party aborts, the deposit would be +refunded to all parties. +9. D ensures all clients sent their inputs to SCJUS. If the check fails, it halts and the deposit would be +refunded to all parties. It allocates a fresh pseudorandom polynomial γ′ of degree 3d, to each bin. To +do so, it uses mk to derive a key for each bin: kindx = PRF(mk, indx) and then uses the derived key to +generate 3d + 1 pseudorandom coefficients gj,indx = PRF(kindx, j) where 0 ≤ j ≤ 3d. Also, for each bin, it +allocates a fresh random polynomial: ω′(D) of degree d. +10. D, for every bin, computes a polynomial of the form: +ν +(D) = ζ · ω′(D) · π +(D) − +Am +� +C=A1 +(γ +(D,C) + δ +(D,C)) + ζ · γ′ +It sends to SCJUS polynomials ν(D) and ζ, for each bin. +11. SCJUS takes the following steps: +(a) for every bin, sums all related polynomials provided by all clients in ¯P: +φ = ν +(D) + +Am +� +C=A1 +ν +(C) += ζ · +� +ω′(D) · π +(D) + +Am +� +C=A1 +(ω +(D,C) · ω +(C,D) · π +(C)) + π +(D) · +Am +� +C=A1 +(ρ +(D,C) · ρ +(C,D)) + γ′ +� +(b) checks whether, for every bin, ζ divides φ. If the check passes, it sets Flag = True. Otherwise, it +sets Flag = False. +12. If the above check passes (i.e., Flag = True), then the following steps are taken: +(a) SCJUS sends back each party’s deposit, i.e., ¨y + ¨ch amount. +(b) each client (given ζ and mk) finds the elements in the intersection as follows. +i. derives a bin’s pseudorandom polynomial, γ′, from mk. +ii. removes the blinding polynomial from each bin’s polynomial: +φ′ = φ − ζ · γ′ +iii. evaluates each bin’s unblinded polynomial at every element si belonging to that bin and considers +the element in the intersection if the evaluation is zero: i.e., φ′(si) = 0. +13. If the check does not pass (i.e., Flag = False), then the following steps are taken. +(a) Aud asks every client C to send to it the PRF’s key (generated in step 3), for every bin. It inserts +the keys to #»k . It generates a list ¯L initially empty. Then, for every bin, Aud takes step 2 of ZSPA-A, +i.e., invokes Audit(#»k , q, ζ, 3d + 3, g) → (L, #»µ). Every time it invokes Audit, it appends the elements +of returned L to ¯L. Aud for each bin sends #»µ to SCJUS. It also sends to SCJUS the list ¯L of all +misbehaving clients detected so far. +(b) to help identify further misbehaving clients, D takes the following steps, for each bin of client C +whose ID is not in ¯L. +i. computes polynomial χ(D,C) as follows. +χ +(D,C) = ζ · η +(D,C) − (γ +(D,C) + δ +(D,C)) +where η(D,C) is a fresh random polynomial of degree 3d + 1. +ii. sends polynomial χ(D,C) to SCJUS. +Note, if ¯L contains all clients’ IDs, then D does not need to take the above steps 13(b)i and 13(b)ii. +(c) SCJUS, takes the following steps to check if the client misbehaved, for each bin of client C whose ID +is not in ¯L. +23 + +i. computes polynomial ι(C) as follows: +ι +(C) = χ +(D,C) + ν +(C) + µ +(C) += ζ · (η +(D,C) + ω +(D,C) · ω +(C,D) · π +(C) + ρ +(D,C) · ρ +(C,D) · π +(D) + ξ +(C)) +where µ(C) ∈ #»µ generated and sent to SCJUS by Aud in step 13a. +ii. checks if ζ divides ι(C). If the check fails, it appends the client’s ID to a list L′. +If ¯L contains all clients’ IDs, then SCJUS does not take the above two steps. +(d) SCJUS refunds the honest parties’ deposit. Also, it retrieves the total amount of ¨ch from the deposit +of dishonest clients (i.e., those clients whose IDs are in ¯L or L′) and sends it to Aud. It also splits +the remaining deposit of the misbehaving parties among the honest ones. Thus, each honest client +receives ¨y + ¨ch + m′·(¨y+ ¨ +ch)− ¨ +ch +m−m′ +amount in total, where m′ is the total number of misbehaving parties. +One may be tempted to replace Justitia with a scheme in which all clients send their encrypted sets to +a server (potentially semi-honest and plays Aud’s role) which computes the result in a privacy-preserving +manner. We highlight that the main difference is that in this (hypothetical) scheme the server is always +involved; whereas, in our protocol, Aud remains offline as long as the clients behave honestly and it is +invoked only when the contract detects misbehaviours. +Next, we present a theorem that formally states the security of JUS. +Theorem 8. Let polynomials ζ, ω, and γ be three secret uniformly random polynomials. If θ = ζ · ω · π + +γ mod p is an unforgeable polynomial (w.r.t. Theorem 6), ZSPA-A, VOPR, PRF, and smart contracts are secure, +then JUS securely realises f PSI with Q-fairness (w.r.t. Definition 7) in the presence of m − 1 active-adversary +clients (i.e., Ajs) or a passive dealer client, passive auditor, or passive public. +6.3 +Proof of JUS +In this section, we prove Theorem 8, i.e., the security of JUS. +Proof. We prove Theorem 8 by considering the case where each party is corrupt, at a time. +Case 1: Corrupt m − 1 clients in {A1, ..., Am}. Let P ′ be a set of at most m − 1 corrupt clients, where +P ′ ⊂ {A1, ..., Am}. Let set ˆP be defined as follows: ˆP = {A1, ..., Am} \ P ′. Also, let Sim +JUS +A be the simulator, +which uses a subroutine adversary, A. Below, we explain how Sim +JUS +A (which receives the input sets of honest +dealer D and honest client(s) in ˆP) works. +1. constructs and deploys a smart contract. It sends the contract’s address to A. +2. simulates CT and receives the output value, mk, from its functionality, fCT. +3. simulates ZSPA-A for each bin and receives the output value, (k, g, q), from its functionality, f ZSPA-A. +4. deposits in the contract the total amount of (¨y + ¨ch) · (m − |P ′| + 1) coins on behalf of D and honest +client(s) in ˆP. It sends to A the amount deposited in the contract. +5. checks if A has deposited (¨y + ¨ch) · |P ′| amount. If the check fails, it instructs the ledger to refund the +coins that every party deposited and sends message abort1 to TTP (and accordingly to all parties); it +outputs whatever A outputs and then halts. +6. picks a random polynomial ζ of degree 1, for each bin. Also, Sim +JUS +A , for each client C ∈ {A1, ..., Am} +allocates to each bin two random polynomials: (ω(D,C), ρ(D,C)) of degree d and two random polynomials: +(γ(D,C), δ(D,C)) of degree 3d + 1. Moreover, Sim +JUS +A for every honest client C′ ∈ ˆP, for each bin, picks two +random polynomials: (ω(C′,D), ρ(C′,D)) of degree d. +7. simulates VOPR using inputs ζ ·ω(D,C) and γ(D,C) for each bin. Accordingly, it receives the inputs of clients +C′′ ∈ P ′, i.e., ω(C′′,D) · π(C′′), from its functionality f VOPR, for each bin. +8. extracts the roots of polynomial ω(C′′,D) · π(C′′) for each bin and appends those roots that are in the sets +universe to a new set S(C′′). +9. simulates VOPR again using inputs ζ · ρ(D,C) · π(D) and δ(D,C), for each bin. +24 + +10. sends to TTP the input sets of all parties; namely, (i) client D’s input set: S(D), (ii) honest clients’ input +sets: S(C′) for all C′ in ˆP, and (iii) A’s input sets: S(C′′), for all C′′ in P ′. For each bin, it receives the +intersection set, S∩, from TTP. +11. represents the intersection set for each bin as a polynomial, π, as follows: π = +|S∩| +� +i=1 +(x−si), where si ∈ S∩. +12. constructs polynomials θ(C′) +1 += ζ · ω(D,C′) · ω(C′,D) · π + γ(D,C′), θ(C′) +2 += ζ · ρ(D,C′) · ρ(C′,D) · π + δ(D,C′), and +ν(C′) = θ(C′) +1 ++ θ(C′) +2 ++ τ (C′), for each bin and each honest client C′ ∈ ˆP, where τ (C) = +3d+2 +� +i=0 +zi,c · xi and +each zi,c is derived from k. +13. sends to A polynomial ν(C′) for each bin and each client C′. +14. receives ν(C′′) from A, for each bin and each client C′′ ∈ P ′. It ensures that the output for every C′′ has +been provided. Otherwise, it halts. +15. if there is any abort within steps 7–14, then it sends abort2 to TTP and instructs the ledger to refund +the coins that every party deposited. It outputs whatever A outputs and then halts. +16. constructs polynomial ν(D) = ζ · ω′(D) · π − +Am +� +C=A1 +(γ(D,C) + δ(D,C)) + ζ · γ′ for each bin, where ω′(D) is a +fresh random polynomial of degree d and γ′ is a pseudorandom polynomial derived from mk. +17. sends to A polynomials ν(D) and ζ for each bin. +18. given each ν(C′′), computes polynomial φ′ as follows φ′ = +� +∀C′′∈P ′ ν(C′′) − +� +∀C′′∈P ′(γ(D,C′′) + δ(D,C′′)), for +every bin. Then, Sim +JUS +A checks whether ζ divides φ′, for every bin. If the check passes, it sets Flag = True. +Otherwise, it sets Flag = False. +19. if Flag = True: +(a) instructs the ledger to send back each party’s deposit, i.e., ¨y + ¨ch amount. It sends a message deliver +to TTP. +(b) outputs whatever A outputs and then halts. +20. if Flag = False: +(a) receives |P ′| keys of the PRF from A, i.e., #»k ′ = [k′ +1, ..., k′ +|P ′|], for every bin. +(b) checks whether the following equation holds: k′ +j = k, for every k′ +j ∈ #»k ′. Note that k is the output of +f ZSPA-A generated in step 3. It constructs an empty list L′ and appends to it the indices (e.g., j) of the +keys that do not pass the above check. +(c) simulates ZSPA-A and receives from f ZSPA-A the output that contains a vector of random polynomials, +#»µ ′, for each valid key. +(d) sends to A, the list L′ and vector #»µ ′, for every bin. +(e) for each bin of client C whose index (or ID) is not in L′ computes polynomial χ(D,C) as follows: +χ(D,C) = ζ · η(D,C) − (γ(D,C) + δ(D,C)), where η(D,C) is a fresh random polynomial of degree 3d + 1. +Note, C includes both honest and corrupt clients, except those clients whose index is in L′. Sim +JUS +A +sends every polynomial χ(D,C) to A. +(f) given each ν(C′′) (by A in step 14), computes polynomial φ′(C′′) as follows: φ′(C′′) = ν(C′′) −γ(D,C′′) − +δ(D,C′′), for every bin. Then, Sim +JUS +A checks whether ζ divides φ′(C′′), for every bin. It appends the +index of those clients that did not pass the above check to a new list, L′′. Note that L′ ∩ L′′ = ⊥. +(g) if L′ or L′′ is not empty, then it instructs the ledger: (i) to refund the coins of those parties whose +index is not in L′ and L′′, (ii) to retrieve ¨ch amount from the adversary (i.e., one of the parties +whose index is in one of the lists) and send the ¨ch amount to the auditor, and (iii) to compensate +each honest party (whose index is not in the two lists) m′·(¨y+ ¨ +ch)− ¨ +ch +m−m′ +amount, where m′ = |L′| + |L′′|. +Then, it sends message abort3 to TTP. +(h) outputs whatever A outputs and halts. +Next, we show that the real and ideal models are computationally indistinguishable. We first focus on +the adversary’s output. In the real and ideal models, the adversary sees the transcripts of ideal calls to fCT as +well as this functionality outputs, i.e., mk. Due to the security of CT (as we are in the fCT-hybrid world), the +transcripts of fCT in both models have identical distribution, so have the random output of fCT, i.e., mk. The +25 + +same holds for (the transcripts and) outputs (i.e., (k, g, q)) of f ZSPA-A that the adversary observes in the two +models. Also, the deposit amount is identical in both models. Thus, in the case where abort1 is disseminated +at this point; the adversary’s output distribution in both models is identical. +The adversary also observes (the transcripts and) outputs of ideal calls to f VOPR in both models, i.e., +output (θ(C′′) +1 += ζ · ω(D,C′′) · ω(C′′,D) · π(C′′) + γ(D,C′′), θ(C′′) +2 += ζ · ρ(D,C′′) · ρ(C′′,D) · π(D) + δ(D,C′′)) for each +corrupted client C′′. However, due to the security of VOPR, the A’s view, regarding VOPR, in both models +have identical distribution. In the real model, the adversary observes the polynomial ν(C) that each honest +client C stores in the smart contract. Nevertheless, this is a blinded polynomial comprising of two uniformly +random blinding polynomials (i.e., γ(D,C) and δ(D,C)) unknown to the adversary. In the ideal model, A is +given polynomial ν(C′) for each honest client C′. This polynomial has also been blinded via two uniformly +random polynomials (i.e., γ(D,C′) and δ(D,C′)) unknown to A. Thus, ν(C) in the real model and ν(C) in the +ideal model have identical distributions. As a result, in the case where abort2 is disseminated at this point; +the adversary’s output distribution in both models is identical. +Furthermore, in the real world, the adversary observes polynomials ζ and ν(D) that D stores in the +smart contract. Nevertheless, ζ is a uniformly random polynomial, also polynomial ν(D) has been blinded; +its blinding factors are the additive inverse of the sum of the random polynomials γ(D,C) and δ(D,C) unknown +to the adversary, for every client C ∈ {A1, ..., Am} and D. In the ideal model, A is given ζ and ν(D), where +the former is a random polynomial and the latter is a blinded polynomial that has been blinded with the +additive inverse of the sum of random polynomials γ(D,C) and δ(D,C) unknown to it, for all client C. Therefore, +(ζ, ν(D)) in the real model and (ζ, ν(D)) in the ideal model component-wise have identical distribution. +Also, the sum of less than m + 1 blinded polynomials ν(A1), ..., ν(Am), ν(D) in the real model has identical +distribution to the sum of less than m + 1 blinded polynomials ν(A1), ..., ν(Am), ν(D) in the ideal model, +as such a combination would still be blinded by a set of random blinding polynomials unknown to the +adversary. Now we discuss why the two polynomials φ +ζ − γ′ in the real model and φ +ζ − γ′ in the ideal model +are indistinguishable. Note that we divide and then subtract polynomials φ because the adversary already +knows (and must know) polynomials (ζ, γ′). In the real model, polynomial φ +ζ − γ′ has the following form: +φ +ζ −γ′ = ω′(D)·π +(D)+ +Am +� +C=A1 +(ω +(D,C)·ω +(C,D)·π +(C))+π +(D)· +Am +� +C=A1 +(ρ +(D,C)·ρ +(C,D)) = µ·gcd(π +(D), π +(A1), ..., π +(Am)) (2) +In Equation 2, every element of [ω′(D), ..., ω(D,C), ρ(D,C)] is a uniformly random polynomial for every client +C ∈ {A1, ..., Am} (including corrupt ones) and client D; because it has been picked by (in this case honest) +client D. Thus, as shown in Section 3.9, φ +ζ − γ′ has the form µ · gcd(π(D), π(A1), ..., π(Am)), where µ is a +uniformly random polynomial and gcd(π(D), π(A1), ..., π(Am)) represents the intersection of the input sets. +In the ideal model, A can construct polynomial φ using its (well-formed) inputs ν(C′′) and polynomials +ν(C′) that the simulator has sent to it, for all C′ ∈ ˆP and all C′′ ∈ P ′. Thus, in the ideal model, polynomial +φ +ζ − γ′ has the following form: +φ +ζ − γ′ = π · +� � +∀C′∈ ˆ +P +(ω +(D,C′) · ω +(C′,D)) + +� +∀C′∈ ˆ +P +(ρ +(D,C′) · ρ +(C′,D)) +� ++ ++ +� � +∀C′′∈P ′ +(ω +(D,C′′) · ω +(C′′,D) · π +(C′′)) + π +(D) · +� +∀C′′∈P ′ +(ρ +(D,C′′) · ρ +(C′′,D)) +� += µ · gcd(π, π +(D), π +(C′′)) +(3) +In Equation 3, every element of the vector [ω(D,C′), ω(D,C′′), ρ(D,C′), ρ(D,C′′)] is a uniformly random poly- +nomial for all C′ ∈ ˆP and all C′′ ∈ P ′, as they have been picked by Sim +JUS +A . Therefore, +φ +ζ − γ′ equals +µ · gcd(π, π(D), π(C′′)), such that µ is a uniformly random polynomial and gcd(π, π(D), π(C′′)) represents the +intersection of the input sets. We know that gcd(π(D), π(A1), ..., π(Am)) = gcd(π, π(D), π(C′′)), as π includes +the intersection of all clients’ sets. Also, µ has identical distribution in the two models, because they are +uniformly random polynomials. Thus, φ +ζ − γ′ in the real model and φ +ζ − γ′ in the ideal model are indistin- +guishable. +Now we focus on the case where Flag = False. In the real model, the adversary observes the output of +Audit(.) which is a list of indices L and a vector of random polynomials #»µ picked by an honest auditor. In +26 + +the ideal model, A is given a list L′ of indices and a vector of random polynomials #»µ ′ picked by the simulator. +Thus, the pair (L, #»µ) in the real model has identical distribution to the pair (L′, #»µ ′) in the ideal model. +Moreover, in the real model, the adversary observes each polynomial χ(D,C) = ζ · η(D,C) − (γ(D,C) + δ(D,C)) +that D stores in the contract, for each bin and each client C whose index is not in L. This is a blinded +polynomial with blinding factor η(D,C) which itself is a uniformly random polynomial picked by D. In the +ideal model, A is given a polynomial of the form χ(D,C) = ζ · η(D,C) − (γ(D,C) + δ(D,C)), for each bin and each +client C whose index is not in L′. This is also a blinded polynomial whose blinding factor is η(D,C) which +itself is a random polynomial picked by the simulator. Therefore, χ(D,C) in the real model has identical +distribution to χ(D,C) in the ideal model. In the real model, the adversary observes polynomial ι(C) = +ζ · (η(D,C) + ω(D,C) · ω(C,D) · π(C) + ρ(D,C) · ρ(C,D) · π(D) + ξ(C)) which is a blinded polynomial whose blinding +factor is the sum of the above random polynomials, i.e., η(D,C) + ξ(C). In the ideal model, A already has +polynomials χ(D,C), ν(C), and µ(C), where µ(C) ∈ #»µ ′; this lets A compute ι(C) = χ(D,C) + ν(C) + µ(C) = +ζ ·(η(D,C) +ω(D,C′) ·ω(C′,D) ·π +ρ(D,C′) ·ρ(C′,D) ·π +ξ(C)), where ξ(C) is a random blinding polynomial used in +µ(C). Nevertheless, ι(C) itself is a blinded polynomial whose blinding factor is the sum of random polynomials, +i.e., η(D,C) + ξ(C). Hence, the distribution of polynomial ι(C) in the real model and ι(C) in the ideal model +are identical. Moreover, the integer ¨y + ¨ch + m′·(¨y+ ¨ +ch)− ¨ +ch +m−m′ +has identical distribution in both models. +Next, we show that the honest party aborts with the same probability in the real and ideal models. Due +to the security of CT, an honest party (during the execution of CT) aborts with the same probability in both +models; in this case, the adversary learns nothing about the parties’ input set and the sets’ intersection as +the parties have not sent out any encoded input set yet. The same holds for the probability that honest +parties abort during the execution of ZSPA-A. In this case, an aborting adversary also learns nothing about +the parties’ input set and the sets’ intersection. Since all parties’ deposit is public, an honest party can +look up and detect if not all parties have deposited a sufficient amount with the same probability in both +models. If parties halt because of insufficient deposit, no one learns about the parties’ input set and the sets’ +intersection because the inputs (representation) have not been sent out at this point. +Due to the security of VOPR, honest parties abort with the same probability in both models. In the case +of an abort during VOPR execution, the adversary would learn nothing (i) about its counter party’ input set +due to the security of VOPR, and (ii) about the rest of the honest parties’ input sets and the intersection +as the other parties’ input sets are still blinded by random blinding factors known only to D. In the real +model, D can check if all parties provided their encoded inputs, by reading from the smart contract. The +simulator also can do the same check to ensure A has provided the encoded inputs of all corrupt parties. +Therefore, in both models, an honest party with the same probability would detect the violation of such +a requirement, i.e., providing all encoded inputs. Even in this case, if an adversary aborts (by not proving +its encoded inputs), then it learns nothing about the honest parties’ input sets and the intersection for the +reason explained above. +In the real model, the smart contract sums every client C’s polynomial ν(C) with each other and with D’s +polynomial ν(D), which removes the blinding factors that D initially inserted (during the execution of VOPR), +and then checks whether the result is divisible by ζ. Due to (a) Theorem 7 (i.e., unforgeable polynomials’ +linear combination), (b) the fact that the smart contract is given the random polynomial ζ in plaintext, (c) +no party (except honest client D) knew anything about ζ before they send their input to the contract, and +(d) the security of the contract (i.e., the adversary cannot influence the correctness of the above verification +performed by the contract), the contract can detect if a set of outputs of VOPR has been tampered with, +with a probability at least 1 − ϵ(λ). In the ideal model, Sim +JUS +A also can remove the blinding factors and it +knows the random polynomial ζ, unlike the adversary who does not know ζ when it sends the outputs of +VOPR to Sim +JUS +A . So, Sim +JUS +A can detect when A modifies a set of the outputs of VOPR that were sent to Sim +JUS +A +with a probability at least 1 − ϵ(λ), due to Theorem 7. Hence, the smart contract in the real model and the +simulator in the ideal model would abort with a similar probability. +Moreover, due to the security of ZSPA-A, the probability that an invalid key ki ∈ #»k is added to the list L +in the real world is similar to the probability that Sim +JUS +A detects an invalid key k′ +i ∈ #» +k′ in the ideal world. In +the real model, when Flag = False, the smart contract can identify each ill-structured output of VOPR (i.e., +ν(C)) with a probability of at least 1 − ϵ(λ) by checking whether ζ divides ι(C), due to (a) Theorem 6 (i.e., +27 + +unforgeable polynomial), (b) the fact that the smart contract is given ζ in plaintext, (c) no party (except +honest client D) knew anything about ζ before they send their input to the contract, and (d) the security +of the contract. In the ideal model, when Flag = False, given each ν(C′′), Sim +JUS +A can remove its blinding +factors from ν(C′′) which results in φ′(C′′) and then check if ζ divides φ′(C′′). The simulator can also detect +an ill-structured ν(C′′) with a probability of at least 1 − ϵ(λ), due to Theorem 6, the fact that the simulator +is given ζ in plaintext, and the adversary is not given any knowledge about ζ before it sends to the simulator +the outputs of VOPR. Hence, the smart contract in the real model and Sim +JUS +A in the ideal model would detect +an ill-structured input of an adversary with the same probability. +Now, we analyse the output of the predicates (QInit, QDel, QUF-A, QF-A) in the real and ideal models. In +the real model, all clients proceed to prepare their input set only if the predefined amount of coins has been +deposited by the parties; otherwise, they will be refunded and the protocol halts. In the ideal model also the +simulator proceeds to prepare its inputs only if a sufficient amount of deposit has been put in the contract; +otherwise, it would send message abort1 to TTP. Thus, in both models, the parties proceed to prepare their +inputs only if QInit(.) → 1. In the real model, if there is an abort after the parties ensure there is enough +deposit and before client D provides its encoded input to the contract, then all parties would be able to +retrieve their deposit in full; in this case, the aborting adversary would not be able to learn anything about +honest parties input sets, because the parties’ input sets are still blinded by random blinding polynomials +known only to client D. In the ideal model, if there is any abort during steps 7–14, then the simulator sends +abort2 to TTP and instructs the ledger to refund the coins that every party deposited. Also, in the case of an +abort (within the above two points of time), the auditor is not involved. Thus, in both models, in the case +of an abort within the above points of time, we would have QF-A(.) → 1. In the real model, if Flag = True, +then all parties would be able to learn the intersection and the smart contract refunds all parties, i.e., sends +each party ¨y + ¨ch amount which is the amount each party initially deposited. +In the ideal model, when Flag = True, then Sim +JUS +A can extract the intersection (by summing the output +of VOPR provided by all parties and removing the blinding polynomials) and sends back each party’s deposit, +i.e., ¨y + ¨ch amount. Hence, in both models in the case of Flag = True, when all of the parties receive the +result, we would have QDel(.) → 1. In the real model, when Flag = False, only the adversary which might +corrupt m′ clients would be able to learn the result; in this case, the contract sends (i) ¨ch amount to the +auditor, and (ii) m′·(¨y+ ¨ +ch)− ¨ +ch +m−m′ +amount as a compensation, to each honest party, in addition to each party’s +deposit ¨y + ¨ch. In the ideal model, when Flag = False, Sim +JUS +A sends abort3 to TTP and instructs the ledger +to distribute the same amount among the auditor (e.g., with address adrj) and every honest party (e.g., with +address adri) as the contract does in the real model. Thus, in both models when Flag = False, we would +have QUF-A(., ., ., ., adri) → (a = 1, .) and QUF-A(., ., ., ., adrj) → (., b = 1). +We conclude that the distributions of the joint outputs of the honest client C ∈ ˆP, client D, Aud, and +the adversary in the real and ideal models are computationally indistinguishable. +Case 2: Corrupt dealer D. In the real execution, the dealer’s view is defined as follows: +View +JUS +D +� +S +(D), (S +(1), ..., S +(m)) +� += +{S +(D), adrsc, m · (¨y + ¨ch), rD, View +CT +D, k, g, q, View +VOPR +D , ν +(A1), ..., ν +(Am), S∩} +where View +CT +D and View +VOPR +D +refer to the dealer’s real-model view during the execution of CT and VOPR respec- +tively. Also, rD is the outcome of internal random coins of client D and adrsc is the address of contract SCJUS. +The simulator Sim +JUS +D , which receives all parties’ input sets, works as follows. +1. receives from the subroutine adversary polynomials ζ, (γ(A1), δ(A1)), ..., (γ(Am), δ(Am)), (ω′(A1), ρ′(A1)), ..., +(ω′(Am), ρ′(Am)), where deg(γ(C)) = deg(δ(C)) = 3d + 1, deg(ω′(C)) = deg(ρ′(C)) = d, and deg(ζ) = 1, +where C ∈ {A1, ..., Am}. +2. generates an empty view. It appends to the view, the input set S(D). It constructs and deploys a smart +contract. Let adrsc be the contract’s address. It appends adrsc to the view. +3. appends to the view integer m · (¨y + ¨ch) and coins r′ +D chosen uniformly at random. +28 + +4. extracts the simulation of CT from CT’s simulator for client D. Let Sim +CT +D be the simulation. It appends +Sim +CT +D to the view. +5. picks a random key, k′, and derives pseudorandom values z′ +i,j from the key (the same way is done in +Figure 2). It constructs a Merkle tree on top of all values z′ +i,j. Let g′ be the root of the resulting tree. It +appends k′, g′, and q′ = H(k′) to the view. +6. invokes VOPR’s functionality twice and extracts the simulation of VOPR from VOPR’s simulator for client +D. Let Sim +VOPR +D +be the simulation. It appends Sim +VOPR +D +to the view. +7. given the parties’ input sets, computes a polynomial π that represents the intersection of the sets. +8. picks m random polynomials τ (A1), ..., τ (Am) of degree 3d + 1 such that their sum is 0. +9. picks m pairs of random polynomials (ω(A1), ρ(A1)), ..., (ω(Am), ρ(Am)), where each polynomial is of degree +d. Then, Sim +JUS +D for each client C ∈ {A1, ..., Am} computes polynomial ν(C) = ζ · π · (ω(C) · ω′(C) + ρ(C) · +ρ′(C)) + δ(C) + γ(C) + τ (C). +10. appends ν(A1), ..., ν(Am) and the intersection of the sets S′ +∩ to the view. +Next, we will show that the two views are computationally indistinguishable. D’s input S(D) is identi- +cal in both models; therefore, they have identical distributions. Also, the contract’s address has the same +distribution in both views, and so has the integer m · (¨y + ¨ch). Since the real-model semi-honest adversary +samples its randomness according to the protocol’s description, the random coins in both models (i.e., rD +and r′ +D) have identical distributions. Moreover, due to the security of CT, View +CT +D and Sim +CT +D have identical +distributions. Keys k and k′ have identical distributions, as both have been picked uniformly at random from +the same domain. In the real model, each element of the pair (g, p) is the output of a deterministic function +on a random value k. We know that k in the real model has identical distribution to k′ in the ideal model, +so do the evaluations of deterministic functions (i.e., Merkle tree, H, and PRF) on them. Therefore, each pair +(g, q) in the real model component-wise has an identical distribution to each pair (g′, q′) in the ideal model. +Furthermore, due to the security of the VOPR, View +VOPR +D +and Sim +VOPR +D +have identical distributions. +In the real model, each ν(C) has been blinded by a pseudorandom polynomial (i.e., derived from PRF’s +output) unknown to client D. In the ideal model, however, each ν(C) has been blinded by a random polynomial +unknown to client D. Due to the security of PRF, its outputs are computationally indistinguishable from +truly random values. Therefore, ν(C) in the real model and ν(C) in the ideal model are computationally +indistinguishable. Now we focus on the sum of all ν(C) in the real model and the sum of all ν(C) in the +ideal model, as adding them together would remove the above blinding polynomials that are unknown to +client D. Specifically, in the real model, after client D sums all ν(C) and removes the blinding factors and +ζ that it initially imposed, it would get a polynomial of the form ˆφ = +Am +� +C=A1 +ν(C)− +Am +� +C=A1 +(γ(D,C)+δ(D,C)) +ζ += +Am +� +C=A1 +(ω(D,C) · ω(C,D) · π(C)) + π(D) · +Am +� +C=A1 +(ρ(D,C) · ρ(C,D)), where ω(C,D) and ρ(C,D) are random polynomials +unknown to client D. In the ideal model, after summing all ν(C) and removing the random polynomials +that it already knows, it would get a polynomial of the following form: ˆφ′ = +Am +� +C=A1 +ν(C)− +Am +� +C=A1 +(γ(C)+δ(C)) +ζ += +π · +Am +� +C=A1 +(ω′(C) · ω(C)) + (ρ′(C) · ρ(C)). +As shown in Section 3.9, polynomial ˆφ has the form µ·gcd(π(D), π(A1), ..., π(Am)), where µ is a uniformly +random polynomial and gcd(π(D), π(A1), ..., π(Am)) represents the intersection of the input sets. Moreover, it +is evident that ˆφ′ has the form µ · π, where µ is a random polynomial and π represents the intersection. We +know that both gcd(π(D), π(A1), ..., π(Am)) and π represent the same intersection, also µ in the real model +and µ in the ideal model have identical distribution as they are uniformly random polynomials. Thus, two +polynomials ˆφ and ˆφ′ are indistinguishable. Also, the output S∩ is identical in both views. We conclude that +the two views are computationally indistinguishable. +Case 3: Corrupt auditor. In this case, by using the proof that we have already provided for Case 1 (i.e., +m − 1 client Ajs are corrupt), we can easily construct a simulator that generates a view computationally +29 + +distinguishable from the real-model semi-honest auditor. The reason is that, in the worst-case scenario +where m − 1 malicious client Ajs reveal their input sets and randomness to the auditor, the auditor’s view +would be similar to the view of these corrupt clients, which we have shown to be indistinguishable. The +only extra messages the auditor generates, that a corrupt client Aj would not see in plaintext, are random +blinding polynomials (ξ(A1), ..., ξ(Am)) generated during the execution of Audit(.) of ZSPA-A; however, these +polynomials are picked uniformly at random and independent of the parties’ input sets. Thus, if the smart +contract detects misbehaviour and invokes the auditor, even if m − 1 corrupt client Aj reveals their input +sets, then the auditor cannot learn anything about honest parties’ input sets. +Case 4: Corrupt public. In the real model, the view of the public (i.e., non-participants of the protocol) +is defined as below: +View +JUS +P ub +� +⊥, S +(D), (S +(A1), ..., S +(Am)) +� += +{⊥, adrsc, (m + 1) · (¨y + ¨ch), k, g, q, ν +(A1), ..., ν +(Am), ν +(D)} +Now, we describe how the simulator Sim +JUS +P ub works. +1. generates an empty view and appends to it an empty symbol, ⊥. It constructs and deploys a smart +contract. It appends the contract’s address, adrsc and integer (m + 1) · (¨y + ¨ch) to the view. +2. picks a random key, k′, and derives pseudorandom values z′ +i,j from the key, in the same way, done in +Figure 2. It constructs a Merkle tree on top of the z′ +i,j values. Let g′ be the root of the resulting tree. It +appends k′, g′, and q′ = H(k′) to the view. +3. for each client C ∈ {A1, ..., Am} and client D generates a random polynomial of degree 3d + 1 (for each +bin), i.e., ν(A1), ..., ν(Am), ν(D). +Next, we will show that the two views are computationally indistinguishable. In both views, ⊥ is identical. +Also, the contract’s addresses (i.e., adrsc) has the same distribution in both views, and so has the integer +(m + 1) · (¨y + ¨ch). Keys k and k′ have identical distributions as well, because both of them have been picked +uniformly at random from the same domain. In the real model, each element of pair (g, p) is the output of a +deterministic function on the random key k. We know that k in the real model has identical distribution to +k′ in the ideal model, and so do the evaluations of deterministic functions on them. Hence, each pair (g, q) +in the real model component-wise has an identical distribution to each pair (g′, q′) in the ideal model. In +the real model, each polynomial ν(C) is a blinded polynomial comprising of two uniformly random blinding +polynomials (i.e., γ(D,C) and δ(D,C)) unknown to the adversary. In the ideal model, each polynomial ν(C) +is a random polynomial; thus, polynomials ν(A1), ..., ν(Am) in the real model have identical distribution to +polynomials ν(A1), ..., ν(Am) in the ideal model. Similarly, polynomial ν(D) has been blinded in the real model; +its blinding factors are the additive inverse of the sum of the random polynomials γ(D,C) and δ(D,C) unknown +to the adversary. In the ideal model, polynomial ν(D) is a uniformly random polynomial; thus, ν(D) in the +real model and ν(D) in the ideal model have identical distributions. Moreover, in the real model even though +the sum φ of polynomials ν(A1), ..., ν(Am), ν(D) would remove some of the blinding random polynomials, it +is still a blinded polynomial with a pseudorandom blinding factor γ′ (derived from the output of PRF), +unknown to the adversary. In the ideal model, the sum of polynomials ν(A1), ..., ν(Am), ν(D) is also a random +polynomial. Thus, the sum of the above polynomials in the real model is computationally indistinguishable +from the sum of those polynomials in the ideal model. We conclude that the two views are computationally +indistinguishable. +2 +7 +Definition of Multi-party PSI with Fair Compensation and Reward +In this section, we upgrade PSI +FC to “multi-party PSI with Fair Compensation and Reward” (PSI +FCR), +which (in addition to offering the features of PSI +FC) allows honest clients who contribute their set to receive +a reward by a buyer who initiates the PSI computation and is interested in the result. +30 + +In PSI +FCR, there are (1) a set of clients {A1, ..., Am} a subset of which is potentially active adversaries +and may collude with each other, (2) a non-colluding dealer, D, potentially semi-honest, and (3) an auditor +Aud potentially semi-honest, where all clients (except Aud) have input set. Furthermore, in PSI +FCR there +are two “extractor” clients, say A1 and A2, where (A1, A2) ∈ {A1, ..., Am}. These extractor clients volunteer +to extract the (encoded) elements of the intersection and send them to a public bulletin board, i.e., a smart +contract. In return, they will be paid. We assume these two extractors act rationally only when they want +to carry out the paid task of extracting the intersection and reporting it to the smart contract, so they can +maximise their profit.5 For simplicity, we let client Am be the buyer, i.e., the party which initiates the PSI +computation and is interested in the result. +The formal definition of PSI +FCR is built upon the definition of PSI +FC (presented in Section 4); neverthe- +less, in PSI +FCR, we ensure that honest non-buyer clients receive a reward for participating in the protocol +and revealing a portion of their inputs deduced from the result. We: (i) upgrade the predicate QDel to QDel +R +to ensure that when honest clients receive the result, then an honest non-buyer client receives its deposit +back plus a reward and a buyer client receives its deposit back minus the paid reward, and (ii) upgrade the +predicate QUF-A to QUF-A +R +to ensure when an adversary aborts in an unfair manner (i.e., aborts but learns the +result) then an honest party receives its deposit back plus a predefined amount of compensation plus a re- +ward. The other two predicates (i.e., QInit and QF-A) remain unchanged. Given the above changes, we denote +the four predicates as ¯Q := (QInit, QDel +R , QUF-A +R +, QF-A). Below, we present the formal definition of predicates +QDel +R +and QUF-A +R +. +Definition 8 (QDel +R : Delivery-with-Reward predicate). Let G be a stable ledger, adrsc be smart contract +sc’s address, adri ∈ Adr be the address of an honest party, ¨x be a fixed amount of coins, and pram := +(G, adrsc, ¨x). Let R be a reward function that takes as input the computation result: res, a party’s address: +adri, a reward a party should receive for each unit of revealed information: ¨l, and input size: inSize. Then +R is defined as follows, if adri belongs to a non-buyer, then it returns the total amount that adri should be +rewarded and if adri belongs to a buyer client, then it returns the reward’s leftover that the buyer can collect, +i.e., R(res, adri, ¨l, inSize) → +¨ +rewi. Then, the delivery with reward predicate QDel +R (pram, adri, res, ¨l, inSize) +returns 1 if adri has sent ¨x amount to sc and received at least ¨x + +¨ +rewi amount from it. Else, it returns 0. +Definition 9 (QUF-A +R +: UnFair-Abort-with-Reward predicate). Let pram := (G, adrsc, ¨x) be the param- +eters defined above, and Adr′ ⊂ Adr be a set containing honest parties’ addresses, m′ = |Adr′|, and adri ∈ +Adr′. Let also G be a compensation function that takes as input three parameters ( ¨ +deps, adri, m′), where +¨ +deps +is the amount of coins that all m + 1 parties deposit, adri is an honest party’s address, and m′ = |Adr′|; +it returns the amount of compensation each honest party must receive, i.e., G( ¨ +deps, ardi, m′) → ¨xi. Let R +be the reward function defined above, i.e., R(res, adri, ¨l, inSize) → +¨ +rewi, and let +ˆ +pram := (res, ¨l, inSize). +Then, predicate QUF-A +R +is defined as QUF-A +R +(pram, +ˆ +pram, G, R, +¨ +deps, m′, adri) → (a, b), where a = 1 if adri is +an honest party’s address which has sent ¨x amount to sc and received ¨x + ¨xi + +¨ +rewi from it, and b = 1 if +adri is an auditor’s address which received ¨xi from sc. Otherwise, a = b = 0. +Next, we present the formal definition of multi-party PSI with Fair Compensation and Reward, PSI +FCR. +Definition 10 (PSI +FCR). Let f PSI be the multi-party PSI functionality defined in Section 4. We say protocol +Γ realises f PSI with ¯Q-fairness-and-reward in the presence of m−3 static active-adversary clients Ajs and two +rational clients Ais or a static passive dealer D or passive auditor Aud, if for every non-uniform probabilistic +polynomial time adversary A for the real model, there exists a non-uniform probabilistic polynomial-time +adversary (or simulator) Sim for the ideal model, such that for every I ∈ {A1, ..., Am, D, Aud}, it holds that: +{Ideal +W(fPSI, ¯ +Q) +Sim(z),I +(S1, ..., Sm+1)}S1,...,Sm+1,z +c≡ {Real +Γ +A(z),I(S1, ..., Sm+1)}S1,...,Sm+1,z +where z is an auxiliary input given to A and W(f PSI, ¯Q) is a functionality that wraps f PSI with predicates +¯Q := (QInit, QDel +R , QUF-A +R +, QF-A). +5 Thus, similar to any Ai in PSIFC, these extractors might be corrupted by an active adversary during the PSI +computation. +31 + +8 +Anesidora: A Concrete Construction of PSI +FCR +8.1 +Main Challenges to Overcome +Rewarding Clients Proportionate to the Intersection Cardinality. In PSIs, the main private infor- +mation about the clients which is revealed to a result recipient is the private set elements that the clients +have in common. Thus, honest clients must receive a reward proportionate to the intersection cardinality, +from a buyer. To receive the reward, the clients need to reach a consensus on the intersection cardinality. +The naive way to do that is to let every client find the intersection and declare it to the smart contract. +Under the assumption that the majority of clients are honest, then the smart contract can reward the honest +result recipient (from the buyer’s deposit). Nevertheless, the honest majority assumption is strong in the +context of multi-party PSI. Moreover, this approach requires all clients to extract the intersection, which +would increase the overall costs. Some clients may not even be interested in or available to do so. This task +could also be conducted by a single entity, such as the dealer; but this approach would introduce a single +point of failure and all clients have to depend on this entity. To address these challenges, we allow any two +clients to become extractors. Each of them finds and sends to the contract the (encrypted) elements in the +intersection. It is paid by the contract if the contract concludes that it is honest. This allows us to avoid (i) +the honest majority assumption, (ii) requiring all clients to find the intersection, and (iii) relying on a single +trusted/semi-honest party to complete the task. +Dealing with Extractors’ Collusion. Using two extractors itself introduces another challenge; namely, +they may collude with each other (and with the buyer) to provide a consistent but incorrect result, e.g., +both may declare that only s1 is in the intersection while the actual intersection contains 100 set elements, +including s1. This behaviour will not be detected by a verifier unless the verifier always conducts the delegated +task itself too, which would defeat the purpose of delegation. To efficiently address this issue, we use the +counter-collusion smart contracts (outlined in Section 3.4) which creates distrust between the two extractors +and incentivises them to act honestly. +8.2 +Description of Anesidora (ANE) +An Overview. To construct ANE, we mainly use JUS, deterministic encryption, “double-layered” commit- +ments, the hash-based padding technique (from Section 3.9), and the counter-collusion smart contracts. At +a high level, ANE works as follows. First, all clients run step 1 of JUS to agree on a set of parameters and +JUS’s smart contract. They deploy another smart contract, say SCANE. They also agree on a secret key, mk′. +Next, the buyer places a certain deposit into SCANE. This deposit will be distributed among honest clients as +a reward. The extractors and D deploy one of the counter-collusion smart contracts, i.e., SCPC. These three +parties deposit a certain amount on this contract. Each honest extractor will receive a portion of D’s deposit +for carrying out its task honestly and each dishonest extractor will lose a portion of its deposit for acting +maliciously. Then, each client encrypts its set elements (under mk′ using deterministic encryption) and then +represents the encrypted elements as a polynomial. The reason each client encrypts its set elements is to +ensure that the privacy of the plaintext elements in the intersection will be preserved from the public. +Next, the extractors commit to the encryption of their set elements and publish the commitments. All +clients (including D) take the rest of the steps in JUS using their input polynomials. This results in a blinded +polynomial, whose correctness is checked by JUS’s smart contact. +If JUS’s smart contact approves the result’s correctness, then all parties receive the money that they +deposited in JUS’s contract. In this case, each extractor finds the set elements in the intersection. Each +extractor proves to SCANE that the encryptions of the elements in the intersection are among the commitments +that the extractor previously published. If SCANE accepts both extractors’ proofs, then it pays each client +(except the buyer) a reward, where the reward is taken from the buyer’s deposit. The extractors receive their +deposits back and are paid for carrying out the task honestly. Nevertheless, if SCANE does not accept one of +the extractors’ proofs (or one extractor betrays the other), then it invokes the auditor in the counter-collusion +32 + +contracts to identify the misbehaving extractor. Then, SCANE pays each honest client (except the buyer) a +reward, taken from the misbehaving extractor. SCANE also refunds the buyer’s deposit. +If JUS’s smart contact does not approve the result’s correctness and Aud identified misbehaving clients, +then honest clients will receive (1) their deposit back from JUS’s contract, and (2) compensation and reward, +taken from misbehaving clients. Moreover, the buyer and extractors receive their deposit back from SCANE. +Figure 5 outlines the interaction between parties. +ACXicbZDLSsNAFIZP6q3GW9Wlm2BbcFWSgpdlURcuK9oLtKFMpN26GQmz +EwKJfQJxK0+hztx61P4GL6B0zYLbf1h4OP/z2E4fxAzqrTrflm5tfWNza38tr2zu7d/UDg8aiqRSEwaWDAh2wFShFOGpqRtqxJCgKGkFo5tZ3hoTqajgj3oSEz9CA05DipE21kPptQrFN2KO5ezCl4GRchU7xW+u32Bk4hwjRlSquO5sfZTJDXFjEzt8lIsJB0T7KeMcaymdjdRJEZ4hAakY5CjiCg +/nV8ydcrG6TuhkOZx7czd3xspipSaRIGZjJAequVsZv6XdRIdXvkp5XGiCceLj8KEOVo4s1qcPpUEazYxgLCk5hQHD5FEWJvybNORt9zIKjSrFe+icn5fLdaus7bycAKncAYeXEIN7qAODcAwgGd4gVfryXqz3q2PxWjOynaO4Y+szx+6yJo6D +CT +ACnicbZDLSgMxGIUz9VbHW9Wlm2BbcFVmCl6WRTcuK9gLtEPJpJk2NJchyRTK0DcQt/oc7sStL+Fj+AZm2lo64HAxzn/T/hPGDOqjed9OYWNza3tneKu7d/cHhUOj5pa5koTFpYMqm6IdKEUFahpGurEiIeMdMLJXZ3pkRpKsWjmcUk4GgkaEQxMplV4ZPKoFT2at5CcB38HMogV3NQ+u4PJU4EQYzpHXP92ITpEgZihmZu9WVWCo6JThIGRNYz91+okmM8ASNSM+iQJzoIF2cModV6wxhJV9wsCF+3sjRVzrGQ/tJEdmrFezPwv6yUmuglSKuLE 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+HnQiClKDJ85gd1I3VMQmWCFiXErBt2M4uWJXBYHO734We/pu5327ptmWhvgAXgItkEMnoNd8Bbsgz4g3ifvi/fV+Z/9r/75/7PRajvNTn3wQXzf/0B1HP+Jg=ˆx0 := (¯ei, qi) +, +, +, +Extractor +Extractor +AEmXicpVNdaxNBFJ0kq9bVaqOgD30ZzIakIbdgh8IxVZfiBUNG2hG8LsZJKMmdldZ26K6TLg3/QX+Dec3eTBpEkjeGXyz3njtzdm+UCq7B93+VyhXnzt17W/fdBw+3Hz3eqT4508lEUdahiUjURU +Q0EzxmHeAg2EWqGJGRYOfR+EOn18xpXkSf4VpyrqSDGM+4JSALfWq5Z+u64WRDNMRb+IwIipjpeFmiqewvwNU8EwN3v7OfGaAdnIDGk/AZzTh0RK0tjcoIHQsWJiFf7OHPqe67p1L5QERgDZJ2jbW/HB1DRHayRbw42iwaKoVTRNmsjVgq1vpoXDEYHsh2n8q3SWN2Db8PawebsDLYy/rzPHy6WOV6LSFOBtp/YWDuK59f8/yzrPC7CYZE2aTZLjFr5u7BV27C+Epcpxw+vt1Py2XwS+mQTzpIbmcdqrlDYT+hEshioIFpfBn4K3Ywo4FQw49aX4ETxK0a7mRAx1cYNJ5ql9uORIbu0aUwk092sWCWD67bSx4NE2ScGXFT/7siI1HoqI8vM/xq9jOXFVdjl +BAZvuhmP0wmwmM4GDSYCQ4LzvcR9rhgFMbUJsZ7aq2A6IopQsNu7MCWSxnoWLDt0Mzk7aAev2i8/H9SO3s/d20K76AVqogC9RkfoBJ2iDqLl35XtyrPKc2fXOXZOnI8zark073mKFsL58geMXYI9 +�(¯ei) � ⇣(¯ei) · �0(¯ei) +?= 0 +ADinicpVLbtNAFJ3YPIopJYUFSGxGxFESKUR2pJaXKspjwbJIpK0UW9F4MmGzNhm5roiWF7wmfwBW/6AseMFSVM2XMnW0T3nPnx8o1RwDZ73s2HZN27eur1zx7m7e2/vfnP/wal +OMkXZiCYiUecR0UzwmI2Ag2DnqWJERoKdRYv3JX92yZTmSfwZlikLJbmI+YxTAiY12W/8cBw3kATmALkRFl2ayEkeaKp4CvUbloJh3v9S9HEwJ5B/Kzq9QAOhC8XENu2b4sh3Hcdpu3lZgE3Bq6NuEBGVs2J7d9Mbf72G6rlq7dbWVlU5L+2dtcWcZ32/+8yv4atyGqSMWk1S76+HunV9nxbC2MVC467qTZ8gZeFfgq8GvQnWcTJq/g2lCM8lioIJoPfa9FMKcKOBUsMJpb9CJ4peMhrkQMdWFE2SapebfkQs2NjAmkukwr06pwG2TmeJZoswTA6yf1fkRGq9lJFRlkejN7kyuY0bZzB7EeY8TjNgMV0NmUCQ4L +Lu8RTrhgFsTSAGEvNp2A6J4pQMNfrGI/8TUeugtPhwD8cHwato7f1W7toCfoKeoiHz1Hx+gjOkEjRBu/rD3rkfXY3rWH9kv79UpqNeqah2gt7A9/AMptG/M= +ver(comi,j, ˆx0) +?= 1 +AEmXicpVPva9NAGL62UWd0uiroh305bEo76Eoy8AfCcNMvQxAmrtugKeVyvbZn75KYezPswoH/pn+B/4aXtIjt2lXwhYSX93ne5717kjeIBVfguj9L5Yp15+69rfv2g4fbjx7vVJ+ +cqyhNKOvQSETJZUAUEzxkHeAg2GWcMCIDwS6CyYcv7hieJReAbTmPUkGYV8yCkBU+pXyz9su+74gfTjMW9iPyBJxnQ/8xVNeAzN0wFw1zv7efEawZkI9OngwhwTh8RKUljc4MCQicJE6vwd/rQdWzbdnxJYAyQfTprm1vx4VQ3x2sUW6ONml6uWf8jahR1k0ZytWDrq25hf0wg+64b/yqd5Q3YNLw9bN5uQAvjb+u8cXKp45Wo1AV426mdhYM4dv3/z7LO8wIsJhmTZpPkpIWvG3uFHfsLYahy0nD6OzW37RaBbybePKmheZz2qyXkDyKaShYCFUSprufG0MtIApwKpu36Ehwl/IrRXiZESJW2/VSx2Hw8MmJdk4Z +EMtXLilXSuG4qAzyMEvOEgIvq3x0ZkUpNZWCY+V+jlrG8uArpjB808t4GKfAQjobNEwFhgjne4kHPGEUxNQkxHhqroLpmCSEgtnehSmB1MYzb9mhm8n5Qdt71X75+aB29H7u3hbaRS9QE3noNTpCJ+gUdRAt/6psV5Vnlu71rF1Yn2cUculec9TtBDWl9/0vId +MT.verify(hi, g) +?= 1 +Fig. 5: Outline of the interactions between parties in Anesidora +Detailed Description of ANE. Next, we describe the protocol in more detail (Table 1 summarises the main +notations used). +1. All clients in CL = {A1, ..., Am, D} together run step 1 of JUS (in Section 6.2) to deploy JUS’s contract +SCJUS and agree on a master key, mk. +2. All clients in CL deploy a new smart contract, SCANE. The address of SCANE is given to all clients. +3. The buyer, client Am, before time t1 deposits Smin · ¨v amount to SCANE. +4. All clients after time t2 > t1 ensure that the buyer has deposited Smin · ¨v amount on SCANE. Otherwise, +they abort. +5. D signs SCPC with the extractors. SCANE transfers Smin · ¨r amount (from the buyer deposit) to SCPC for +each extractor. This is the maximum amount to be paid to an honest extractor for honestly declaring +the elements of the intersection. Each extractor deposits ¨d′ = ¨d + Smin · ¨f amount in SCPC at time t3. +At time t4 all clients ensure that the extractors deposited enough coins; otherwise, they withdraw their +deposit and abort. +6. D encrypts mk under the public key of the dispute resolver (in SCPC); let ctmk be the resulting ciphertext. +It also generates a commitment of mk as follows: z′ = PRF(mk, 0), commk = Com(mk, z′). It stores ctmk +and commk in SCANE. +7. All clients in CL engage in CT to agree on another key, mk′. +8. Each client in CL maps the elements of its set S : {s1, ..., sc} to random values by encrypting them as: +∀i, 1 ≤ i ≤ c : ei = PRP(mk′, si). Then, it encodes its encrypted set element as ¯ei = ei||H(ei). After +33 + +that, it constructs a hash table HT and inserts the encoded elements into the table. ∀i : H(¯ei) = j, then +¯ei → HTj. It pads every bin with random dummy elements to d elements (if needed). Then, for every +bin, it builds a polynomial whose roots are the bin’s content: π(I) = +d� +i=1 +(x − e′ +i), where e′ +i is either ¯ei, or +a dummy value. +9. Every extractor in {A1, A2}: +(a) for each j-th bin, commits to the bin’s elements: comi,j = Com(e′ +i, qi), where qi is a fresh randomness +used for the commitment and e′ +i is either ¯ei, or a dummy value of the bin. +(b) constructs a Merkel tree on top of all committed values: MT.genTree(com1,1, ..., comd,h) → g. +(c) stores the Merkel tree’s root g on SCANE. +10. All clients in CL run steps 3–11 of JUS, where each client now deposits (in the SCJUS) ¨y′ amount where +¨y′ > Smin · ¨v + ¨ch. Recall, at the end of step 11 of JUS for each j-th bin (i) a random polynomial ζ has +been registered in SCJUS, (ii) a polynomial φ (blinded by a random polynomial γ′) has been extracted +by SCJUS, and (iii) SCJUS has checked this polynomial’s correctness. If the latter check: +• passes (i.e., Flag = True): all parties run step 12 of JUS (with a minor difference, see Section 8.3). +In this case, each party receives ¨y′ amount it deposited in SCJUS. They proceed to step 11 below. +• fails (i.e., Flag = False): all parties run step 13 of JUS. In this case, (as in JUS) Aud is paid ¨ch +amount, and each honest party receives back its deposit, i.e., ¨y′ amount. Also, from the misbehaving +parties’ deposit m′·¨y′− ¨ +ch +m−m′ +amount is sent to each honest client, to reward and compensate the client +Smin · ¨l and m′·¨y′− ¨ +ch +m−m′ +− Smin · ¨l amounts respectively, where m′ is the total number of misbehaving +parties. Moreover, SCANE returns to the buyer its deposit (i.e., Smin · ¨v amount paid to SCANE), and +returns to each extractor its deposit, i.e., ¨d′ amount paid to SCPC. Then, the protocol halts. +11. Every extractor client: +(a) finds the elements in the intersection. To do so, it first encodes each of its set elements to get ¯ei, as +explained in step 8. Then, it determines to which bin the encrypted value belongs, i.e., j = H(¯ei). +Next, it evaluates the resulting polynomial (for that bin) at the encrypted element. It considers the +element in the intersection if the evaluation is zero, i.e., φ(¯ei) − ζ(¯ei) · γ′(¯ei) = 0. If the extractor is +a traitor, by this point it should have signed SCTC with D and provided all the inputs (e.g., correct +result) to SCTC. +(b) proves that every element in the intersection is among the elements it has committed to. Specifically, +for each element in the intersection, say ¯ei, it sends to SCANE: +• commitment comi,j (generated in step 9a, for ¯ei) and its opening ˆx′ := (¯ei, qi). +• proof hi that asserts comi,j is a leaf node of a Merkel tree with root g. +(c) sends the opening of commitment commk, i.e., pair ˆx := (mk, z′), to SCANE. This is done only once +for all elements in the intersection. +12. Contract SCANE: +(a) verifies the opening of the commitment for mk, i.e., Ver(commk, ˆx) = 1. If the verification passes, +it generates the index of the bin to which ¯ei belongs, i.e., j = H(¯ei). It uses mk to derive the +pseudorandom polynomial γ′ for j-th bin. +(b) checks whether (i) the opening of commitment is valid, (ii) the Merkle tree proof is valid, and (iii) +the encrypted element is the resulting polynomial’s root. Specifically, it ensures that the following +relation holds: +� +Ver(comi,j, ˆx′) = 1 +� +∧ +� +MT.verify(hi, g) = 1 +� +∧ +� +φ(¯ei) − ζ(¯ei) · γ′(¯ei) = 0 +� +13. The parties are paid as follows. +• if all proofs of both extractors are valid, both extractors provided identical elements of the intersec- +tions (for each bin), and there is no traitor, then SCANE: +(a) takes |S∩| · m · ¨l amount from the buyer’s deposit (in SCANE) and distributes it among all clients, +except the buyer. +34 + +(b) calls SCPC which returns the extractors’ deposit (i.e., ¨d′ amount each) and pays each extractor +|S∩| · ¨r amount, for doing their job correctly. +(c) checks if |S∩| < Smin. If the check passes, then it returns (Smin − |S∩|) · ¨v amount to the buyer. +• if both extractors failed to deliver any result, then SCANE: +(a) refunds the buyer, by sending Smin · ¨v amount (deposited in SCANE) back to the buyer. +(b) retrieves each extractor’s deposit (i.e., ¨d amount) from the SCPC and distributes it among the +rest of the clients (except the buyer and extractors). +• Otherwise (e.g., if some proofs are invalid, if an extractor’s result is inconsistent with the other +extractor’s result, or there is a traitor), SCANE invokes (steps 8.c and 9 of) SCPC and its auditor to +identify the misbehaving extractor, with the help of ctmk after decrypting it. Then, SCANE asks SCPC +to pay the auditor the total amount of ¨ch taken from the deposit of the extractor(s) who provided +incorrect result to SCANE. Moreover, +(a) if both extractors cheated: +i. if there is no traitor, then SCANE refunds the buyer, by sending Smin · ¨v amount (deposited in +SCANE) back to the buyer. It also distributes 2 · ¨d′ − ¨ch amount (taken from the extractors’ +deposit in SCPC) among the rest of clients (except the buyer and extractors). +ii. if there is a traitor, then: +A. if the traitor delivered a correct result in SCTC, SCANE retrieves ¨d′ − ¨d amount from the +other dishonest extractor’s deposit (in SCPC) and distributes it among the rest of the clients +(except the buyer and dishonest extractor). Also, it asks SCPC to send |S∩| · ¨r + ¨d′ + ¨d − ¨ch +amount to the traitor (via SCTC). SCTC refunds the traitor’s deposit, i.e., ¨ch amount. It +refunds the buyer, by sending Smin · ¨v − |S∩| · ¨r amount (deposited in SCANE) back to it. +B. if the traitor delivered an incorrect result in SCTC, SCANE pays the buyer and rest of clients +in the same way it does in step 13(a)i. SCTC refunds the traitor, i.e., ¨ch amount. +(b) if one of the extractors cheated: +i. if there is no traitor, then SCANE calls SCPC that (a) returns the honest extractor’s deposit +(i.e., ¨d′ amount), (b) pays this extractor |S∩| · ¨r amount, for doing its job honestly, and (c) +pays this extractor ¨d − ¨ch amount taken from the dishonest extractor’s deposit. SCANE pays +the buyer and the rest of the clients in the same way it does in step 13(a)iiA. +ii. if there is a traitor +A. if the traitor delivered a correct result in SCTC (but it cheated in SCANE), then SCANE calls +SCPC that (a) returns the other honest extractor’s deposit (i.e., ¨d′ amount), (b) pays the +honest extractor |S∩| · ¨r amount taken from the buyer’s deposit, for doing its job honestly, +(c) pays the honest extractor ¨d− ¨ch amount taken from the traitor’s deposit, (d) pays to the +traitor |S∩| · ¨r amount taken from the buyer’s deposit (via the SCTC), and (e) refunds the +traitor ¨d′ − ¨d amount taken from its own deposit. SCTC refunds the traitor’s deposit (i.e., ¨ch +amount). SCANE takes |S∩|·m·¨l amount from the buyer’s deposit (in SCANE) and distributes +it among all clients, except the buyer. If |S∩| < Smin, then SCANE returns (Smin − |S∩|) · ¨v +amount (deposited in SCANE) back to the buyer. +B. if the traitor delivered an incorrect result in SCTC (and it cheated in SCANE), then SCANE pays +the honest extractor in the same way it does in step 13(b)iiA. SCTC refunds the traitor’s +deposit, i.e., ¨ch amount. Also, SCANE pays the buyer and the rest of the clients in the same +way it does in step 13(a)iiA. +Theorem 9. If PRP, PRF, the commitment scheme, smart contracts, the Merkle tree scheme, JUS and the +counter-collusion contracts are secure and the public key encryption is semantically secure, then ANE realises +f PSI with ¯Q-fairness-and-reward (w.r.t. Definition 10) in the presence of m−3 static active-adversary clients +Ajs and two rational clients Ais or a static passive dealer D or passive auditor Aud, or passive public which +sees the intersection cardinality. +Before we prove Theorem 9 in Section 8.4, we present several remarks on the ANE. +35 + +8.3 +Further Discussion on Anesidora +There is a simpler but costlier approach to finding the intersection without involving the extractors; that +is the smart contract finds the (encoded) elements of the intersection and distributes the parties’ deposit +according to the number of elements it finds. This approach is simpler, as we do not need the involvement +of (i) the extractors and (ii) the three counter collusion contracts. Nevertheless, it is costlier, because the +contract itself needs to factorise the unblinded resulting polynomial and find the roots, which would cost it +O(d2) for each bin, where d is the size of each bin. Our proposed approach however moves such a computation +off-chain, leading to a lower monetary computation cost. +The reason each client uses the hash-based padding to encode each encrypted element ei as ¯ei = ei||H(ei) +is to allow the auditor in the counter collusion contracts to find the error-free intersection, without having +to access to one of the original (encrypted) sets. +Compared to JUS, there is a minor difference in finding the result in ANE. Specifically, because in ANE +each set element si is encoded as (i) ei = PRP(mk′, si) and then (ii) ¯ei = ei||H(ei) by a client, then when the +client wants to find the intersection it needs to first regenerate ¯ei as above and then treat it as a set element +to check if φ′(¯ei) = 0, in step 12(b)iii of JUS. +In ANE, each extractor uses double-layered commitments (i.e., it first commits to the encryption of each +element and then constructs a Merkle tree on top of all commitments) for efficiency and privacy purposes. +Constructing a Merkle tree on top of the commitments allows the extractor to store only a single value in +SCANE would impose a much lower storage cost compared to the case where it would store all commitments +in SCANE. Also, committing to the elements’ encryption allows it to hide from other clients the encryption +of those elements that are not in the intersection. Recall that encrypting each element is not sufficient to +protect one client’s elements from the rest of the clients, as they all know the decryption key. +To increase their reward, malicious clients may be tempted to insert “garbage” elements into their sets +with the hope that those garbage elements appear in the result and accordingly they receive a higher reward. +However, they would not succeed as long as there exists a semi-honest client (e.g., dealer D) which uses actual +set elements. In this case, by the set intersection definition, those garbage elements will not appear in the +intersection. +In ANE, for the sake of simplicity, we let each party receive a fixed reward, i.e., ¨l, for every element it +contributes to the intersection. However, it is possible to make the process more flexible/generic. For instance, +we could define a Reward Function RF that takes ¨l, an (encoded) set element ei in the intersection, its +distribution/value valei, and output a reward rewei that each party should receive for contributing that +element to the intersection, i.e., RF(¨l, ei, valei) → rewei. +8.4 +Proof of ANE +In this section, we prove Theorem 9, i.e., the security of ANE. +Proof. We prove the theorem by considering the case where each party is corrupt, at a time. +Case 1: Corrupt extractors {A1, A2} and m − 3 clients in {A3, ..., Am}. Let set G include extractors +{A1, A2} and a set of at most m − 3 corrupt clients in {A3, ..., Am}. Let set ˆG be a set of honest clients +in {A3, ..., Am}. Also, let Sim +ANE +A be the simulator. We let the simulator interact with (i) active adversary +A′ that may corrupt m − 3 clients in {A3, ..., Am}, and (ii) two rational adversaries A′′ := (A1, A2) that +corrupt extractors (A1, A2) component-wise. In the simulation, before the point where the extractors are +invoked to provide proofs and results, the simulator directly deals with active adversary A′. However, when +the extractors are involved (to generate proofs and extract the result) we require the simulator to interact +with each rational adversary A1 and A2. We allow these two subroutine adversaries (A′, A′′) to internally +interact with each other. Now, we explain how Sim +ANE +A , which receives the input sets of honest dealer D and +honest client(s) in ˆG, works. +1. constructs and deploys two smart contracts (for JUS and ANE). It sends the contracts’ addresses to A′. +It also simulates CT and receives the output value, mk, from its functionality, fCT. +36 + +2. deposits Smin · ¨v amount to SCANE if buyer Am is honest, i.e., Am ∈ ˆG. Otherwise, Sim +ANE +A checks if A′ has +deposited Smin ·¨v amount in SCANE. If the check fails, it instructs the ledger to refund the coins that every +party deposited and sends message abort1 to TTP (and accordingly to all parties); it outputs whatever +A′ outputs and then halts. +3. constructs and deploys a (Prisoner’s) contract and transfers ¨w = Smin · ¨r amount for each extractor. +Sim +ANE +A ensures that each extractor deposited ¨d′ = ¨d+Smin · ¨f coins in this contract; otherwise, it instructs +the ledger to refund the coins that every party deposited and sends message abort1 to TTP; it outputs +whatever A′ outputs and then halts. +4. encrypts mk under the public key of the dispute resolver; let ctmk be the resulting ciphertext. It also +generates a commitment of mk as follows: commk = Com(mk, PRF(mk, 0)). It stores ctmk and commk in +SCANE. +5. simulates CT again and receives the output value mk′ from fCT. +6. receives from A′ a Merkle tree’s root g′ for each extractor. +7. simulates the steps of 3–11 in JUS. For completeness, we include the steps that the simulator takes in +this proof. Specifically, Sim +ANE +A : +(a) simulates ZSPA-A for each bin and receives the output value (k, g, q) from f ZSPA-A. +(b) deposits in the contract the amount of ¨y′ = Smin · ¨v + ¨ch for client D and each honest client in ˆG. It +sends to A′ the amount deposited in the contract. +(c) checks if A′ has deposited ¨y′ ·|G| amount (in addition to ¨d′ amount deposited in step 3 above). If the +check fails, it instructs the ledger to refund the coins that every party deposited and sends message +abort1 to TTP (and accordingly to all parties); it outputs whatever A′ outputs and then halts. +(d) picks a random polynomial ζ of degree 1, for each bin. Sim +ANE +A , for each client C ∈ {A1, ..., Am} +allocates to each bin two degree d random polynomials: (ω(D,C), ρ(D,C)), and two degree 3d + 1 +random polynomials: (γ(D,C), δ(D,C)). Also, Sim +ANE +A for each honest client C′ ∈ ˆG, for each bin, picks +two degree d random polynomials: (ω(C′,D), ρ(C′,D)). +(e) simulates VOPR using inputs ζ ·ω(D,C) and γ(D,C) for each bin. It receives the inputs of clients C′′ ∈ G, +i.e., ω(C′′,D) · π(C′′), from its functionality f VOPR, for each bin. +(f) extracts the roots of polynomial ω(C′′,D) · π(C′′) for each bin and appends those roots that are in the +sets universe to a new set S(C′′). +(g) simulates again VOPR using inputs ζ · ρ(D,C) · π(D) and δ(D,C), for each bin. +(h) sends to TTP the input sets of all parties; specifically, (i) client D’s input set: S(D), (ii) honest clients’ +input sets: S(C′) for all C′ in ˆG, and (iii) A′’s input sets: S(C′′), for all C′′ in G. For each bin, it +receives the intersection set, S∩, from TTP. +(i) represents the intersection set for each bin as a polynomial, π, as follows. First, it encrypts each +element si of S∩ as ei = PRP(mk′, si). Second, it encodes each encrypted element as ¯ei = ei||H(ei). +Third, it constructs π as π = +|S∩| +� +i=1 +(x − si) · +d−|S∩| +� +j=1 +(x − uj), where uj is a dummy value. +(j) constructs polynomials θ(C′) +1 += ζ·ω(D,C′)·ω(C′,D)·π+γ(D,C′), θ(C′) +2 += ζ·ρ(D,C′)·ρ(C′,D)·π+δ(D,C′), and +ν(C′) = θ(C′) +1 ++ θ(C′) +2 ++ τ (C′), for each bin and each honest client C′ ∈ ˆG, such that τ (C) = +3d+2 +� +i=0 +zi,c · xi +and each value zi,c is derived from key k generated in step 7a. It sends to A′ polynomial ν(C′) for +each bin and each honest client C′ ∈ ˆG. +(k) receives ν(C′′) from A′, for each bin and each corrupt client C′′ ∈ G. It checks whether the output +for every C′′ has been provided. Otherwise, it halts. +(l) if there is any abort within steps 7e–7k, then it sends abort2 to TTP and instructs the ledger to +refund the coins that every party deposited. It outputs whatever A′ outputs and then halts. +(m) constructs polynomial ν(D) = ζ · ω′(D) · π − +Am +� +C=A1 +(γ(D,C) + δ(D,C)) + ζ · γ′ for each bin on behalf of +client D, where ω′(D) is a fresh random polynomial of degree d and γ′ is a pseudorandom polynomial +derived from mk. +(n) sends to A′ polynomials ν(D) and ζ for each bin. +37 + +(o) computes polynomial φ′ as φ′ = +� +∀C′′∈G +ν(C′′)− � +∀C′′∈G +(γ(D,C′′)+δ(D,C′′)), for every bin. Next, it checks if +ζ divides φ′, for every bin. If the check passes, it sets Flag = True. Otherwise, it sets Flag = False. +(p) if Flag = True, then instructs the ledger to send back each party’s deposit, i.e., ¨y′ amount. It sends +a message deliver to TTP. It proceeds to step 8 below. +(q) if Flag = False: +i. receives |G| keys of the PRF from A′, i.e., #»k ′ = [k′ +1, ..., k′ +|G|], for every bin. +ii. checks if k′ +j = k, for every k′ +j ∈ #»k ′. Recall, k was generated in step 3. It constructs an empty list +L′ and appends to it the indices (e.g., j) of the keys that do not pass the above check. +iii. receives from f ZSPA-A the output containing a vector of random polynomials, #»µ ′, for each valid key. +iv. sends to A′, L′ and #»µ ′, for every bin. +v. for each bin of client C whose index is not in L′ computes polynomial χ(D,C) as χ(D,C) = ζ · +η(D,C) − (γ(D,C) + δ(D,C)), where η(D,C) is a fresh random polynomial of degree 3d + 1. Note that +C includes both honest and corrupt clients, except those clients whose index is in L′. Sim +ANE +A sends +every polynomial χ(D,C) to A′. +vi. given each ν(C′′) (by A′ in step 7k), computes polynomial φ′(C′′) as follows: φ′(C′′) = ν(C′′) − +γ(D,C′′) − δ(D,C′′), for every bin. Sim +ANE +A checks if ζ divides φ′(C′′), for every bin. It appends the +index of those clients that did not pass the above check to a new list, L′′. +vii. if L′ or L′′ is not empty, then instructs the ledger: (a) to refund ¨y′ amount to each client whose +index is not in L′ and L′′, (b) to retrieve ¨ch amount from the adversary (i.e., one of the parties +whose index is in one of the lists) and send the ¨ch amount to Aud, and (c) to reward and +compensate each honest party (whose index is not in the two lists) +m′·¨y′− ¨ +ch +m−m′ +amount, where +m′ = |L′| + |L′′|. Then, it sends message abort3 to TTP. +viii. outputs whatever A′ outputs and halts. +8. for each I ∈ {1, 2}, receives from AI (1) a set E(I) of encoded encrypted elements, e.g., ¯ei, in the +intersection, (2) each ¯ei’s commitment comi,j, (3) each comi,j’s opening ˆx′, (4) a proof hi that each +comi,j is a leaf node of a Merkle tree with root g′ (given to simulator in step 6 above), and (5) the +opening ˆx of commitment commk. +9. encrypts each element si of S∩ as ei = PRP(mk′, si). Then, it encodes each encrypted element as ¯ei = +ei||H(ei). Let set S′ include all encoded encrypted elements in the intersection. +10. sings a SCTC with AI, if AI decides to be a traitor extractor. In this case, AI, provides the intersection +to SCTC. Sim +ANE +A checks this intersection’s validity. Shortly (in step 16b), we will explain how Sim +ANE +A acts +based on the outcome of this check. +11. checks if each set E(I) equals set S′. +12. checks if comi,j matches the opening ˆx′ and the opening corresponds to a unique element in S′. +13. verifies each commitment’s proof, hi. Specifically, given the proof and root g, it ensures the commitment +comi,j is a leaf node of a Merkle tree with a root node g′. It also checks whether the opening ˆx matches +commk. +14. if all the checks in steps 11–13 pass, then instructs the ledger (i) to take |S∩| · m · ¨l amount from the +buyer’s deposit and distributes it among all clients, except the buyer, (ii) to return the extractors deposit +(i.e., ¨d′ amount each) and pay each extractor |S∩| · ¨r amount, and (iii) to return (Smin − |S∩|) · ¨v amount +to the buyer. +15. if neither extractor sends the extractor’s set intersection (E(A1), E(A2)) in step 8, then instructs the ledger +(i) to refund the buyer, by sending Smin · ¨v amount back to the buyer and (ii) to retrieve each extractor’s +deposit (i.e., ¨d′ amount) from SCPC and distribute it among the rest of the clients (except the buyer and +extractors). +16. if the checks in step 14 fail or in step 15 both A1 and A2 send the extractors’ set intersection but they +are inconsistent with each other, then it tags the extractor whose proof or set intersection was invalid as +a misbehaving extractor. Sim +ANE +A instructs the ledger to pay the auditor (of SCPC) the total amount of ¨ch +coins taken from the misbehaving extractor(s) deposit. Furthermore, Sim +ANE +A takes the following steps. +(a) if both extractors cheated and there is no traitor, then instructs the ledger (i) to refund the buyer +Smin · ¨v amount, and (ii) to take 2 ¨d′ − ¨ch amount from the misbehaving extractors’ deposit and +distribute it to the rest of the clients except the buyer and extractors. +38 + +(b) if both extractors cheated, there is a traitor, and the traitor delivered a correct result (in step 10), +then instructs the ledger (i) to take ¨d′ − ¨d amount from the other misbehaving extractor’s deposit +and distribute it among the rest of the clients (except the buyer and dishonest extractor), (ii) to +distribute |S∩| · ¨r + ¨d′ + ¨d − ¨ch amount to the traitor, (iii) to refund the traitor ¨ch amount, and (iv) +to refund the buyer Smin · ¨v − |S∩| · ¨r amount. +(c) if both extractors cheated, there is a traitor, and the traitor delivered an incorrect result (in step +10), then instructs the ledger to distribute coins the same way it does in step 16a. +(d) if one of the extractors cheated and there is no traitor, then instructs the ledger (i) to return the +honest extractor’s deposit (i.e., ¨d′ amount), (ii) to pay the honest extractor |S∩| · ¨r amount, (iii) to +pay this extractor ¨d − ¨ch amount taken from the dishonest extractor’s deposit, and (iv) to pay the +buyer and the rest of the clients the same way it does in step 16b. +(e) if one of the extractors cheated, there is a traitor, and the traitor delivered a correct result (in step +10), then instructs the ledger (i) to return the other honest extractor’s deposit (i.e., ¨d′ amount), (ii) +to pay the honest extractor |S∩| · ¨r amount, taken from the buyer’s deposit, (iii) to pay the honest +extractor ¨d − ¨ch amount, taken from the traitor’s deposit, (iv) to pay to the traitor |S∩| · ¨r amount, +taken from the buyer’s deposit, (v) to refund the traitor ¨d′ − ¨d amount, (vi) to refund the traitor ¨ch +amount, (vii) to take |S∩| · m · ¨l amount from the buyer’s deposit and distribute it among all clients, +except the buyer, and (viii) to return (Smin − |S∩|) · ¨v amount back to the buyer. +(f) if one of the extractors cheated, there is a traitor, and the traitor delivered an incorrect result (in step +10), then instructs the ledger (i) to pay the honest extractor the same way it does in step 13(b)iiA, +(ii) to refund the traitor ¨ch amount, and (iii) to pay the buyer and the rest of the clients in the same +way it does in step 16b. +(g) outputs whatever A outputs and then halts. +Next, we show that the real and ideal models are computationally indistinguishable. We first focus on +the adversary’s output. The addresses of the smart contracts have identical distribution in both models. In +the real and ideal models, the adversary sees the transcripts of ideal calls to fCT as well as the functionality +outputs (mk, mk′). Due to the security of CT (as we are in the fCT-hybrid world), the transcripts of fCT in +both models have identical distribution, so have the random outputs of fCT, i.e., (mk, mk′). Also, the deposit +amounts Smin · ¨v and ¨w have identical distributions in both models. Due to the semantical security of the +public key encryption, the ciphertext ctmk in the real model is computationally indistinguishable from the +ciphertext ctmk in the ideal model. Due to the hiding property of the commitment scheme, commitment +commk in the real model is computationally indistinguishable from commitment commk in the ideal model. +Moreover, due to the security of JUS, all transcripts and outputs produced in the ideal model, in step 7 above, +have identical distribution to the corresponding transcripts and outputs produced in JUS in the real model. +The address of SCTC has the same distribution in both models. The amounts each party receives in the real +and ideal models are the same, except when both extractors produce an identical and incorrect result (i.e., +intersection) in the real model, as we will shortly discuss, this would not occur under the assumption that +the extractors are rational and due to the security of the counter collusion smart contracts. +Now, we show that an honest party aborts with the same probability in the real and ideal models. As +before, for the sake of completeness, we include the JUS in the following discussion as well. Due to the +security of CT, an honest party, during CT invocation, aborts with the same probability in both models; in +this case, the adversary learns nothing about the parties’ input set and the sets’ intersection as the parties +have not sent out any encoded input set yet. In both models, an honest party can read the smart contract +and check if sufficient amounts of coins have been deposited. Thus, it would halt with the same probability +in both models. If the parties halt because of insufficient amounts of deposit, no one could learn about (i) the +parties’ input set and (ii) the sets’ intersection because the inputs (representation) have not been dispatched +at this point. Due to the security of ZSPA-A, an honest party during ZSPA-A execution aborts with the same +probability in both models. In this case, an aborting adversary also learns nothing about the parties’ input +set and the sets’ intersection. +Due to the security of VOPR, honest parties abort with the same probability in both models. In the case +where a party aborts during the execution of VOPR, the adversary would learn nothing (i) about its counter +39 + +party’s input set, and (ii) about the rest of the honest parties’ input sets and the intersection as the other +parties’ input sets remain blinded by random blinding factors known only to client D. In the real model, +client D can check if all parties provided their encoded inputs via reading the state of the smart contract. +The simulator can perform the same check to ensure A′ has provided the encoded inputs of all corrupt +parties. So, in both models, an honest party with the same probability detects if not all encoded inputs have +been provided. In this case, if an adversary aborts and does not provide its encoded inputs (i.e., polynomials +ν(C′′)), then it learns nothing about the honest parties’ input sets and the intersection, for the same reason +explained above. +In the real model, the contract sums every client C’s polynomial ν(C) with each other and with client +D’s polynomial ν(D), that ultimately removes the blinding factors that D initially inserted (during the VOPR +execution), and then checks if the result is divisible by ζ. Due to (a) Theorem 7, (b) the fact that the smart +contract is given the random polynomial ζ in plaintext, (c) no party (except honest D) knew polynomial ζ +before they send their input to the contract, and (d) the security of the contract (i.e., the adversary cannot +influence the correctness of the smart contract’s verifications), the contract can detect if a set of outputs of +VOPR were tampered with, with a probability at least 1 − ϵ(λ). In the ideal model, Sim +ANE +A (in step 7o) can +remove the blinding factors and it knows the random polynomial ζ. So, Sim +ANE +A can detect when A′ tampers +with a set of the outputs of VOPR (sent to Sim +ANE +A ) with a probability at least 1 − ϵ(λ), due to Theorem 7. +Therefore, the smart contract in the real model and the simulator in the ideal model would abort with a +similar probability. +Due to the security of ZSPA-A, the probability that in the real model an invalid ki ∈ #»k is appended to +L is similar to the probability that Sim +ANE +A detects an invalid k′ +i ∈ #» +k′ in the ideal model. In the real model, +when Flag = False, the smart contract can identify each ill-structured output of VOPR (i.e., ν(C)) with a +probability of at least 1 − ϵ(λ) by checking whether ζ divides ι(C), due to (a) Theorem 6 (i.e., unforgeable +polynomial), (b) the fact that the smart contract is given ζ in plaintext, (c) no party (except honest client D) +knew anything about ζ before they send their input to the contract, and (d) the security of the contract. In +the ideal model, when Flag = False, given each ν(C′′), Sim +ANE +A can remove its blinding factors from ν(C′′) which +results in φ′(C′′) and then can check if ζ divides φ′(C′′), in step 7(q)vi. Sim +ANE +A can detect an ill-structured ν(C′′) +with a probability of at least 1 − ϵ(λ), due to Theorem 6, the fact that the simulator is given ζ in plaintext, +and the adversary is not given any knowledge about ζ before it sends to the simulator the outputs of VOPR. +Therefore, the smart contract in the real model and Sim +ANE +A in the ideal model can detect an ill-structured +input of an adversary with the same probability. The smart contract in the real model and Sim +ANE +A in the ideal +model can detect and abort with the same probability if the adversary provides an invalid opening to each +commitment comi,j and commk, due to the binding property of the commitment scheme. Also, the smart +contract in the real model and Sim +ANE +A in the ideal model, can abort with the same probability if a Merkle tree +proof is invalid, due to the security of the Merkle tree, i.e., due to the collision resistance of Merkle tree’s +hash function. +Note that in the ideal model, Sim +ANE +A can detect and abort with a probability of 1, if AI does not send to +the simulator all encoded encrypted elements of the intersection, i.e., when E(I) ̸= S′. Because the simulator +already knows all elements in the intersection (and the encryption key). Thus, it can detect with a probability +of 1 if both the intersection sets that the extractors provide are identical but incorrect. In the real world, if +the extractors collude with each other and provide identical but incorrect intersections, then an honest client +(or the smart contract) cannot detect it. Thus, the adversary can distinguish the two models, based on the +probability of aborting. However, under the assumption that the smart contracts (of Dong et al. [18]) are +secure (i.e., are counter-collusion), and the extractors are rational, such an event (i.e., providing identical +but incorrect result without one extractor betraying the other) would not occur in either model, as the +real model and (A1, A2) rational adversaries follow the strategy that leads to a higher payoff. Specifically, +as shown in [18], providing incorrect but identical results is not the preferred strategy of the extractors; +instead, the betrayal of one extractor by the other is the most profitable strategy in the case of (enforceable) +collusion between the two extractors. This also implies that the amounts that the extractors would receive +in both models are identical. +40 + +Now, we analyse the output of the predicates ¯Q := (QInit, QDel +R , QUF-A +R +, QF-A) in the real and ideal models. +In the real model, all clients proceed to prepare their input set only if the predefined amount of coins have +been deposited by the parties; otherwise (if in steps 2,4,5 of ANE and step 4 of JUS there is not enough +deposit), they will be refunded and the protocol halts. In the ideal model, the simulator proceeds to prepare +its inputs only if enough deposit has been placed in the contract. Otherwise, it would send message abort1 to +TTP, during steps 2–7c. Thus, in both models, the parties proceed to prepare their inputs only if QInit(.) → 1. +In the real model, if there is an abort after the parties ensure there is enough deposit and before client D +provides its encoded input to the contract, then all parties can retrieve their deposit in full. In this case, the +aborting adversary cannot learn anything about honest parties’ input sets, as the parties’ input sets have +been blinded by random blinding polynomials known only to client D. In the ideal model, if there is any +abort during steps 7e–7k, then the simulator sends abort2 to TTP and instructs the ledger to refund every +party’s deposit. In the case of an abort, within the above two steps, the auditor is not involved, and paid. +Therefore, in both models, in the case of an abort within the above steps, we would have QF-A(.) → 1. +In the real model, if Flag = True, then all parties can locally extract the intersection, regardless of the +extractors’ behaviour. In this case, each honest party receives ¨y′ amount that it initially deposited in SCJUS. +Moreover, each honest party receives at least |S∩|·¨l amount as a reward, for contributing to the result. In this +case, the honest buyer always collects the leftover of its deposit. Specifically, if both extractors act honestly, +and the intersection cardinality is smaller than |Smin|, then the buyer collects its deposit leftover, after +paying all honest parties. If any extractor misbehaves, then the honest buyer fully recovers its deposit (and +the misbehaving extractor pays the rest). Even in the case that an extractor misbehaves and then becomes a +traitor to correct its past misbehaviour, the buyer collects its deposit leftover if the intersection cardinality +is smaller than |Smin|. In the ideal model, when Flag = True, then Sim +ANE +A can extract the intersection by +summing the output of VOPR provided by all parties and removing the blinding polynomials. In this case, it +sends back each party’s deposit placed in SCJUS, i.e., ¨y′ amount. Also, in this case, each honest party receives +at least |S∩| · ¨l amount as a reward and the honest buyer always collects the leftover of its deposit. Thus, in +both models in the case of Flag = True, we would have QDel +R (.) → 1. +In the real model, when Flag = False, only the adversary can learn the result. In this case, the contract +sends (i) ¨ch amount to Aud, and (ii) m′·¨y′− ¨ +ch +m−m′ +amount, as compensation and reward, to each honest party, in +addition to each party’s initial deposit. In the ideal model, when Flag = False, Sim +ANE +A sends abort3 to TTP +and instructs the ledger to distribute the same amount the contract distributes among the auditor (e.g., with +address adrj) and every honest party (e.g., with address adri) in the real model. Thus, in both models when +Flag = False, we would have QUF-A +R +(., ., ., ., adri) → (a = 1, .) and QUF-A +R +(., ., ., ., adrj) → (., b = 1). +We conclude that the distribution of the joint outputs of the honest client C ∈ ˆG, client D, Aud, and the +adversary in the real and ideal models are computationally indistinguishable. +Case 2: Corrupt dealer D. In the real execution, the dealer’s view is defined as follows: +View +ANE +D +� +S +(D), (S +(1), ..., S +(m)) +� += +{S +(D), adrsc, Smin · ¨v, 2 · ¨d′, rD, View +JUS +D , View +CT +D, (com1,j, ¯e1, q1, h1)..., (comsz,j′, ¯esz, qsz, hsz), g, ˆx := (mk, z′), S∩} +where View +CT +D and View +VOPR +D +refer to D’s real-model view during the execution of CT and VOPR respectively. Also, +rD is the outcome of internal random coins of D, adrsc is the address of contract, SCANE, (j, ..., j′) ∈ {1, ..., h}, +z′ = PRF( ¯ +mk, 0), sz = |S∩|, and hi is a Merkle tree proof asserting that comi,j is a leaf node of a Merkle tree +with root node g. The simulator Sim +ANE +D , which receives all parties’ input sets, works as follows. +1. generates an empty view. It appends to the view, the input set S(D). It constructs and deploys a smart +contract. It appends the contract’s address, adrsc, to the view. +2. appends to the view integer Smin · ¨v, and 2 · ¨d′. Also, it appends uniformly random coins r′ +D to the view. +3. extracts the simulation of JUS from JUS’s simulator for client D. Let Sim +JUS +D be the simulation, that also +includes a random key mk. It appends Sim +JUS +D to the view. +4. extracts the simulation of CT from CT’s simulator, yielding the simulation Sim +CT +D that includes its output +mk′. It appends Sim +CT +D to the view. +41 + +5. encrypts each element si,j in the intersection set S∩ as ei,j = PRP(mk′, si,j) and then encodes the result +as ¯ei,j = ei,j||H(ei,j). It commits to each encoded value as comi,j = Com(¯ei,j, qi,j), where j is the index of +the bin to which ¯ei,j belongs and qi,j is a random value. +6. It constructs (com′ +1,1, ..., com′ +d,h) where each com′ +i,j is a value picked uniformly at random from the +commitment scheme output range. For every j-th bin, it sets each com′ +i′,j to unique comi,j if value comi,j +for j-th bin has been generated in step 5. Otherwise, the original values of com′ +i′,j remains unchanged. +7. constructs a Merkle tree on top of the values (com′ +1,1, ..., com′ +d,h) generated in step 6. Let g be the resulting +tree’s root. +8. for each element si,j in the intersection, it constructs (com′ +i′,j, ¯ei,j, qi,j, hi,j), where com′ +i′,j is the commit- +ment of ¯ei,j, com′ +i′,j ∈ com, (¯ei,j qi,j) is the commitment’s opening generated in step 5, and hi,j is a Merkle +tree proof asserting that com′ +i′,j is a leaf of a Merkle tree with root g. It appends all (com′ +i′,j, ¯ei,j, qi,j, hi,j) +along with g to the view. +9. generates a commitment to mk as commk = Com(mk, z′) where z′ = PRF(mk, 0). It appends (mk, z′) +along with S∩ to the view. +Now, we will discuss why the two views are computationally indistinguishable. D’s input S(D) is identical +in both models, so they have identical distributions. The contract’s address has the same distribution in +both models. The same holds for the integers Smin · ¨v and 2 · ¨d′. Also, because the real-model semi-honest +adversary samples its randomness according to the protocol’s description, the random coins in both models +(i.e., rD and r′ +D) have identical distribution. Due to the security of JUS, View +JUS +D and Sim +JUS +D have identical +distribution, so do View +CT +D and Sim +CT +D due to the security of CT. The intersection elements in both models +have identical distributions and the encryption scheme is schematically secure. Therefore, each intersection +element’s representation (i.e., ¯ei in the real model and ¯ei,j in the deal model) are computationally indistin- +guishable. Each qi in the real model and qi,j in the ideal model have identical distributions as they have been +picked uniformly at random. Each commitment comi,j in the real model is computationally indistinguishable +from commitment com′ +i′,j in the ideal model. +In the real model, each Merkle tree proof hi contains two leaf nodes (along with intermediary nodes that +are the hash values of a subset of leaf nodes) that are themselves the commitment values. Also, for each +hi, only one of the leaf node’s openings (that contains an element in the intersection) is seen by D. The +same holds in the ideal model, with the difference that for each Merkle tree proof hi,j the leaf node whose +opening is not provided is a random value, instead of an actual commitment. However, due to the hiding +property of the commitment scheme, in the real and ideal models, these two leaf nodes (whose openings +are not provided) and accordingly the two proofs are computationally indistinguishable. In both models, +values mk and z′ are random values, so they have identical distributions. Furthermore, the intersection S∩ +is identical in both models. Thus, we conclude that the two views are computationally indistinguishable. +Case 3: Corrupt auditor. In this case, the real-model view of the auditor is defined as +View +ANE +Aud +� +⊥, S +(D), (S +(1), ..., S +(m)) +� += +{View +JUS +Aud, adrsc, Smin · ¨v, 2 · ¨d′, (com1,j, ¯e1, q1, h1)..., (comsz,j′, ¯esz, qsz, hsz), g, ˆx := (mk, z′)} +Due to the security of JUS, Aud’s view View +JUS +Aud during the execution of JUS can be easily simulated. As +we have shown in Case 2, for the remaining transcript, its real-model view can be simulated too. However, +there is a difference between Case 3 and Case 2; namely, in the former case, Aud does not have the PRP’s +key mk′ used to encrypt each set element. However, due to the security of PRP, it cannot distinguish each +encrypted encoded element from a uniformly random element and cannot distinguish PRP(mk′, .) from a +uniform permutation. Therefore, each value ¯ej in the real model has identical distribution to each value ¯ei,j +(as defined in Case 2) in the ideal model, as both are the outputs of PRP. +Case 4: Corrupt public. In the real model, the view of the public (i.e., non-participants of the protocol) +is defined as below: +42 + +View +ANE +P ub +� +⊥, S +(D), (S +(A1), ..., S +(Am)) +� += +{View +JUS +P ub, adrsc, Smin · ¨v, 2 · ¨d′, (com1,j, ¯e1, q1, h1)..., (comsz,j′, ¯esz, qsz, hsz), g, ˆx := (mk, z′)} +Due to the security of JUS, the public’s view View +JUS +P ub during JUS’s execution can be simulated in the +same way which is done in Case 4, in Section 6.3. The rest of the public’s view overlaps with Aud’s view in +Case 3, excluding View +JUS +Aud. Therefore, we can use the argument provided in Case 3 to show that the rest of +the public’s view can be simulated. We conclude that the public’s real and ideal views are computationally +indistinguishable. +2 +9 +Evaluation +In this section, we analyse the asymptotic costs of ANE. We also compare its costs and features with the fastest +two and multiple parties PSIs in [2,34,38,41]) and with the fair PSIs in [15,17]. Tables 2 and 3 summarise +the result of the cost analysis and the comparison respectively. +Table 2: Asymptotic costs of different parties in ANE. In the table, h is the total number of bins, d is a bin’s capacity +(i.e., d = 100), m is the total number of clients (excluding D), |S| is a set cardinality, and ¯ξ is OLE’s security parameter. +Party +Computation Cost +Communication Cost +Client A3, ..., Am +O +� +h · d(m + d) + |S|( d2+d +2 +) +� +O +� +h · d2 · ¯ξ +� +Dealer D +O +� +h · m(d2 + d) + |S|( d2+d +2 +) +� +O +� +h · d2 · ¯ξ · m +� +Auditor Aud +O +� +h · m · d +� +O +� +h · d +� +Extractor A1, A2 +O +� +h · d(m + d) + |S|( d2+d +2 +) +� +O +� +|S∩| · log2 |S| +� +Smart contract SCANE & SCJUS O +� +|S∩|(d + log2 |S|) + h · m · d +� +— +Overal Complexity +O +� +h · d2 · m +� +O +� +h · d2 · ¯ξ · m +� +Table 3: Comparison of the asymptotic complexities and features of state-of-the-art PSIs. In the table, t is a parameter +that determines the maximum number of colluding parties, κ is a security parameter, and c is a set cardinality. +Schemes +Asymptotic Cost +Features +Computation +Communication Fairness +Rewarding +Sym-key based Multi-party Active Adversary +[2] +O(h · d2 · m) +O(h · d · m) +× +× +✓ +✓ +× +[15] +O(c) +O(c) +✓ +× +× +× +✓ +[17] +O(c2) +O(c) +✓ +× +× +× +✓ +[34] +O(c · m2 + c · m) +O(c · m2) +× +× +✓ +✓ +× +[38] +O(c · κ(m + t2 − t(m + 1))) +O(c · m · κ) +× +× +✓ +✓ +✓ +[41] +O(c) +O(c · κ) +× +× +✓ +× +✓ +Ours: ANE +O(h · d2 · m) +O(h · d2 · ¯ξ · m) +✓ +✓ +✓ +✓ +✓ +43 + +9.1 +Computation Cost +In step 1, each client’s cost is O(m) and mainly involves an invocation of CT. In steps 2–5, the clients’ +cost is negligible as it involves deploying smart contracts and reading from the deployed contracts. Step 6 +involves only D whose cost in this step is constant, as it involves invoking a public key encryption, PRF, and +commitment only once. In step 7, the clients’ cost is O(m), as they need to invoke an instance of CT. In +step 8, each client invokes PRP and H linear with its set’s cardinality. In the same step, it also constructs h +polynomials, where the construction of each polynomial involves d modular multiplications and additions. +Thus, its complexity in this step is O(h · d). It has been shown in [2] that O(h · d) = O(c) and d = 100 +for all set sizes. In step 9, each extractor invokes the commitment scheme linear with the number of its set +cardinality |S| and constructs a Merkle tree on top of the commitments. Therefore, its complexity is O(|S|). +In step 10, each client A1, ..., Am: (i) invokes an instance of ZSPA-A which involves O(h · m) invocation +of CT, 3h · m(d + 1) invocation of PRF, 3h · m(d + 1) addition, and O(h · m · d) invocation of H (in step 3 of +subroutine JUS), (ii) invokes 2h instances of VOPR, where each VOPR invocation involves 2d(1+d) invocations +of OLE+, multiplications, and additions (in steps 6 and 7 of JUS), and (iii) performs h(3d+2) modular addition +(in step 8 of JUS). The dealer D: (a) invokes 2h · m instances of VOPR (in steps 6 and 7 of JUS), (b) invokes +PRF h(3d + 1) times (in step 9 of JUS), and (c) performs h(d2 + 1) multiplications and 3h · m · d additions (in +step 10 of JUS). In the same step, the subroutine smart contract SCJUS performs h · m(3d + 1) additions and +h polynomial divisions, where each division includes dividing a polynomial of degree 3d + 1 by a polynomial +of degree 1 (in step 11 of JUS). Moreover, if Flag = True, then each client invokes PRF h(3d + 1) times, and +performs h(3d+1) additions, and performs polynomial evaluations linear with its set cardinality, where each +evaluation involves O(d) additions and O( d2+d +2 +) multiplications (in step 12 of JUS). If Flag = False, then +(a) Aud invokes PRF 3h · m(d + 1) times and invokes H O(h · m · d) times, and (b) D performs O(h · m · d) +multiplications and additions (in step 13 of JUS). +In step 11, each extractor invokes H linear with its set cardinality |S|; it also performs polynomial eval- +uations linear with |S|. In step 12a, SCANE invokes the commitment’s verification algorithm Ver once, H at +most |S∩| times, and PRF |S∩|(3d + 1) times. In step 12b, SCANE invokes Ver at most |S∩| times, and calls +H O(|S∩| · log2 |S|) times. In the same step, it performs polynomial evaluation |S∩| times. Thus, its overall +complexity is O(|S∩|(d + log2 |S|)). +9.2 +Communication Cost +In steps 1 and 7, the communication cost of the clients is dominated by the cost of CT which is O(m). In +steps 2–6, the clients’ cost is negligible, as it involves sending a few transactions to the smart contracts, e.g., +SCJUS, SCANE, and SCPC. Step 9 involves only extractors whose cost is O(h) as each of them only sends to +SCANE a single value for each bin. In step 10, the clients’ cost is dominated by VOPR’s cost; specifically, each +pair of client and D invokes VOPR O(d2) times for each bin; therefore, the cost of each client (excluding D) is +O(h · d2 · ¯ξ) while the cost of D is O(h · d2 · ¯ξ · m), where ¯ξ is the subroutine OLE’s security parameter. Step 11 +involves only the extractors, where each extractor’s cost is dominated by the size of the Merkle tree’s proof +it sends to SCANE, i.e., O(|S∩| · log2 |S|), where |S| is the extractor’s set cardinality. In step 13, Aud sends h +polynomials of degree 3d+1 to SCJUS; thus, its complexity is O(h·d). The rest of the steps impose negligible +communication costs. +9.3 +Comparison +Below we show that ANE offers various features that the state-of-the-art PSIs do not offer simultaneously +while keeping its overall overheads similar to the efficient PSIs. +Computation Complexity. The computation complexity of ANE is similar to that of PSI in [2], but is +better than the multiparty PSI’s complexity in [34] as the latter’s complexity is quadratic with the number +of parties. Also, ANE’s complexity is better than the complexity of the PSI in [38] that is quadratic with +44 + +parameter t, i.e., the total number of parties that may collude. Similar to the two-party PSIs in [15,41], ANE’s +complexity is linear with c. The two-party PSI in [17] imposes a higher computation overhead than ANE +does, as its complexity is quadratic with sets’ cardinality. Hence, the complexity of ANE is: (i) linear with the +set cardinality, similar to the above schemes except the one in [17] and (ii) linear with the total number of +parties, similar to the above multi-party schemes, except the one in [34]. Hence, the computation complexity +of ANE is linear with the set cardinality and the number of parties, similar to the above schemes except for +the ones in [34,17] whose complexities are quadratic with the set cardinality or the number of parties. +Communication Complexity. ANE’s communication complexity is slightly higher than the complexity of +the PSI in [2], by a factor of d · ¯ξ. However, it is better than the PSI’s complexity in [34] as the latter has +a complexity quadratic with the number of parties. ANE’s complexity is slightly higher than the one in [38], +by a factor of d. Similar to the two-party PSIs in [15,41,17], ANE’s complexity is linear with c. Therefore, +the communication complexity of ANE is linear with the set cardinality and number of parties, similar to the +above schemes except the one in [34] whose complexity is quadratic with the number of parties. +Features. ANE is the only scheme that offers all the five features, i.e., supports fairness, rewards participants, +is based on symmetric key primitives, supports multi-party, and is secure against active adversaries. After +ANE is the scheme in [38] which offers three of the above features. The rest of the schemes support only two +of the above features. +10 +Conclusion and Future Direction +Private Set Intersection (PSI) is a crucial protocol with numerous real-world applications. In this work, +we proposed, Justitia, the first multi-party fair PSI that ensures that either all parties get the result or if +the protocol aborts in an unfair manner, then honest parties will receive financial compensation. We then +upgraded it to Anesidora, the first PSI ensuring that honest parties who contribute their private sets receive +a reward proportionate to the number of elements they reveal. Since an MPC that rewards participants for +contributing their private inputs would help increase its real-world adoption, an interesting open question is: +How can we generalise the idea of rewarding participants to MPC? +Acknowledgements +Aydin Abadi was supported in part by REPHRAIN: The National Research Centre on Privacy, Harm +Reduction and Adversarial Influence Online, under UKRI grant: EP/V011189/1. Steven J. Murdoch was +supported by REPHRAIN and The Royal Society under grant UF160505. We would like to thank Yvo +Desmedt for his suggestion of having a flexible reward mechanism. +References +1. Abadi, A., Terzis, S., Metere, R., Dong, C.: Efficient delegated private set intersection on outsourced private +datasets. IEEE TDSC (2018) +2. 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(Upper Tail in Chernoff Bounds) Let Xi be a random variable defined as Xi = +c� +i=1 +Yi, +where Pr[Yi = 1] = pi, Pr[Yi = 0] = 1 − pi, and all Yi are independent. Let the expectation be µ = E[Xi] = +h� +i=1 +pi, then Pr[Xi > d = (1 + σ) · µ] < +� +eσ +(1+σ)(1+σ) +�µ +, ∀σ > 0 +In this model, the expectation is µ = c +h, where c is the number of balls and h is the number of bins. The +above inequality provides the probability that bin i gets more than (1 + σ) · µ balls. Since there are h bins, +the probability that at least one of them is overloaded is bounded by the union bound: +Pr[∃i, Xi > d] ≤ +h +� +i=1 +Pr[Xi > d] = h · +� +eσ +(1 + σ)(1+σ) +� c +h +(4) +Thus, for a hash table of length h = O(c), there is always an almost constant expected number of +elements, d, mapped to the same bin with a high probability [39], e.g., 1 − 2−40. +B +Enhanced OLE’s Ideal Functionality and Protocol +The PSIs proposed in [23] use an enhanced version of the OLE. The enhanced OLE ensures that the receiver +cannot learn anything about the sender’s inputs, in the case where it sets its input to 0, i.e., c = 0. The +enhanced OLE’s protocol (denoted by OLE+) is presented in Figure 6. +47 + +1. Receiver (input c ∈ F): Pick a random value, r +$← F, and send (inputS, (c−1, r)) to +the first FOLE. +2. Sender (input a, b ∈ F): Pick a random value, u +$← F, and send (inputR, u) to the +first FOLE, to learn t = c−1 · u + r. Send (inputS, (t + a, b − u)) to the second FOLE. +3. Receiver: Send (inputR, c) to the second FOLE and obtain k = (t + a) · c + (b − u) = +a · c + b + r · c. Output s = k − r · c. +Fig. 6: Enhanced Oblivious Linear function Evaluation (OLE+) [23]. +C +Counter Collusion Contracts +In this section, we present Prisoner’s Contract (SCPC), Colluder’s Contract (SCCC), Traitor’s Contract (SCTC) +originally proposed by Dong et al. [18]. As we previously stated, we have slightly adjusted the contracts. We +will highlight the adjustments in blue. For the sake of completeness, below we restate the parameters used +in these contracts and their relation. +• ¨b: the bribe paid by the ringleader of the collusion to the other server in the collusion agreement, in SCCC. +• ¨c: a server’s cost for computing the task. +• ¨ch: the fee paid to invoke an auditor for recomputing a task and resolving disputes. +• ¨d: the deposit a server needs to pay to be eligible for getting the job. +• ¨t: the deposit the colluding parties need to pay in the collusion agreement, in SCCC. +• ¨w: the amount that a server receives for completing the task. +• ¨w ≥ ¨c: the server would not accept underpaid jobs. +• ¨ch > 2 ¨w: If it does not hold, then there would be no need to use the servers and the auditor would do +the computation. +• (pk, sk): an asymmetric-key encryption’s public-private key pair belonging to the auditor. +The following relations need to hold when setting the contracts in order for the desirable equilibria to hold: +(i) ¨d > ¨c + ¨ch, (ii) ¨b < ¨c, and (iii) ¨t < ¨w − ¨c + 2 ¨d − ¨ch − ¨b. +C.1 +Prisoner’s Contract (SCPC) +SCPC has been designed for outsourcing a certain computation. It is signed by a client who delegates the +computation and two other parties (or servers) who perform the computation. This contract tries to incen- +tivize correct computation by using the following idea. It requires each server to pay a deposit before the +computation is delegated. If a server behaves honestly, then it can withdraw its deposit. However, if a server +cheats (and is detected), its deposit is transferred to the client. When one of the servers is honest and the +other one cheats, the honest server receives a reward. This reward is taken from the deposit of the cheating +server. Hence, the goal of SCPC is to create a Prisoner’s dilemma between the two servers in the following +sense. Although the servers may collude with each other (to cut costs and provide identical but incorrect +computation results) which leads to a higher payoff than both behaving honestly, there is an even higher +payoff if one of the servers manages to persuade the other server to collude and provide an incorrect result +while itself provides a correct result. In this setting, each server knows that collusion is not stable as its +counterparty will always try to deviate from the collusion to increase its payoff. If a server tries to convince +its counterparty (without a credible and enforceable promise), then the latter party will consider it as a trap; +consequently, collusion will not occur. Below, we restate the contract. +1. The contract is signed by three parties; namely, client D and two other parties E1 and E2. A third-party +auditor will resolve any dispute between D and the servers. The address of another contract, called SCANE, +is hardcoded in this contract. +2. The servers agree to run computation f on input x, both of which have been provided by D. +48 + +3. The parties agree on three deadlines (T1, T2, T3), where Ti+1 > Ti. +4. D agrees to pay ¨w to each server for the correct and on-time computation. Therefore, D deposits 2 · ¨w +amount in the contract. This deposit is transferred from SCANE to this contract. +5. Each server deposits ¨d′ = ¨d + ¨X amount in the contract. +6. The servers must pay the deposit before T1. If a server fails to meet the deadline, then the contract would +refund the parties’ deposit (if any) and terminates. +7. The servers must deliver the computation’s result before T2. +8. The following steps are taken when (i) both servers provided the computation’s result or (ii) deadline T2 +elapsed. +(a) if both servers failed to deliver the computation’s result, then the contract transfers their deposits +to SCANE. +(b) if both servers delivered the result, and the results are equal, then (after verifying the results) this +contract must pay the agreed amount ¨w and refund the deposit ¨d′ to each server. +(c) otherwise, D raises a dispute with the auditor. +9. When a dispute is raised, the auditor (which is independent of Aud in JUS) re-generates the computation’s +result, by using algorithm resComp(.) described shortly in Appendix C.1. Let yt, y1, and y2 be the result +computed by the auditor, E1, and E2 respectively. The auditor uses the following role to identify the +cheating party. +• if Ei failed to deliver the result (i.e., yi is null), then it has cheated. +• if a result yi has been delivered before the deadline and yi ̸= yt, then Ei has cheated. +The auditor sends its verdict to SCPC. +10. Given the auditor’s decision, the dispute is settled according to the following rules. +• if none of the servers cheated, then this contract transfers to each server (i) ¨w amount for performing +the computation and (ii) its deposit, i.e., ¨d′ amount. The client also pays the auditor ¨ch amount. +• if both servers cheated, then this contract (i) pays the auditor the total amount of ¨ch, taken from +the servers’ deposit, and (ii) transfers to SCANE the rest of the deposit, i.e., 2 · ¨d′ − ¨ch amount. +• if one of the servers cheated, then this contract (i) pays the auditor the total amount of ¨ch, taken +from the misbehaving server’s deposit, (ii) transfers the honest server’s deposit (i.e., ¨d′ amount) back +to this server, (iii) transfers to the honest server ¨w + ¨d − ¨ch amount (which covers its computation +cost and the reward), and (iv) transfers to SCANE the rest of the misbehaving server’s deposit, i.e., ¨X +amount. The cheating server receives nothing. +11. After deadline T3, if D has neither paid nor raised a dispute, then this contract pays ¨w to each server +which delivered a result before deadline T2 and refunds each server its deposit, i.e., ¨d′ amount. Any +deposit left after that will be transferred to SCANE. +Now, we explain why we have made the above changes to the original SCPC of Dong et al. [18]. In the +original SCPC (a) the client does not deposit any amount in this contract; instead, it directly sends its coins to +a server (and auditor) according to the auditor’s decision, (b) the computation correctness is determined only +within this contract (with the involvement of the auditor if required), and (c) the auditor simply re-generates +the computation’s result given the computation’s plaintext inputs. Nevertheless, in ANE, (1) all clients need +to deposit a certain amount in SCANE and only the contracts must transfer the parties’ deposit, (2) SCANE +also needs to verify a part of the computation’s correctness without the involvement of the auditor and +accordingly distribute the parties deposit based on the verification’s outcome, and (3) the auditor must be +able to re-generate the computation’s result without being able to learn the computation’s plaintext input, +i.e., elements of the set. Therefore, we have included the address of SCANE in SCPC to let the parties’ deposit +move between the two contracts (if necessary) and allowed SCPC to distribute the parties’ deposit; thus, the +requirements in points (1) and (2) are met. To meet the requirement in point (3) above, we have included +a new algorithm, called resComp(.), in SCPC. Shortly, we will provide more detail about this algorithm. +Moreover, to make this contract compatible with ANE, we increased the amount of each server’s deposit by +¨X. Nevertheless, this adjustment does not change the logic behind SCPC’s design and its analysis. +49 + +Auditor’s Result-Computation Algorithm. In this work, we use SCPC to delegate the computation +of intersection cardinality to two extractor clients, a.k.a. servers in the original SCPC. In this setting, the +contract’s auditor is invoked when an inconsistency is detected in step 13 of ANE. For the auditor to recompute +the intersection cardinality, we have designed resComp(.) algorithm. The auditor uses this algorithm for every +bin’s index indx, where 1 ≤ indx ≤ h and h is the hash table’s length. We present this algorithm in Figure +7. The auditor collects the inputs of this algorithm as follows: (a) reads random polynomial ζ, and blinded +polynomial φ from contract SCJUS, (b) reads the ciphertext if secret key mk from SCANE, and (c) fetches +public parameters (desH, h) from the hash table’s public description. +Note that in the original SCPC of Dong et al. [18], the auditor is assumed to be fully trusted. However, in +this work, we have relaxed such an assumption. We have designed ANE and resComp(.) in such a way that +even a semi-honest auditor cannot learn anything about the actual elements of the sets, as they have been +encrypted under a key unknown to the auditor. +resComp(ζ, φ, sk, ctmk, indx, desH) → R +– Input. ζ: a random polynomial of degree 1, φ: a blinded polynomial of the form ζ·(ϵ+γ′) +where ϵ and γ′ are arbitrary and pseudorandom polynomials respectively, deg(φ)−1 = +deg(γ′), sk: the auditor’s secret key, ctmk: ciphertext of mk which is a key of PRF, indx: +an input of PRF, and desH: a description of hash function H. +– Output. R: a set containing valid roots of unblinded φ. +1. decrypts the ciphertext ctmk under key sk. Let mk be the result. +2. unblinds polynomial φ, as follows: +(a) re-generates pseudorandom polynomial γ′ using key mk. Specifically, it uses mk to +derive a key: k = PRF(mk, indx). Then, it uses the derived key to generate 3d + 1 +pseudorandom coefficients, i.e., ∀j, 0 ≤ j ≤ deg(φ) − 1 : gj = PRF(k, j). Next, it +uses these coefficients to construct polynomial γ′, i.e., γ′ = +deg(φ)−1 +� +j=0 +gj · xj. +(b) removes the blinding factor from φ. Specifically, it computes polynomial φ′ of the +following form φ′ = φ − ζ · γ′. +3. extracts roots of polynomial φ′. +4. finds valid roots, by (i) parsing each root ¯e as (e1, e2) with the assistance of desH and +(ii) checking if e2 = H(e1). It considers a root valid, if this equation holds. +5. returns set R containing all valid roots. +Fig. 7: Auditor’s result computation, resComp(.), algorithm +C.2 +Colluder’s Contract (SCCC) +Recall that SCPC aimed at creating a dilemma between the two servers. However, this dilemma can be +addressed if they can make an enforceable promise. This enforceable promise can be another smart contract, +called Colluder’s Contract (SCCC). This contract imposes additional rules that ultimately would affect the +parties’ payoffs and would make collusion the most profitable strategy for the colluding parties. In SCCC, the +party who initiates the collusion would pay its counterparty a certain amount (or bribe) if both follow the +collusion and provide an incorrect result of the computation to SCPC. Note, SCCC requires both servers to +send a fixed amount of deposit when signing the contract. The party who deviates from collusion will be +punished by losing the deposit. Below, we restate SCCC. +50 + +1. The contract is signed between the server who initiates the collusion, called ringleader (LDR) and the +other server called follower (FLR). +2. The two agree on providing to SCPC a different result res′ than a correct computation of f on x would +yield, i.e., res′ ̸= f(x). Parameter res′ is recorded in this contract. +3. LDR and FLR deposit ¨t + ¨b and ¨t amounts in this contract respectively. +4. The above deposit must be paid before the result delivery deadline in SCPC, i.e., before deadline T2. If +this condition is not met, the parties’ deposit in this contract is refunded and this contract is terminated. +5. When SCPC is finalised (i.e., all the results have been provided), the following steps are taken. +(a) both follow the collusion: if both LDR and FLR provided res′ to SCPC, then ¨t and ¨t +¨b amounts are +delivered to LDR and FLR respectively. Therefore, FLR receives its deposit plus the bribe ¨b. +(b) only FLR deviates from the collusion: if LDR and FLR provide res′ and res′′ ̸= res′ to SCPC respec- +tively, then 2 · ¨t + ¨b amount is transferred to LDR while nothing is sent to FLR. +(c) only LDR deviates from the collusion: if LDR and FLR provide res′′ ̸= res′ and res′ to SCPC respec- +tively, then 2 · ¨t + ¨b amount is sent to FLR while nothing is transferred to LDR. +(d) both deviate from the collusion: if LDR and FLR deviate and provide any result other than res′ to +SCPC, then 2 · ¨t + ¨b amount is sent to LDR and ¨t amount is sent to FLR. +We highlight that the amount of bribe a rational LDR is willing to pay is less than its computation cost +(i.e., ¨b < ¨c); otherwise, such collusion would not bring a higher payoff. We refer readers to [18] for further +discussion. +C.3 +Traitor’s Contract (SCTC) +SCTC incentivises a colluding server (who has had signed SCCC) to betray its counterparty and report the +collusion without being penalised by SCPC. The Traitor’s contract promises that the reporting server will not +be punished by SCPC which makes it safe for the reporting server to follow the collusion strategy (of SCCC), +and get away from the punishment imposed by SCPC. Below, we restate SCTC. +1. This contract is signed among D and the traitor server (TRA) who reports the collusion. This contract +is signed only if the parties have already signed SCPC. +2. D signs this contract only with the first server who reports the collusion. +3. The traitor TRA must also provide to this contract the result of the computation, i.e., f(x). The result +provided in this contract could be different than the one provided in SCPC, e.g., when TRA has to follow +SCCC and provide an incorrect result to SCPC. +4. D needs to pay ¨w + ¨d′ + ¨d − ¨ch amount to this contract. This amount equals the maximum amount +TRA could lose in SCPC plus the reward. This deposit will be transferred via SCANE to this contract. TRA +must deposit in this contract ¨ch amount to cover the cost of resolving a potential dispute. +5. This contract should be signed before the deadline T2 for the delivery of the computation result in SCPC. +If it is not signed on time, then this contract would be terminated and any deposit paid will be refunded. +6. It is required that the TRA provide the computation result to this contract before the above deadline +T2. +7. If this contract is fully signed, then during the execution of SCPC, D always raises a dispute, i.e., takes +step 8c in SCPC. +8. After SCPC is finalised (with the involvement of the auditor), the following steps are taken to pay the +parties involved. +(a) if none of the servers cheated in SCPC (according to the auditor), then amount ¨w + ¨d′ + ¨d − ¨ch is +refunded to SCANE and TRA’s deposit (i.e., ¨ch amount) is transferred to D. Nothing is paid to TRA. +(b) if in SCPC, the other server did not cheat and TRA cheated; however, TRA provided a correct result +in this contract, then ¨d′ + ¨d − ¨ch amount is transferred to SCANE. Also, TRA gets its deposit back +(i.e., ¨ch amount) plus ¨w amounts for providing a correct result to this contract. +(c) if in SCPC, both servers cheated; however, TRA delivered a correct computation result to this contract, +then TRA gets its deposit back (i.e., ¨ch amount), it also receives ¨w + ¨d′ + ¨d − ¨ch amount. +51 + +(d) otherwise, ¨w + ¨d′ + ¨d − ¨ch and ¨ch amounts are transferred to SCANE and TRA respectively. +9. If TRA provided a result to this contract, and deadline T3 (in SCPC) has passed, then all deposits, if any +left, will be transferred to TRA. +TRA must take the following three steps to report collusion: (i) it waits until SCCC is signed by the other +server, (ii) it reports the collusion to D before signing SCCC, and (iii) it signs SCCC only after it signed SCTC +with D. Note, the original analysis of SCTC does not require SCTC to remain secret. Therefore, in our smart +PSI, parties TRA and D can sign this contract and then store its address in SCANE. Alternatively, to keep +this contract confidential, D can deploy a template SCTC to the blockchain and store the commitment of +the contract’s address (instead of the plaintext address) in SCANE. When a traitor (TRA) wants to report +collusion, it signs the deployed SCTC with D which provides the commitment opening to TRA. In this case, +at the time when the deposit is distributed, either D or TRA provides the opening of the commitment to +SCANE which checks whether it matches the commitment. If the check passes, then it distributes the deposit +as before. +D +Proof of Theorem 2 +Below, we restate the proof of Theorem 2, taken from [3]. +Proof. Let P = {p1, ..., pt} and Q = {q1, ..., qt′} be the roots of polynomials p and q respectively. By the +Polynomial Remainder Theorem, polynomials p and q can be written as p(x) = g(x) · +t� +i=1 +(x − pi) and +q(x) = g′(x) · +t′� +i=1 +(x − qi) respectively, where g(x) has degree d − t and g′(x) has degree d′ − t′. Let the +product of the two polynomials be r(x) = p(x) · q(x). For every pi ∈ P, it holds that r(pi) = 0. Because +(a) there exists no non-constant polynomial in Fp[X] that has a multiplicative inverse (so it could cancel +out factor (x − pi) of p(x)) and (b) pi is a root of p(x). The same argument can be used to show for every +qi ∈ Q, it holds that r(qi) = 0. Thus, r(x) preserves roots of both p and q. Moreover, r does not have any +other roots (than P and Q). In particular, if r(α) = 0, then p(α) · q(α) = 0. Since there is no non-trivial +divisors of zero in Fp[X] (as it is an integral domain), it must hold that either p(α) = 0 or q(α) = 0. Hence, +α ∈ P or α ∈ Q. +2 +52 + diff --git a/f9E2T4oBgHgl3EQfbweU/content/tmp_files/load_file.txt b/f9E2T4oBgHgl3EQfbweU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..816fd8b99f1157b197ba271077784981ac06b2d8 --- /dev/null +++ b/f9E2T4oBgHgl3EQfbweU/content/tmp_files/load_file.txt @@ -0,0 +1,3864 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf,len=3863 +page_content='Earn While You Reveal: Private Set Intersection that Rewards Participants Aydin Abadi⋆ and Steven J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Murdoch⋆⋆ University College London Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In Private Set Intersection protocols (PSIs), a non-empty result always reveals something about the private input sets of the parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Moreover, in various variants of PSI, not all parties necessarily receive or are interested in the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Nevertheless, to date, the literature has assumed that those parties who do not receive or are not interested in the result still contribute their private input sets to the PSI for free, although doing so would cost them their privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this work, for the first time, we propose a multi-party PSI, called “Anesidora”, that rewards parties who contribute their private input sets to the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Anesidora is efficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' it mainly relies on symmetric key primitives and its computation and communication complexities are linear with the number of parties and set cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It remains secure even if the majority of parties are corrupted by active colluding adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 1 Introduction Secure Multi-Party Computation (MPC) allows multiple mutually distrustful parties to jointly compute a certain functionality on their private inputs without revealing anything beyond the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Private Set Intersection (PSI) is a subclass of MPC that aims to efficiently achieve the same security property as MPC does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' PSI has numerous applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' For instance, it has been used in Vertical Federated Learning (VFL) [35], COVID-19 contact tracing schemes [20], remote diagnostics [11], and finding leaked credentials [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' There exist two facts about PSIs: (i) a non-empty result always reveals something about the parties’ private input sets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', the set elements that are in the intersection), and (ii) various variants of PSIs do not output the result to all parties, even in those PSIs that do, not all of the parties are necessarily interested in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Given these facts, one may ask a natural question: How can we incentivise the parties that do not receive the result or are not interested in it to participate in a PSI?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To date, the literature has not answered the above question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The literature has assumed that all parties will participate in a PSI for free and bear the privacy cost (in addition to computation and computation overheads imposed by the PSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this work, we answer the above question for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We present a multi-party PSI, called “Anesidora”, that allows a buyer who initiates the PSI computation (and is interested in the result) to pay other parties proportionate to the number of elements it learns about other parties’ private inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='1 Anesidora is efficient and mainly relies on symmetric key primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Its computation and communication complexities are linear with the number of parties and set cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Anesidora remains secure even if the majority of parties are corrupt by active adversaries which may collude with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We develop Anesidora in a modular fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Specifically, we propose the formal notion of “PSI with Fair Compensation” (PSI FC) and devise the first construction, called “Justitia”, that realises the notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='2 PSI FC ensures that either all parties get the result or if the protocol aborts in an unfair manner (where only ⋆ aydin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='abadi@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='uk ⋆⋆ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='murdoch@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='uk 1 Anesidora is in Greek and Roman mythology an epithet of several goddesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It means sender of gifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We call our protocol which sends rewards (or gifts) to honest parties Anesidora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2 Justitia is the ancient Roman personification of justice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We call our protocol which ensures that parties are treated fairly Justitia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='03889v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='CR] 10 Jan 2023 dishonest parties learn the result), then honest parties will receive financial compensation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', adversaries are penalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, we enhance PSI FC to the notion of “PSI with Fair Compensation and Reward” (PSI FCR) and develop Anesidora that realises PSI FCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The latter notion ensures that honest parties (a) are rewarded regardless of whether all parties are honest, or a set of them aborts in an unfair manner and (b) are compensated in the case of an unfair abort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We formally prove the two PSIs using the simulation-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To devise efficient PSIs, we have developed a primitive, called “unforgeable polynomial” that might be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' A PSI, like Anesidora, that supports more than two parties and rewards set contributors can create opportunities for much richer analytics and incentivise parties to participate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It can be used (1) by an advertiser who wants to conduct advertisements targeted at certain customers by first finding their common shopping patterns distributed across different e-commerce companies’ databases [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' (2) by a malware detection service that allows a party to send a query to a collection of malware databases held by independent antivirus companies to find out whether all of them consider a certain application as malware [42],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' or (3) by a bank,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' like “WeBank”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' that uses VFL and PSI to gather information about certain customers from various partners (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', national electronic invoice and other financial institutions) to improve its risk management of loans [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In all these cases, the set contributors will be rewarded by such a PSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We hope that our work initiates future research on developing reward mechanisms for participants of generic MPC, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Such reward mechanisms have the potential to increase MPC’s real-world adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Our Contributions Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this work, we: (1) devise Anesidora, the first PSI that lets participants receive a reward for contributing their set elements to the intersection, (2) develop Justitia, the first PSI that lets either all parties receive the result or if the protocol aborts in an unfair manner, honest parties receive compensation, and (3) propose formal definitions of the above constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2 Related Work Since their introduction in [21], various PSIs have been designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' PSIs can be divided into traditional and delegated ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In traditional PSIs, data owners interactively compute the result using their local data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Very recently, Raghuraman and Rindal [41] proposed two two-party PSIs, one secure against semi-honest/passive and the other against malicious/active adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To date, these two protocols are the fastest two-party PSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' They mainly rely on Oblivious Key-Value Stores (OKVS) data structure and Vector Oblivious Linear Evaluation (VOLE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The protocols’ computation cost is O(c), where c is a set’s cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' They also impose O(c log c2+ κ) and O(c · κ) communication costs in the semi-honest and malicious security models respectively, where l is a set element’s bit-size, and κ is a security parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Also, researchers designed PSIs that allow multiple (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', more than two) parties to efficiently compute the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The multi-party PSIs in [26,34] are secure against passive adversaries while those in [7,23,45,34,38] were designed to remain secure against active ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' However, Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' [3] showed that the PSIs in [23] are susceptible to several attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To date, the protocols in [34] and [38] are the most efficient multi-party PSIs designed to be secure against passive and active adversaries respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' These protocols are secure even if the majority of parties are corrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The former relies on inexpensive symmetric key primitives such as Programmable Pseudorandom Function (OPPRF) and Cuckoo Hashing, while the latter mainly uses OPPRF and OKVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The overall computation and communication complexities of the PSI in [34] are O(c · m2 + c · m) and O(c · m2) respectively, as each client needs to interact with the rest (in the “Conditional Zero-Sharing” phase), where m is the number of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Later, to achieve efficiency, Chandran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' [12] proposed a multi- party PSI that remains secure only if the minority of the parties is corrupt by a semi-honest adversary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' thus, it offers a weaker security guarantee than the one in [34] does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The PSI in [38] has a parameter t that determines how many parties can collude with each other and must be set before the protocol’s execution, where t ∈ {2, m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The protocol divides the parties into three groups, clients: A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='.Am−t−1, leader: Am−t, and servers: Am−t+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='.Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Each client needs to send a set of messages to every server and the leader which 2 jointly compute the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Hence, this protocol’s overall computation and communication complexities are O(c · κ(m + t2 − t(m + 1))) and O(c · m · κ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' proposed a “fair” two-party PSIs [17] that ensure either both parties receive the result or neither does, even if a malicious party aborts prematurely during the protocol’s execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The protocol relies on homomorphic encryption, zero-knowledge proofs, and polynomial representation of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The protocol’s computation and communication complexities are O(c2) and O(c) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Since then, various fair two- party PSIs have been proposed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', in [14,16,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To date, the fair PSI in [15] is the most efficient one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It mainly relies on a combination of ElGamal encryption, verifiable encryption, and zero-knowledge proofs, which often impose a significant overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The protocol’s computation and communication cost is O(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' So far, there exists no fair multi-party PSI in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Our Justitia is the first fair multi-party PSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Delegated PSIs use cloud computing for computation and/or storage, while preserving the privacy of the computation inputs and outputs from the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' They can be divided further into protocols that support one-off and repeated delegation of PSI computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The former like [28,30,46] cannot reuse their outsourced encrypted data and require clients to re-encode their data locally for each computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The most efficient such protocol is [28], which has been designed for the two-party setting and its computation and commu- nication complexity is O(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In contrast, those protocols that support repeated PSI delegation let clients outsource the storage of their encrypted data to the cloud only once, and then execute an unlimited number of computations on the outsourced data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To date, the protocol in [1] is the most efficient PSI that supports re- peated delegation in the semi-honest model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It relies on the polynomial representation of sets, pseudorandom function, and hash table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Its overall communication and computation complexities are O(h · d2) and O(h · d) respectively, where h is the total number of bins in the hash table, d is a bin’s capacity (often d = 100), and h · d is linear with c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Recently, a multi-party PSI that supports repeated delegation and efficient updates has been proposed in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It allows a party to efficiently update its outsourced set securely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It is also in the semi-honest model and uses a pseudorandom function, hash table, and Bloom filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The protocol imposes O(h·d2 ·m) and O(h·d·m) computation and communication costs respectively, during the PSI computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It also imposes O(d2) computation and communication overheads, during the update phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3 Notations and Preliminaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='1 Notations Table 1 summarises the main notations used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='2 Security Model In this paper, we use the simulation-based paradigm of secure computation [25] to define and prove the proposed protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Since both types of (static) active and passive adversaries are involved in our protocols, we will provide formal definitions for both types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this work, we consider a static adversary, we assume there is an authenticated private (off-chain) channel between the clients and we consider a standard public blockchain, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Two-party Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' A two-party protocol Γ problem is captured by specifying a random process that maps pairs of inputs to pairs of outputs, one for each party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Such process is referred to as a functionality denoted by f : {0, 1}∗ ×{0, 1}∗ → {0, 1}∗ ×{0, 1}∗, where f := (f1, f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' For every input pair (x, y), the output pair is a random variable (f1(x, y), f2(x, y)), such that the party with input x wishes to obtain f1(x, y) while the party with input y wishes to receive f2(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' When f is deterministic, then f1 = f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the setting where f is asymmetric and only one party (say the first one) receives the result, f is defined as f := (f1(x, y), ⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Security in the Presence of Passive Adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the passive adversarial model, the party corrupted by such an adversary correctly follows the protocol specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Nonetheless, the adversary obtains the 3 Table 1: Notation Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Setting Symbol Description CL Set of all clients, {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Am, D} D Dealer client Am Buyer client m Total number of clients (excluding D) p Large prime number H Hash function |S∩| Intersection size Smin Smallest set’s size Smax Largest set’s size | Divisible \\ Set subtraction c Set’s cardinality h Total number of bins in a hash table d A bin’s capacity λ Security parameter OLE Oblivious Linear Evaluation OLE+ Advanced OLE Com Commitment algorithm of commitment Ver Verification algorithm of commitment MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='genTree Tree construction algorithm of Merkle tree MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='prove Proof generation algorithm of Merkle tree MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='verify Verification algorithm of Merkle tree CT Coin tossing protocol VOPR Verifiable Oblivious Poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Randomization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='ZSPA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Zero-sum Pseudorandom Values Agreement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='ZSPA-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='ZSPA with an External Auditor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='PSIFC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Multi-party PSI with Fair Compensation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='PSIFCR Multi-party PSI with Fair Compensation and Reward ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='JUS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Protocol that realises PSIFC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='ANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Protocol that realises PSIFCR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='PRF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Pseudorandom function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='PRP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Pseudorandom permutation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='gcd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Greatest common divisor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Generic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Negligible function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Setting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Symbol ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='SCPC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Prisoner’s Contract ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='SCCC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Colluder’s Contract ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='SCTC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Traitor’s Contract ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='¨c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Server’s cost for computing a task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='ch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Auditor’s cost for resolving disputes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='¨d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Deposit a server pays to get the job ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Amount a server receives for completing the task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Counter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Collusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Contracts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='(pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sk) SCJUS’s auditor’s public-private key pair SCJUS JUS’s smart contract ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' ω′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' ρ Random poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' of degree d γ, δ Random poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' of degree d + 1 ν(C) Blinded poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sent by each C to SCJUS φ Blinded poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' encoding the intersection χ Poly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sent to SCJUS to identify misbehaving parties ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='¯L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='List of identified misbehaving parties ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='A portion of a party’s deposit into SCJUS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='¨y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='transferred to honest clients if it misbehaves ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='mk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Master key of PRF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='QInit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Initiation predicate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='QDel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Delivery predicate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='QUF-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='UnFair-Abort predicate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Justitia (JUS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='QF-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Fair-Abort predicate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='SCANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='ANE’s smart contract ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='¨d′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Extractor’s deposit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='¨y′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Each client’s deposit into SCJUS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='¨l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Reward a client earns for an intersection element ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='¨r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Extractor’s cost for extracting an intersection element ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Shorthand for ¨l(m − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Price a buyer pays for an intersection element ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='¨v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='¨v = m · ¨l + 2¨r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='mk′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Another master key of PRF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='ctmk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Encryption of mk under pk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='QDel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Delivery-with-Reward predicate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='Anesidora (ANE) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='QUF-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='UnFair-Abort-with-Reward predicate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='internal state of the corrupted party,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' including the transcript of all the messages received,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' and tries to use this to learn information that should remain private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Loosely speaking, a protocol is secure if whatever can be computed by a party in the protocol can be computed using its input and output only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the simulation- based model, it is required that a party’s view in a protocol’s execution can be simulated given only its input and output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This implies that the parties learn nothing from the protocol’s execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' More formally, party i’s view (during the execution of Γ) on input pair (x, y) is denoted by View Γ i (x, y) and equals (w, ri, mi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', mi t), where w ∈ {x, y} is the input of ith party, ri is the outcome of this party’s internal random coin tosses, and mi j represents the jth message this party receives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The output of the ith party during the execution of Γ on (x, y) is denoted by Output Γ 1 (x, y) and can be generated from its own view of the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The joint output of both parties is denoted by Output Γ(x, y) := (Output Γ 1 (x, y), Output Γ 2 (x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let f be the deterministic functionality defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Protocol Γ security computes f in the presence of a static passive adversary if there exist polynomial-time algorithms (Sim1, Sim2) such that: {Sim1(x, f1(x, y))}x,y c≡ {View Γ 1 (x, y)}x,y {Sim2(x, f2(x, y))}x,y c≡ {View Γ 2 (x, y)}x,y Security in the Presence of Active Adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this adversarial model, the corrupted party may arbitrarily deviate from the protocol specification, to learn the private inputs of the other parties or to influence the outcome of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this case, the adversary may not use the input provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Therefore, beyond the possibility that a corrupted party may learn more than it should, correctness is also required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This means that a corrupted party must not be able to cause the output to be incorrectly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Moreover, we require independence of inputs meaning that a corrupted party cannot make its input depend on the other party’s input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To capture the threats, the security of a protocol is analyzed by comparing what an adversary can do in the real protocol to what it can do in an ideal scenario that is secure by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This is formalized by considering an ideal computation involving an incorruptible Trusted 4 Third Party (TTP) to whom the parties send their inputs and receive the output of the ideal functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Below, we describe the executions in the ideal and real models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' First, we describe the execution in the ideal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let P1 and P2 be the parties participating in the protocol, i ∈ {0, 1} be the index of the corrupted party, and A be a non-uniform probabilistic polynomial- time adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Also, let z be an auxiliary input given to A while x and y be the input of party P1 and P2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The honest party, Pj, sends its received input to TTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The corrupted party Pi may either abort (by replacing the input with a special abort message Λi), send its received input or send some other input of the same length to TTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This decision is made by the adversary and may depend on the input value of Pi and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If TTP receives Λi, then it sends Λi to the honest party and the ideal execution terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Upon obtaining an input pair (x, y), TTP computes f1(x, y) and f2(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It first sends fi(x, y) to Pi which replies with “continue” or Λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the former case, TTP sends fj(x, y) to Pj and in the latter it sends Λi to Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The honest party always outputs the message that it obtained from TTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' A malicious party may output an arbitrary function of its initial inputs and the message it has obtained from TTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The ideal execution of f on inputs (x, y) and z is denoted by Ideal f A(z),i(x, y) and is defined as the output pair of the honest party and A from the above ideal execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the real model, the real two-party protocol Γ is executed without the involvement of TTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this setting, A sends all messages on behalf of the corrupted party and may follow an arbitrary strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The honest party follows the instructions of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The real execution of Γ is denoted by Real Γ A(z),i(x, y), it is defined as the joint output of the parties engaging in the real execution of Γ (on the inputs), in the presence of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, we define security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' At a high level, the definition states that a secure protocol in the real model emulates the ideal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This is formulated by stating that adversaries in the ideal model can simulate executions of the protocol in the real model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let f be the two-party functionality defined above and Γ be a two-party protocol that computes f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Protocol Γ securely computes f with abort in the presence of static active adversaries if for every non- uniform probabilistic polynomial time adversary A for the real model, there exists a non-uniform probabilistic polynomial-time adversary (or simulator) Sim for the ideal model, such that for every i ∈ {0, 1}, it holds that: {Ideal f Sim(z),i(x, y)}x,y,z c≡ {Real Γ A(z),i(x, y)}x,y,z 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='3 Smart Contracts Cryptocurrencies, such as Bitcoin [37] and Ethereum [44], beyond offering a decentralised currency, support computations on transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this setting, often a certain computation logic is encoded in a computer program, called a “smart contract”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To date, Ethereum is the most predominant cryptocurrency framework that enables users to define arbitrary smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this framework, contract code is stored on the blockchain and executed by all parties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', miners) maintaining the cryptocurrency, when the program inputs are provided by transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The program execution’s correctness is guaranteed by the security of the underlying blockchain components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To prevent a denial-of-service attack, the framework requires a transaction creator to pay a fee, called “gas”, depending on the complexity of the contract running on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='4 Counter Collusion Smart Contracts In order to let a party, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', a client, efficiently delegate a computation to a couple of potentially colluding third parties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', servers, Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' [18] proposed two main smart contracts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' namely, “Prisoner’s Contract” (SCPC) and “Traitor’s Contract” (SCTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The Prisoner’s contract is signed by the client and the servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This contract tries to incentivize correct computation by using the following idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It requires each server to pay a deposit before the computation is delegated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It is equipped with an external auditor that is invoked to detect a misbehaving server only when the servers provide non-equal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If a server behaves honestly, then it can withdraw its deposit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Nevertheless, if a cheating server is detected by the auditor, then (a portion) of its deposit is transferred to the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If one of the servers is honest and 5 the other one cheats, then the honest server receives a reward taken from the cheating server’s deposit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' However, the dilemma, created by SCPC between the two servers, can be addressed if they can make an enforceable promise, say via a “Colluder’s Contract” (SCCC), in which one party, called “ringleader”, would pay its counterparty a bribe if both follow the collusion and provide an incorrect computation to SCPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To counter SCCC, Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' proposed SCTC, which incentivises a colluding server to betray the other server and report the collusion without being penalised by SCPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this work, we slightly adjust and use these contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We have stated the related parameters of these tree contracts in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We refer readers to Appendix C for the full description of the parameters and contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='5 Pseudorandom Function and Permutation Informally, a pseudorandom function is a deterministic function that takes a key of length λ and an input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' and outputs a value indistinguishable from that of a truly random function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this paper, we use pseudorandom functions: PRF : {0, 1}λ × {0, 1}∗ → Fp, where |p| = λ is the security parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In practice, a pseudorandom function can be obtained from an efficient block cipher [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The definition of a pseudorandom permutation, PRP : {0, 1}λ × {0, 1}∗ → Fp, is very similar to that of a pseudorandom function, with a difference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' namely, it is required the keyed function PRP(k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=') to be indistinguishable from a uniform permutation, instead of a uniform function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In cryptographic schemes that involve PRP, sometimes honest parties may require to compute the inverse of pseudorandom permutation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', PRP−1(k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' ), as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this case, it would require that PRP(k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=') be indistinguishable from a uniform permutation even if the distinguisher is additionally given oracle access to the inverse of the permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='6 Commitment Scheme A commitment scheme involves a sender and a receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It also involves two phases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' namely, commit and open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the commit phase, the sender commits to a message: x as Com(x, r) = com, that involves a secret value: r $← {0, 1}λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' At the end of the commit phase, the commitment com is sent to the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the open phase, the sender sends the opening ˆx := (x, r) to the receiver who verifies its correctness: Ver(com, ˆx) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='= 1 and accepts if the output is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' A commitment scheme must satisfy two properties: (a) hiding: it is infeasible for an adversary (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', the receiver) to learn any information about the committed message x, until the commitment com is opened, and (b) binding: it is infeasible for an adversary (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', the sender) to open a commitment com to different values ˆx′ := (x′, r′) than that was used in the commit phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', infeasible to find ˆx′, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Ver(com, ˆx) = Ver(com, ˆx′) = 1, where ˆx ̸= ˆx′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' There exist efficient commitment schemes both in (a) the standard model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Pedersen scheme [40], and (b) the random oracle model using the well-known hash-based scheme such that committing is : H(x||r) = com and Ver(com, ˆx) requires checking: H(x||r) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='= com, where H : {0, 1}∗ → {0, 1}λ is a collision-resistant hash function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', the probability to find x and x′ such that H(x) = H(x′) is negligible in the security parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='7 Hash Tables A hash table is an array of bins each of which can hold a set of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It is accompanied by a hash function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To insert an element, we first compute the element’s hash, and then store the element in the bin whose index is the element’s hash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this paper, we set the table’s parameters appropriately to ensure the number of elements in each bin does not exceed a predefined capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Given the maximum number of elements c and the bin’s maximum size d, we can determine the number of bins, h, by analysing hash tables under the balls into the bins model [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In Appendix A, we explain how the hash table parameters are set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='8 Merkle Tree A Merkle tree is a data structure that supports a compact commitment of a set of values/blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' As a result, it includes two parties, prover P and verifier V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The Merkle tree scheme includes three algorithms (MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='genTree, MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='prove, MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='verify), defined as follows: 6 The algorithm that constructs a Merkle tree, MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='genTree, is run by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It takes blocks, u := u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', un, as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, it groups the blocks in pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, a collision-resistant hash function, H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' ), is used to hash each pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' After that, the hash values are grouped in pairs and each pair is further hashed, and this process is repeated until only a single hash value, called “root”, remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This yields a tree with the leaves corresponding to the input blocks and the root corresponding to the last remaining hash value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' V sends the root to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The proving algorithm, MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='prove, is run by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It takes a block index, i, and a tree as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It outputs a vector proof, of log2(n) elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The proof asserts the membership of i-th block in the tree, and consists of all the sibling nodes on a path from the i-th block to the root of the Merkle tree (including i-th block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The proof is given to V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The verification algorithm, MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='verify, is run by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It takes as an input i-th block, a proof, and the tree’s root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It checks if the i-th block corresponds to the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If the verification passes, it outputs 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' otherwise, it outputs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The Merkle tree-based scheme has two properties: correctness and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Informally, the correctness requires that if both parties run the algorithms correctly, then a proof is always accepted by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The security requires that a computationally bounded malicious P cannot convince V into accepting an incorrect proof, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', proof for a non-member block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The security relies on the assumption that it is computationally infeasible to find the hash function’s collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Usually, for the sake of simplicity, it is assumed that the number of blocks, n, is a power of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The height of the tree, constructed on m blocks, is log2(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='9 Polynomial Representation of Sets The idea of using a polynomial to represent a set’s elements was proposed by Freedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Since then, the idea has been widely used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', in [4,5,24,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this representation, set elements S = {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', sd} are defined over Fp and set S is represented as a polynomial of form: p(x) = d� i=1 (x − si), where p(x) ∈ Fp[X] and Fp[X] is a polynomial ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Often a polynomial, p(x), of degree d is represented in the “coefficient form” as follows: p(x) = a0+a1·x+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='+ad·xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The form d� i=1 (x−si) is a special case of the coefficient form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' As shown in [10,33], for two sets S(A) and S(B) represented by polynomials pA and pB respectively, their product, which is polynomial pA · pB, represents the set union, while their greatest common divisor, gcd(pA, pB), represents the set intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' For two polynomials pA and pB of degree d, and two random polynomials γA and γB of degree d, it is proven in [10,33] that: θ = γA · pA + γB · pB = µ · gcd(pA, pB), where µ is a uniformly random polynomial, and polynomial θ contains only information about the elements in S(A) ∩ S(B), and contains no information about other elements in S(A) or S(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Given a polynomial θ that encodes sets intersection, one can find the set elements in the intersection via one of the following approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' First, via polynomial evaluation: the party who already has one of the original input sets, say pA, evaluates θ at every element si of pA and considers si in the intersection if pA(si) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Second, via polynomial root extraction: the party who does not have one of the original input sets, extracts the roots of θ, which contain the roots of (i) random polynomial µ and (ii) the polynomial that represents the intersection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', gcd(pA, pB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this approach, to distinguish errors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', roots of µ) from the intersection, PSIs in [1,33] use the “hash-based padding technique”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this technique, every element ui in the set universe U, becomes si = ui||H(ui), where H is a cryptographic hash function with a sufficiently large output size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Given a field’s arbitrary element, s ∈ Fp and H’s output size |H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' )|, we can parse s into x1 and x2, such that s = x1||x2 and |x2| = |H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=')|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In a PSI that uses polynomial representation and this padding technique, after we extract each root of θ, say s, we parse it into (x1, x2) and check x2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='= H(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If the equation holds, then we consider s as an element of the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='10 Horner’s Method Horner’s method [19] allows for efficiently evaluating polynomials at a given point, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Specifically, given a polynomial of the form: τ(x) = a0+a1·x+a2·x2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='+an·xn and a point: x0, one can efficiently evaluate the 7 polynomial at x0 iteratively, in the following fashion: τ(x0) = a0 + x0(a1 + x0(a2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' + x0(an−1 + x0 · an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Evaluating a polynomial of degree n naively requires n additions and (n2+n) 2 multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' However, using Horner’s method the evaluation requires only n additions and n multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We use this method throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='11 Oblivious Linear Function Evaluation Oblivious Linear function Evaluation (OLE) is a two-party protocol that involves a sender and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In OLE, the sender has two inputs a, b ∈ Fp and the receiver has a single input, c ∈ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The protocol allows the receiver to learn only s = a · c + b ∈ Fp, while the sender learns nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' [22] proposed an efficient OLE that has O(1) overhead and involves mainly symmetric key operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Later, in [23] an enhanced OLE, called OLE+ was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The latter ensures that the receiver cannot learn anything about the sender’s inputs, even if it sets its input to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this paper, we use OLE+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We refer readers to Appendix B, for its construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='12 Coin-Tossing Protocol A Coin-Tossing protocol, CT, allows two mutually distrustful parties, say A and B, to jointly generate a single random bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Formally, CT computes the functionality fCT(inA, inB) → (outA, outB), which takes inA and inB as inputs of A and B respectively and outputs outA to A and outB to B, where outA = outB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' A basic security requirement of a CT is that the resulting bit is (computationally) indistinguishable from a truly random bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Blum proposed a simple CT in [9] that works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Party A picks a random bit inA $← {0, 1}, commits to it and sends the commitment to B which sends its choice of random input, inB $← {0, 1}, to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, A sends the opening of the commitment (including inA) to B, which checks whether the commitment matches its opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If so, each party computes the final random bit as inA ⊕ inB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' There have also been fair coin-tossing protocols, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', in [36], that ensure either both parties learn the result or nobody does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' These protocols can be generalised to multi-party coin-tossing protocols to generate a random string (rather than a single bit), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', see [6,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The overall computation and communication complexities of (fair) multi-party coin-tossing protocols are often linear with the number of participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this paper, any secure multi-party CT that generates a random string can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' For the sake of simplicity, we let a multi-party fCT take m inputs and output a single value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', fCT(in1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', inm) → out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 4 Definition of Multi-party PSI with Fair Compensation In this section, we present the notion of multi-party PSI with Fair Compensation (PSI FC) which allows either all clients to get the result or the honest parties to be financially compensated if the protocol aborts in an unfair manner, where only dishonest parties learn the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In a PSI MFC, three types of parties are involved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' namely, (1) a set of clients {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Am} potentially malicious (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', active adversaries) and all but one may collude with each other, (2) a non-colluding dealer, D, potentially semi-honest (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', a passive adversary) and has an input set, and (3) an auditor Aud potentially semi-honest, where all parties except Aud have input set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' For simplicity, we assume that given an address one can determine whether it belongs to Aud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The basic functionality that any multi-party PSI computes can be defined as f PSI(S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Sm+1) → (S∩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', S∩) � �� � m+1 , where S∩ = S1 ∩ S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', ∩ Sm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To formally define a PSI FC, we equip the above PSI func- tionality with four predicates, Q := (QInit, QDel, QUF-A, QF-A), which ensure that certain financial conditions are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We borrow three of these predicates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', QInit, QDel, QUF-A) from the “fair and robust multi-party computation” initially proposed in [32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' nevertheless, we will (i) introduce an additional predicate QF-A and (ii) provide more formal accurate definitions of these predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Predicate QInit specifies under which condition a protocol that realises PSI FC should start executing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', when all set owners have enough deposit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Predicate QDel determines in which situation parties receive their 8 output, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', when honest parties receive their deposit back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Predicate QUF-A specifies under which condition the simulator can force parties to abort if the adversary learns the output, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', when an honest party receives its deposit back plus a predefined amount of compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Predicate QF-A specifies under which condition the simulator can force parties to abort if the adversary receives no output, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', when honest parties receive their deposits back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We observed that the latter predicate should have been defined in the generic framework in [32] too;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' as the framework should have also captured the cases where an adversary may abort without learning any output after the onset of the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Intuitively, by requiring any protocol that realises PSI FC to implement a wrapped version of f PSI that includes Q, we will ensure that an honest set owner only aborts in an unfair manner if QUF-A returns 1, it only aborts in a fair manner if QF-A returns 1, and outputs a valid value if QDel returns 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Now, we formally define each of these predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Definition 3 (QInit: Initiation predicate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let G be a stable ledger, adrsc be smart contract sc’s address, Adr be a set of m+1 distinct addresses, and ¨x be a fixed amount of coins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, predicate QInit(G, adrsc, m+ 1, Adr, ¨x) returns 1 if every address in Adr has at least ¨x coins in sc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' otherwise, it returns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Definition 4 (QDel: Delivery predicate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let pram := (G, adrsc, ¨x) be the parameters defined above, and adri ∈ Adr be the address of an honest party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, predicate QDel(pram, adri) returns 1 if adri has sent ¨x amount to sc and received ¨x amount from it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' thus, its balance in sc is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Otherwise, it returns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Definition 5 (QUF-A: UnFair-Abort predicate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let pram := (G, adrsc, ¨x) be the parameters defined above, and Adr′ ⊂ Adr be a set containing honest parties’ addresses, m′ = |Adr′|, and adri ∈ Adr′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let also G be a compensation function that takes as input three parameters ( ¨ deps, adri, m′), where ¨ deps is the amount of coins that all m+1 parties deposit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It returns the amount of compensation each honest party must receive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', G( ¨ deps, ardi, m′) → ¨xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, predicate QUnF-Abt is defined as QUF-A(pram, G, ¨ deps, m′, adri) → (a, b), where a = 1 if adri is an honest party’s address and adri has sent ¨x amount to sc and received ¨x + ¨xi from it, and b = 1 if adri is Aud’s address and adri received ¨xi from sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Otherwise, a = b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Definition 6 (QF-A: Fair-Abort predicate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let pram := (G, adrsc, ¨x) be the parameters defined above, and Adr′ ⊂ Adr be a set containing honest parties’ addresses, m′ = |Adr′|, adri ∈ Adr′, and adrj be Aud’s address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let G be the compensation function, defined above and let G(deps, ardj, m′) → ¨xj be the compensation that the auditor must receive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, predicate QF-A(pram, G, ¨ deps, m′, adri, adrj) returns 1, if adri (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' adri ̸= adrj) has sent ¨x amount to sc and received ¨x from it, and adrj received ¨xj from sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Otherwise, it returns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, we present a formal definition of PSI FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Definition 7 (PSI FC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let f PSI be the multi-party PSI functionality defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We say protocol Γ realises f PSI with Q-fairness in the presence of m − 1 static active-adversary clients (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Ajs) or a static passive dealer D or passive auditor Aud, if for every non-uniform probabilistic polynomial time adversary A for the real model, there exists a non-uniform probabilistic polynomial-time adversary (or simulator) Sim for the ideal model, such that for every I ∈ {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Am, D, Aud}, it holds that: {Ideal W(fPSI,Q) Sim(z),I (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Sm+1)}S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=',Sm+1,z c≡ {Real Γ A(z),I(S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Sm+1)}S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=',Sm+1,z where z is an auxiliary input given to A and W(f PSI, Q) is a functionality that wraps f PSI with predicates Q := (QInit, QDel, QUF-A, QF-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 5 Other Subroutines Used in Justitia In this section, we present three subroutines and a primitive that we developed and are used in the instan- tiation of PSI FC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Justitia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='1 Verifiable Oblivious Polynomial Randomisation (VOPR) In the VOPR, two parties are involved, (i) a sender which is potentially a passive adversary and (ii) a receiver that is potentially an active adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The protocol allows the receiver with input polynomial β (of degree e′) and the sender with input random polynomials ψ (of degree e) and α (of degree e + e′) to compute: θ = ψ · β + α, such that (a) the receiver learns only θ and nothing about the sender’s input even if it sets β = 0, (b) the sender learns nothing, and (c) the receiver’s misbehaviour is detected in the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Thus, the functionality that VOPR computes is defined as f VOPR((ψ, α), β) → (⊥, ψ · β + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We will use VOPR in Justitia for two main reasons: (a) to let a party re-randomise its counterparty’s polynomial (representing its set) and (b) to impose a MAC-like structure to the randomised polynomial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' such a structure will allow a verifier to detect if VOPR’s output has been modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Now, we outline how we design VOPR without using any (expensive) zero-knowledge proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='3 In the setup phase, both parties represent their input polynomials in the regular coefficient form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' therefore, the sender’s polynomials are defined as ψ = e� i=0 gi · xi and α = e+e′ � j=0 aj · xj and the receiver’s polynomial is defined as β = e′� i=0 bi · xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' However, the sender computes each coefficient aj (of polynomial α) as follows, aj = k=e′ t=e � t,k=0 at,k, where t + k = j and each at,k is a random value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' For instance, if e = 4 and e′ = 3, then a3 = a0,3 + a3,0 + a1,2 + a2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Shortly, we explain why polynomial α is constructed this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the computation phase, to compute polynomial θ, the two parties interactively multiply and add the related coefficients in a secure way using OLE+ (presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Specifically, for every j (where 0 ≤ j ≤ e′) the sender sends gi and ai,j to an instance of OLE+, while the receiver sends bj to the same instance, which returns ci,j = gi · bj + ai,j to the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This process is repeated for every i, where 0 ≤ i ≤ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, the receiver uses ci,j values to construct the resulting polynomial, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The reason that the sender imposes the above structure to (the coefficients of) α in the setup, is to let the parties securely compute θ via OLE+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Specifically, by imposing this structure (1) the sender can blind each product gi · bj with random value ai,j which is a component of α’s coefficient and (2) the receiver can construct a result polynomial of the form θ = ψ · β + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Now, we outline how the verification works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To check the result’s correctness, the sender picks and sends a random value z to the receiver which computes θ(z) and β(z) and sends these two values to the sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The sender computes ψ(z) and α(z) and then checks if equation θ(z) = ψ(z) · β(z) + α(z) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It accepts the result if the check passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Figure 1 describes VOPR in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let f VOPR be the functionality defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If the enhanced OLE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', OLE+) is secure against malicious (or active) adversaries, then the Verifiable Oblivious Polynomial Randomisation (VOPR), presented in Figure 1, securely computes f VOPR in the presence of (i) a semi-honest sender and honest receiver or (ii) a malicious receiver and honest sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Before proving Theorem 1, we present Lemma 1 and Theorem 2 that will be used in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Informally, Lemma 1 states that the evaluation of a random polynomial at a fixed value results in a uniformly random value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let xi be an element of a finite field Fp, picked uniformly at random and µ(x) be a random polynomial of constant degree d and defined over Fp[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, the evaluation of µ(x) at xi is distributed uniformly at random over the non-zero elements of the field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Pr[µ(xi) = y] = 1 p−1, where y is arbitrary elements of F∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let µ(x) = a0 + d� j=1 ajxj, where the coefficients are distributed uniformly at random over the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We know that if xi is a root of µ(x), then because the polynomial can have at most d roots, we have 3 Previously, Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' [23] designed a protocol called Oblivious Polynomial Addition (OPA) to meet similar security requirements that we laid out above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' But, as shown in [3], OPA is susceptible to several serious attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 10 Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Public Parameters: upper bound on input polynomials’ degree: e and e′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Sender Input: random polynomials: ψ = e� i=0 gi · xi and α = e+e′ � j=0 aj · xj, where gi $← Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Each aj has the form: aj = k=e′ t=e � t,k=0 at,k, such that t + k = j and at,k $← Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Receiver Input: polynomial β = β1·β2 = e′ � i=0 bi·xi, where β1 is a random polynomial of degree 1 and β2 is an arbitrary polynomial of degree e′ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The receiver gets θ = ψ · β + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Computation: (a) Sender and receiver together for every j, 0 ≤ j ≤ e′, invoke e+1 instances of OLE+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In particular, ∀j, 0 ≤ j ≤ e′: sender sends gi and ai,j while the receiver sends bj to OLE+ that returns: ci,j = gi · bj + ai,j to the receiver (∀i, 0 ≤ i ≤ e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' (b) The receiver sums component-wise values ci,j that results in polynomial: θ = ψ · β + α = i=e j=e′ � i,j=0 ci,j · x i+j 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Verification: (a) Sender: picks a random value z and sends it to the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' (b) Receiver: sends θz = θ(z) and βz = β(z) to the sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' (c) Sender: computes ψz = ψ(z) and αz = α(z) and checks if equation θz = ψz·βz +αz holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If the equation holds, it concludes that the computation was performed correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Otherwise, it aborts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 1: Verifiable Oblivious Polynomial Randomization (VOPR) Pr[µ(xi) = 0] = d p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, we focus on the case where xi is not a root of the polynomial (thus y ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' For any choice of xi, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', ad, there exists exactly one value of a0 that makes µ(xi) = y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', µ(xi) = y iff a0 = y − d� j=1 ajxj i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' As a0 is picked uniformly at random, the probability that it equals a certain value that makes µ(xi) = y is 1 p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Thus, Pr[µ(xi) = y] = 1 p−1, ∀y ∈ F∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2 Informally, Theorem 2 states that the product of two arbitrary polynomials (in coefficient form) is a polynomial whose roots are the union of the two original polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Below, we formally state it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The theorem has been taken from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let p and q be two arbitrary non-constant polynomials of degree d and d′ respectively, such that p, q ∈ Fp[X] and they are in coefficient form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, the product of the two polynomials is a polynomial whose roots include precisely the two polynomials’ roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We refer readers to Appendix D for the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, we prove the main theorem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Theorem 1, by considering the case where each party is corrupt, in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Case 1: Corrupt sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the real execution, the sender’s view is defined as follows: View VOPR S � (ψ, α), β � = {ψ, α, rS, β(z), θ(z), View OLE+ S , ⊥} where rS is the outcome of internal random coins of the sender and View OLE+ S refers to the sender’s real-model view during the execution of OLE+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The simulator Sim VOPR S , which receives ψ and α, works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' generates an empty view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It appends to the view polynomials (ψ, α) and coins r′ S chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' computes polynomial β = β1 · β2, where β1 is a random polynomial of degree 1 and β2 is an arbitrary polynomial of degree e′ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, it constructs polynomial θ as follows: θ = ψ · β + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' picks value z $← Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, it evaluates polynomials β and θ at point z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This results in values βz and θz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It appends these two values to the view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' extracts the sender-side simulation of OLE+ from OLE+’s simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let Sim OLE+ S be this simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Note, the latter simulation is guaranteed to exist, as OLE+ has been proven secure (in [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It appends Sim OLE+ S and ⊥ to its view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Now, we are ready to show that the two views are computationally indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The sender’s inputs are identical in both models, so they have identical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Since the real-model semi-honest adversary samples its randomness according to the protocol’s description, the random coins in both models have identical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, we explain why values β(z) in the real model and βz in the ideal model are (computationally) indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the real model, β(z) is the evaluation of polynomial β = β1 · β2 at random point z, where β1 is a random polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We know that β(z) = β1(z) · β2(z), for any (non-zero) z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Moreover, by Lemma 1, we know that β1(z) is a uniformly random value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Therefore, β(z) = β1(z) · β2(z) is a uniformly random value as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the ideal world, polynomial β has the same structure as β has (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', β = β1 · β2, where β1 is a random polynomial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' That means βz is a uniformly random value too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Thus, β(z) and βz are computationally indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, we turn our attention to values θ(z) in the real model and θz in the ideal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We know that θ(z) is a function of β1(z), as polynomial θ has been defined as θ = ψ · (β1 · β2) + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Similarly, θz is a function of βz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' As we have already discussed, β(z) and βz are computationally indistinguishable, so are their functions θ(z) and θz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Moreover, as OLE+ has been proven secure, View OLE+ S and Sim OLE+ S are computationally indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It is also clear that ⊥ is identical in both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We conclude that the two views are computationally indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Case 2: Corrupt receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let Sim VOPR R be the simulator, in this case, which uses a subroutine adversary, AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Sim VOPR R works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' simulates OLE+ and receives AR’s input coefficients bj for all j, 0 ≤ j ≤ e′, as we are in fOLE+-hybrid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' reconstructs polynomial β, given the above coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' simulates the honest sender’s inputs as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It picks two random polynomials: ψ = e� i=0 gi · xi and α = e+e′ � j=0 aj · xj, such that gi $← Fp and every aj has the form: aj = k=e′ t=e � t,k=0 at,k, where t + k = j and at,k $← Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sends to OLE+’s functionality values gi and ai,j and receives ci,j from this functionality (for all i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sends all ci,j to TTP and receives polynomial θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' picks a random value z from Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, it computes ψz = ψ(z) and αz = α(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sends z and all ci,j to AR which sends back θz and βz to the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sends ψz and αz to AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' checks if the following relation hold: β¯z = β(z) ∧ θz = θ(z) ∧ θ(z) = ψz · βz + αz (1) If Relation 1 does not hold, it aborts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', sends abort signal Λ to the sender) and still proceeds to the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' outputs whatever AR outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We first focus on the adversary’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Both values of z in the real and ideal models have been picked uniformly at random from Fp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' therefore, they have identical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the real model, values ψz and αz are the result of the evaluations of two random polynomials at (random) point z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the ideal model, values ψz and αz are also the result of the evaluations of two random polynomials (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', ψ and α) at point z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 12 By Lemma 1, we know that the evaluation of a random polynomial at an arbitrary value yields a uniformly random value in Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Therefore, the distribution of pair (ψz, αz) in the real model is identical to that of pair (ψz, αz) in the ideal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Moreover, the final result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', values ci,j) in the real model has the same distribution as the final result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', values ci,j) in the ideal model, as they are the outputs of the ideal calls to fOLE+, as we are in the fOLE+-hybrid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, we turn our attention to the sender’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We will show that the output distributions of the honest sender in the ideal and real models are statistically close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Our focus will be on the probability that it aborts in each model, as it does not receive any other output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the ideal model, Sim VOPR R is given the honestly generated result polynomial θ (computed by TTP) and the adversary’s input polynomial β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Sim VOPR R aborts with a probability of 1 if Relation 1 does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' However, in the real model, the honest sender (in addition to its inputs) is given only βz and θz and is not given polynomials β and θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' it wants to check if the following equation holds, θz = ψz · βz + αz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Note, polynomial θ = ψ · β + α (resulted from ci,j) is well-structured, as it satisfies the following three conditions, regardless of the adversary’s input β to OLE+, (i) deg(θ) = Max � deg(β) + deg(ψ), deg(α) � , as Fp[X] is an integral domain and (ψ, α) are random polynomials, (ii) the roots of the product polynomial ν = ψ · β contains exactly both polynomials’ roots, by Theorem 2, and (iii) the roots of ν + α is the intersection of the roots of ν and α, as shown in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Furthermore, polynomial θ reveals no information (about ψ and α except their degrees) to the adversary and the pair (ψz, αz) is given to the adversary after it sends the pair (θz, βz) to the sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' There are exactly four cases where pair (θz, βz) can be constructed by the real-model adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Below, we state each case and the probability that the adversary is detected in that case during the verification, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', θz ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='= ψz · βz + αz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' θz = θ(z) ∧ βz = β(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This is a trivial non-interesting case, as the adversary has behaved honestly, so it can always pass the verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' θz ̸= θ(z) ∧ βz = β(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this case, the adversary is detected with a probability of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' θz = θ(z) ∧ βz ̸= β(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this case, the adversary is also detected with a probability of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' θz ̸= θ(z) ∧ βz ̸= β(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this case, the adversary is detected with an overwhelming probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', 1 − 1 22λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' As we illustrated above, in the real model, the lowest probability that the honest sender would abort in case of adversarial behaviour is 1 − 1 22λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Thus, the honest sender’s output distributions in the ideal and real models are statistically close, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', 1 vs 1 − 1 22λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We conclude that the distribution of the joint outputs of the honest sender and adversary in the real and ideal models are computationally indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='2 Zero-sum Pseudorandom Values Agreement Protocol (ZSPA) The ZSPA allows m parties (the majority of which is potentially malicious) to efficiently agree on (a set of vectors, where each vector has) m pseudorandom values such that their sum equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' At a high level, the parties first sign a smart contract and then run a coin-tossing protocol CT to agree on a key: k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, one of the parties generates m − 1 pseudorandom values zj (where 1 ≤ j ≤ m − 1) using key k and PRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It sets the last value as the additive inverse of the sum of the values generated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' zm = − m−1 � j=1 zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, it constructs a Merkel tree on top of the pseudorandom values and stores only the tree’s root g and the key’s hash value q in the smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, each party (using the key) locally checks if the values (on the contract) have been constructed correctly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' if so, then it sends a signed “approved” message to the contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Hence, the functionality that ZSPA computes is defined as f ZSPA (⊥, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', ⊥) � �� � m → ((k, g, q), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', (k, g, q)) � �� � m , where g is the Markle tree’s root built on the pseudorandom values zi,j, q is the hash value of the key used to generate the pseudorandom values, and m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Figure 2 presents ZSPA in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Briefly, ZSPA will be used in Justitia to allow clients {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Am} to provably agree on a set of pseudo- random polynomials whose sum is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Each of these polynomials will be used by a client to blind/encrypt the messages it sends to the smart contract, to protect the privacy of the plaintext message (from Aud, D, 13 and the public).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To compute the sum of the plaintext messages, one can easily sum the blinded messages, which removes the blinding polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' A set of clients {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Am}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' m: the total number of participants, adr: a deployed smart contract’s address, and b: the total number of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let b′ = b − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' k: a secret key that generates b vectors [z0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', z0,m], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', [zb′,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', zb′,m] of pseu- dorandom values, h: hash of the key, g: a Merkle tree’s root, and a vector of signed messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Coin-tossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' CT(in1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', inm) → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' All participants run a coin-tossing protocol to agree on PRF’s key, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Encode(k, m) → (g, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' One of the parties takes the following steps: (a) for every i (where 0 ≤ i ≤ b′), generates m pseudorandom values as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' ∀j, 1 ≤ j ≤ m − 1 : zi,j = PRF(k, i||j), zi,m = − m−1 � j=1 zi,j (b) constructs a Merkel tree on top of all pseudorandom values, MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='genTree(z0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', zb′,m) → g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' (c) sends the Merkel tree’s root: g, and the key’s hash: q = H(k) to adr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Verify(k, g, q, m) → (a, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Each party checks if, all zi,j values, the root g, and key’s hash q have been correctly generated, by retaking step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If the checks pass, it sets a = 1, sets s to a singed “approved” message, and sends s to adr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Otherwise, it aborts by returning a = 0 and s = ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2: Zero-sum Pseudorandom Values Agreement (ZSPA) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let f ZSPA be the functionality defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If CT is secure against a malicious adversary and the correctness of PRF, H, and Merkle tree holds, then ZSPA, in Figure 2, securely computes f ZSPA in the presence of m − 1 malicious adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' For the sake of simplicity, we will assume the sender, which generates the result, sends the result directly to the rest of the parties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', receivers, instead of sending it to a smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We first consider the case in which the sender is corrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Case 1: Corrupt sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let Sim ZSPA S be the simulator using a subroutine adversary, AS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Sim ZSPA S works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' simulates CT and receives the output value k from fCT, as we are in fCT-hybrid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sends k to TTP and receives back from it m pairs, where each pair is of the form (g, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sends k to AS and receives back from it m pairs where each pair is of the form (g′, q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' checks whether the following equations hold (for each pair): g = g′ ∧ q = q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If the two equations do not hold, then it aborts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', sends abort signal Λ to the receiver) and proceeds to the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' outputs whatever AS outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We first focus on the adversary’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the real model, the only messages that the adversary receives are those messages it receives as the result of the ideal call to fCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' These messages have identical distribution to the distribution of the messages in the ideal model, as the CT is secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Now, we move on to the receiver’s 14 output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We will show that the output distributions of the honest receiver in the ideal and real models are computationally indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the real model, each element of pair (g, p) is the output of a deterministic function on the output of fCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We know the output of fCT in the real and ideal models have an identical distribution, and so do the evaluations of deterministic functions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Merkle tree, H, and PRF) on them, as long as these three functions’ correctness holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Therefore, each pair (g, q) in the real model has an identical distribution to pair (g, q) in the ideal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' For the same reasons, the honest receiver in the real model aborts with the same probability as Sim ZSPA S does in the ideal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We conclude that the distributions of the joint outputs of the adversary and honest receiver in the real and ideal models are (computationally) indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Case 2: Corrupt receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let Sim ZSPA R be the simulator that uses subroutine adversary AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Sim ZSPA R works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' simulates CT and receives the output value k from fCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sends k to TTP and receives back m pairs of the form (g, q) from TTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sends (k, g, q) to AR and outputs whatever AR outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the real model, the adversary receives two sets of messages, the first set includes the transcripts (including k) it receives when it makes an ideal call to fCT and the second set includes pair (g, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' As we already discussed above (because we are in the fCT-hybrid model) the distributions of the messages it receives from fCT in the real and ideal models are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Moreover, the distribution of fCT’s output (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', ¯k and k) in both models is identical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' therefore, the honest sender’s output distribution in both models is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' As we already discussed, the evaluations of deterministic functions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Merkle tree, H, and PRF) on fCT’s outputs have an identical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Therefore, each pair (g, q) in the real model has an identical distribution to the pair (g, q) in the ideal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Hence, the distribution of the joint outputs of the adversary and honest receiver in the real and ideal models is indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2 In addition to the security guarantee (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', computation’s correctness against malicious sender or receiver) stated by Theorem 3, ZSPA offers (a) privacy against the public, and (b) non-refutability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Informally, privacy here means that given the state of the contract (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', g and q), an external party cannot learn any information about any of the pseudorandom values, zj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' while non-refutability means that if a party sends “approved” then in future cannot deny the knowledge of the values whose representation is stored in the contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If H is preimage resistance, PRF is secure, the signature scheme used in the smart contract is secure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', existentially unforgeable under chosen message attacks), and the blockchain is secure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', offers liveness property and the hash power of the adversary is lower than those of honest miners) then ZSPA offers (i) privacy against the public and (ii) non-refutability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' First, we focus on privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Since key k, for PRF, has been picked uniformly at random and H is preimage resistance, the probability that given g the adversary can find k is negligible in the security parameter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', ϵ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Furthermore, because PRF is secure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', its outputs are indistinguishable from random values) and H is preimage resistance, given the Merkle tree’s root g, the probability that the adversary can find a leaf node, which is the output of PRF, is ϵ(λ) too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='3 ZSPA’s Extension: ZSPA with an External Auditor (ZSPA-A) In this section, we present an extension of ZSPA, called ZSPA-A which lets a (trusted) third-party auditor, Aud, help identify misbehaving clients in the ZSPA and generate a vector of random polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Informally, ZSPA-A requires that misbehaving parties are always detected, except with a negligible probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Aud of this protocol will be invoked by Justitia when Justitia’s smart contract detects that a combination of the messages sent by the clients is not well-formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Later, in Justitia’s proof, we will show that even a semi- honest Aud who observes all messages that clients send to Justitia’s smart contracts, cannot learn anything about their set elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We present ZSPA-A in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 15 Parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' A set of clients {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Am} and an external auditor, Aud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' m: the total number of participants (excluding the auditor), ζ: a random poly- nomial of degree 1, b: the total number of vectors, and adr: a deployed smart contract’s address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let b′ = b − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Output of each Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' k: a secret key that generates b vectors [z0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', z0,m], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', [zb′,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', zb′,m] of pseudorandom values, h: hash of the key, g: a Merkle tree’s root, and a vector of signed messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Output of Aud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' L: a list of misbehaving parties’ indices, and #»µ: a vector of random polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' ZSPA invocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' ZSPA(⊥, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', ⊥) → � (k, g, q), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', (k, g, q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' All parties in {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Am} call the same instance of ZSPA, which results in (k, g, q), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', (k, g, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Auditor computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Audit(#»k , q, ζ, b, g) → (L, #»µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Aud takes the below steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Note, each kj ∈ #»k is given by Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' An honest party’s input, kj, equals k, where 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' (a) runs the checks in the verification phase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Phase 3) of ZSPA for every j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Verify(kj, g, q, m) → (aj, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' (b) appends j to L, if any checks fails, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', if aj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this case, it skips the next two steps for the current j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' (c) For every i (where 0 ≤ i ≤ b′), it recomputes m pseudorandom values: ∀j, 1 ≤ j ≤ m − 1 : zi,j = PRF(k, i||j), zi,m = − m−1 � j=1 zi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' (d) generates polynomial µ(j) as follows: µ(j) = ζ · ξ(j) − τ (j), where ξ(j) is a random polynomial of degree b′ − 1 and τ (j) = b′ � i=0 zi,j · xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' By the end of this step, a vector #»µ containing at most m polynomials is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' (e) returns list L and #»µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3: ZSPA with an external auditor (ZSPA-A) Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If ZSPA is secure, H is second-preimage resistant, and the correctness of PRF, H, and Merkle tree holds, then ZSPA-A securely computes f ZSPA-A in the presence of m − 1 malicious adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' First, we consider the case where a sender, who (may collude with m−2 senders and) generates pairs (g, q), is corrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Case 1: Corrupt sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let Sim ZSPA-A S be the simulator using a subroutine adversary, AS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Below, we explain how Sim ZSPA-A S works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' simulates CT and receives the output value k from fCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sends k to TTP and receives back from it m pairs, where each pair is of the form (g, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sends k to AS and receives back from it m pairs where each pair is of the form (g′, q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' constructs an empty vector L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Sim ZSPA-A S checks whether the following equations hold for each j-th pair: g = g′ ∧ q = q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If these two equations do not hold, it sends an abort message Λ to other receiver clients, appends the index of the pair (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', j) to L, and proceeds to the next step for the valid pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the case where there are no valid pairs, it moves on to step 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' picks a random polynomial ζ of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Moreover, for every j /∈ L, Sim ZSPA-A S picks a random polynomial ξ(j) of degree b′ − 1, where 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' computes m pseudorandom values for every i, j′, where 0 ≤ i ≤ b′ and j′ /∈ L as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' ∀j′, 1 ≤ j′ ≤ m − 1 : zi,j = PRF(k, i||j′) and zi,m = − m−1 � j′=1 zi,j 16 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' generates polynomial µ(j), for every j /∈ L, as follows: µ(j) = ζ · ξ(j) − τ (j), where τ (j) = b′� i=0 zi,j · xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sends the above ζ, ξ(j), and µ(j) to all parties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', AS and the receivers), for every j /∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' outputs whatever AS outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Now, we focus on the adversary’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the real model, the messages that the adversary receives include those messages it receives as the result of the ideal call to fCT and (ζ, ξ(j), µ(j)), where j /∈ L and 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Those messages yielded from the ideal calls have identical distribution to the distribution of the messages in the ideal model, as CT is secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The distribution of each µ(j) depends on the distribution of its components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' namely, ζ, ξ(j), and τ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' As we are in the fCT-hybrid model, the distributions of τ (j) in the real model and τ (j) in the ideal model are identical, as they were derived from the output of fCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Furthermore, in the real model, each polynomial ζ and ξ(j) has been picked uniformly at random and they are independent of the clients’ and the adversary’s inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The same arguments hold for (ζ, ξ(j), µ(j)) in the ideal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Therefore, (ζ, ξ(j), µ(j)) in the real model and (ζ, ξ(j), µ(j)) in the ideal model have identical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, we turn our attention to the receiver’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We will show that the output distributions of an honest receiver and the auditor in the ideal and real models are computationally indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the real model, each element of the pair (g, p) is the output of a deterministic function on the output of fCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We know the outputs of fCT in the real and ideal models have an identical distribution, and so do the evaluations of deterministic functions (namely Merkle tree, H, and PRF) on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Therefore, each pair (g, q) in the real model has an identical distribution to the pair (g, q) in the ideal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' For the same reasons, the honest receiver in the real model aborts with the same probability as Sim ZSPA-A S does in the ideal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The same argument holds for the arbiter’s output, as it performs the same checks that an honest receiver does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Thus, the distribution of the joint outputs of the adversary, honest receiver, and honest in the real and ideal models is computationally indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Case 2: Corrupt receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let Sim ZSPA-A R be the simulator that uses subroutine adversary AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Below, we explain how Sim ZSPA-A R works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' simulates ZSPA and receives the m output pairs of the form (k, g, q) from f ZSPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sends (k, g, q) to AR and receives m keys, k′ j, where 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' generates an empty vector L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, for every j, Sim ZSPA-A R computes q′ j as H(k′ j) = qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' It generates gj as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' (a) for every i (where 0 ≤ i ≤ b′), generates m pseudorandom values as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' ∀j, 1 ≤ j′ ≤ m − 1 : zi,j = PRF(k′ j, i||j′), zi,m = − m−1 � j=1 zi,j (b) constructs a Merkel tree on top of all pseudorandom values, MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='genTree(z0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', zb′,m) → g′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' checks if the following equations hold for each j-th pair: (k = k′ j) ∧ (g = g′ j) ∧ (q = q′ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If these equations do not hold for j-th value, it appends j to L and proceeds to the next step for the valid value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the case where there is no valid value, it moves on to step 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' picks a random polynomial ζ of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Also, for every j /∈ L, it picks a random polynomial ξ(j) of degree b′ − 1, where 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' generates polynomial µ(j), for every j /∈ L, as follows: µ(j) = ζ · ξ(j) − τ (j), where τ (j) = b′� i=0 zi,j · xi, and values zi,j were generated in step 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' sends the above ζ, ξ(j), and µ(j) to AR, for every j /∈ L and 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' outputs whatever AR outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the real model, the adversary receives two sets of messages, the first set includes the transcripts (including k, g, q) it receives when it makes an ideal call to f ZSPA and the second set includes pairs (ζ, ξ(j), µ(j)), for every j /∈ L and 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Since we are in the f ZSPA-hybrid model and (based on our assumption) there is at least one honest party participated in ZSPA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', there are at most m − 1 malicious participants of ZSPA), the distribution of the messages it receives from f ZSPA in the real and ideal models is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Furthermore, 17 as we discussed in Case 1, (ζ, ξ(j), µ(j)) in the real model and (ζ, ξ(j), µ(j)) in the ideal model have identical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The honest sender’s output distribution in both models is identical, as the distribution of fCT’s output (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', k) in both models is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Now we show that the probability that the auditor aborts in the ideal and real models are statistically close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In the ideal model, Sim ZSPA-A R is given the ideal functionality’s output that includes key k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Therefore, it can check whether the key that AR sends to it equals k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', k ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='= k′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Thus, it aborts with the probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' However, in the real model, an honest auditor is not given the output of CT (say key k) and it can only check whether the key is consistent with the hash value q and the Merkle tree’s root g stored on the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This means the adversary can distinguish the two models if in the real model it sends a key ¨k, such that ¨k ̸= k and still passes the checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Specifically, it sends the invalid key ¨k that can generate valid pair (g, q), as follows: H(¨k) = q and MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='genTree(z′ 0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', z′ b′,m) → g, where each z′ i,j is derived from ¨k using the same technique described in step 3 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Nevertheless, this means that the adversary breaks the second preimage resistance property of H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' however, H is the second-preimage resistance and the probability that the adversary succeeds in finding the second preimage is negligible in the security parameter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', ϵ(λ) where λ is the hash function’s security parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Therefore, in the real model, the auditor aborts if an invalid key is provided with a probability 1−ϵ(λ) which is statically close to the probability that Sim ZSPA-A R aborts in the same situation in the ideal model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', 1 − ϵ(λ) vs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Hence, the distribution of the joint outputs of the adversary, honest sender, and honest auditor in the real and ideal models is indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='4 Unforgeable Polynomials In this section, we introduce the notion of “unforgeable polynomials”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Informally, an unforgeable polynomial has a secret factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To ensure that an unforgeable polynomial has not been tampered with, a verifier can check whether the polynomial is divisible by the secret factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To turn an arbitrary polynomial π of degree d into an unforgeable polynomial θ, one can (i) pick three secret random polynomials (ζ, ω, γ) and (ii) compute θ = ζ ·ω ·π +γ mod p, where deg(ζ) = 1, deg(ω) = d, and deg(γ) = 2d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To verify whether θ has been tampered with, a verifier (given θ, γ, and ζ) can check if θ − γ is divisible by ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Informally, the security of an unforgeable polynomial states that an adversary (who does not know the three secret random polynomials) cannot tamper with an unforgeable polynomial without being detected, except with a negligible probability, in the security parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Below, we formally state it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Theorem 6 (Unforgeable Polynomial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let polynomials ζ, ω, and γ be three secret uniformly random polynomials (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', ζ, ω, γ $← Fp[x]), GCD(ζ, γ) = 1, polynomial π be an arbitrary polynomial, deg(ζ) = 1, deg(ω) = d, deg(γ) = 2d+1, deg(π) = d, and p be a λ-bit prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Also, let polynomial θ be defined as θ = ζ · ω · π + γ mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Given (θ, π), the probability that an adversary (which does not know ζ, ω, and γ) can forge θ to an arbitrary polynomial δ such that δ ̸= θ, deg(δ) = const(λ), and ζ divides δ − γ is negligible in the security parameter λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Pr[ζ | (δ − γ)] ≤ ϵ(λ) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let τ = δ − γ and ζ = a · x + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Since γ is a random polynomial of degree 2d + 1 and unknown to the adversary, given (θ, π), the adversary cannot learn anything about the factor ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' as from its point of view every polynomial of degree 1 in Fp[X] is equally likely to be ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Moreover, polynomial τ has at most Max � deg(δ), 2d + 1 � irreducible non-constant factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' For ζ to divide τ, one of the factors of τ must be equal to ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We also know that ζ has been picked uniformly at random (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', a, b $← Fp) and by definition GCD(ζ, γ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Thus, the probability that ζ divides τ is negligible in the security parameter, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Specifically, Pr[ζ | (δ − γ)] ≤ Max � deg(δ), 2d + 1 � 22λ = ϵ(λ) 2 18 An interesting feature of an unforgeable polynomial is that the verifier can perform the check without needing to know the original polynomial π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Another appealing feature of the unforgeable polynomial is that it supports linear combination and accordingly batch verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Specifically, to turn n arbitrary polynomials [π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', πn] into unforgeable polynomials, one can construct θi = ζ · ωi · πi + γi mod p, where ∀i, 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To check whether all polynomials [θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', θn] are intact, a verifier can (i) compute their sum χ = n� i=1 θi and (ii) check whether χ− n� i=1 γi is divisible by ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Informally, the security of an unforgeable polynomial states that an adversary (who does not know the three secret random polynomials for each θi) cannot tamper with any subset of the unforgeable polynomials without being detected, except with a negligible probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We formally state it, below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Theorem 7 (Unforgeable Polynomials’ Linear Combination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let polynomial ζ be a secret polyno- mial picked uniformly at random;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' also, let #»ω = [ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', ωn] and #»γ = [γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', γn] be two vectors of secret uniformly random polynomials (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', ζ, ωi, γi $← Fp[x]), GCD(ζ, γi) = 1, #»π = [π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', πn] be a vector of arbitrary polynomials, deg(ζ) = 1, deg(ωi) = d, deg(γi) = 2d + 1, deg(πi) = d, p be a λ-bit prime number, and 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Moreover, let polynomial θi be defined as θi = ζ · ωi · πi + γi mod p, and #»θ = [θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', θn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Given (#»θ , #»π), the probability that an adversary (which does not know ζ, #»ω, and #»γ ) can forge t polynomi- als, without loss of generality, say θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', θt ∈ #»θ to arbitrary polynomials δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', δt such that t� j=1 δj ̸= t� j=1 θj, deg(δj) = const(λ), and ζ divides ( t� j=1 δj + n� j=t+1 θj − n� j=1 γj) is negligible in the security parameter λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Pr[ζ | ( t � j=1 δj + n � j=t+1 θj − n � j=1 γj)] ≤ ϵ(λ) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This proof is a generalisation of that of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Let τj = δj − γj and ζ = a · x + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Since every γj is a random polynomial of degree 2d + 1 and unknown to the adversary, given (#»θ , #»π), the adversary cannot learn anything about the factor ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Each polynomial τj has at most Max � deg(δj), 2d + 1 � irreducible non-constant factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We know that ζ has been picked uniformly at random (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', a, b $← Fp), by definition GCD(ζ, γj) = 1, and ζ does divide every θj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Therefore, the probability that ζ divides t� j=1 δj + n� j=t+1 θj − n� j=1 γj equals the probability that ζ equals to one of the factors of every τj, that is negligible in the security parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Concretely, Pr[ζ | ( t � j=1 δj + n � j=t+1 θj − n � j=1 γj)] ≤ t� j=1 Max � deg(δj), 2d + 1 � 22λt = ϵ(λ) 2 It is not hard to see that, Theorem 7 is a generalisation of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Briefly, in Justitia, we will use unforgeable polynomials (and their linear combinations) to allow a smart contract to efficiently check whether the polynomials that the clients send to it are intact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', they are VOPR’s outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 6 Justitia: A Concrete Construction of PSI FC 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='1 Main Challenges to Overcome We need to address several key challenges, to design an efficient scheme that realises PSI FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Below, we outline these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 19 Keeping Overall Complexities Low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In general, in multi-party PSIs, each client needs to send messages to the rest of the clients and/or engage in secure computation with them, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', in [26,34], which would result in communication and/or computation quadratic with the number of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To address this challenge, we (a) allow one of the clients as a dealer to interact with the rest of the clients,4 and (b) we use a smart contract, which acts as a bulletin board to which most messages are sent and also performs lightweight computation on the clients’ messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The combination of these approaches will keep the overall communication and computation linear with the number of clients (and sets’ cardinality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Securely Randomising Input Polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In multi-party PSIs that rely on the polynomial represen- tation, it is essential that an input polynomial of a client be randomised by another client [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To do that securely and efficiently, we require the dealer and each client together to engage in an instance of VOPR, which we developed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Preserving the Privacy of Outgoing Messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Although the use of regular public smart contracts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', Ethereum) will help keep overall complexity low, it introduces another challenge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' namely, if clients do not protect the privacy of the messages they send to the smart contracts, then other clients (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', dealer) and non-participants of PSI (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', the public) can learn the clients’ set elements and/or the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Because standard smart contracts do not automatically preserve messages’ privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To efficiently protect the privacy of each client’s messages (sent to the contracts) from the dealer, we require the clients (except the dealer) to engage in ZSPA-A which lets each of them generate a pseudorandom polynomial with which it can blind its message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To protect the privacy of the intersection from the public, we require all clients to run a coin-tossing protocol to agree on a blinding polynomial, with which the final result that encodes the intersection on the smart contract will be blinded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Ensuring the Correctness of Subroutine Protocols’ Outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In general, any MPC that must remain secure against an active adversary is equipped with a verification mechanism that ensures an adversary is detected if it deviates from the protocol and affects messages’ integrity, during the protocol’s execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This is the case for the subroutine protocols that we use, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', VOPR and ZSPA-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Nevertheless, this type of verification itself is not always sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Because in certain cases, the output of an MPC protocol may be fed as input to another MPC and we need to ensure that the actual/intact output of the first MPC is fed to the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' This is the case in our PSI’s subroutines as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' To address this challenge, we use unforgeable polynomials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' specifically, the output of VOPR is an unforgeable polynomial (that encodes the actual output);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' if the adversary tampers with the VOPR’s output and uses it later, then a verifier can detect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' We will have the same integrity guarantee for the output of ZSPA-A for free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Because (i) VOPR is called before ZSPA-A, and (ii) if clients use intact outputs of ZSPA-A, then the final result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=', the sum of all clients’ messages) will not contain any output of ZSPA-A, as they would cancel out each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Thus, by checking the correctness of the final result, one can ensure the correctness of the outputs of VOPR and ZSPA-A, in one go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='2 Description of Justitia (JUS) An overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' At a high level, Justitia (JUS) works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' First, each client encodes its set elements into a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' All clients sign a smart contract and deposit a predefined amount of coins into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Next, one of the clients as a dealer, D, randomises the rest of the clients’ polynomials and imposes a certain structure to their polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The clients also randomise D’s polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The randomised polynomials reveal nothing about the clients’ original polynomials representing their set elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Then, all clients send their randomised polynomials to the smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' The contract combines all polynomials and checks whether the resulting polynomial still has the structure imposed by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' If the contract concludes that the resulting polynomial does not have the structure, then it invokes an auditor, Aud, to identify misbehaving clients and penalise them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 4 This approach has similarity with the non-secure PSIs in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' 20 Nevertheless, if the resulting polynomial has the structure, then the contract outputs an encoded polynomial and refunds the clients’ deposits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' In this case, all clients can use the resulting polynomial (output by the contract) to locally find the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' Figure 4 outlines the interaction between parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='VOPR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='ADCXicjVJLj9MwEHbCazGP7cIFiYtFt9IioSqpxOPGinJAnBYt3V2pCZHjuq21fkTxZKVi5Rfwa7ghrvy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='K/Tc4aQ602wMjO/o830znonzQgoLUXQdhLdu37l7b+8+fvDw0eP93sGTM2uqkvEJM9KUFzm1XArNJyBA8oui5FTlkp/nl+Mmfn7FSyuM/gqrgqeKLrSYC0bBu7LeNcaDwyRXyXcOlCRsZoA0R6P4gn5ziWlKD7wkpyQpxfRx9fjTP1sq7rQ4IxJuscC6rU/4m8BrcSWPq6OyWeWme7AqO1vL24orBkVLrTcNtTgDu8+TUl8h6/WgYtUZugrgDfdTZSXYQPEtmh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} 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+page_content='w5kELDEAVKGBcWmMol3OX3l2v97gGsE0bf4qATLGFnPBGXpq2nmMol6aq9Quza8qdyKAtsbVxJoRWl1dDlVX38c13Xdi6J+YzcKFmxrYCOR8plBuo4UYrv/X6BZMO1040HcDH0JkhZ0S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='TvX0/2ApDPDSwUauWTOTZK4wKxiFgWXUEf9DdlY8QA8q6TU3NVRWjoGL9nC5h4qJkCl1VNsTXte2ZG58b6o5E27P+JinVir3TsVw6Ta1NblNm5Q4P8sqoYsSQfOnRfNSUjR0/Ut0J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E2T4oBgHgl3EQfbweU/content/2301.03889v1.pdf'} +page_content='ixwlCsPmC/LP4XyJbOMo/LZ1tyVfvOks2GXoLRySD5Pvh2c9I9v2jb2yWH5As5Igk5JefkJ7kmQ8LJn+Aw6AX98Di8Ckfh+MkaBm3mM3k2IfsLJ0DQNg=(Xso-SR(CP(DS(xn) +n =n+1 +end +Output: Xh +(1) Original + +(2) Compressed (PSNR=21.84, SSIM=0.7880) + +(3) Swin2SR (PSNR=22.11, SSIM=0.8828) + +(4) CSwin2SR (PSNR=24.07, SSIM=0.8982) + + + + (5) Original (6)Compressed(PSNR=15.69,SSIM=0.5185) + + + +(7)Swin2SR(PSNR=15.20, SSIM=0.5893) (8)CSwin2SR(PSNR=18.27, SSIM=0.6110) + + + +(9) Swin2SR (10) CSwin2SR +Fig. 4. Comparison between Swin2SR and CSwin2SR (zebra). + + +(1) Original + +(2) Compressed (PSNR=26.34, SSIM=0.8383) + +(3) Swin2SR (PSNR=28.52, SSIM=0.9494) + +(4) CSwin2SR (PSNR=30.29, SSIM=0.9508) + + + (5) Original (6)Compressed(PSNR=22.46,SSIM=0.6873) + + +(7)Swin2SR(PSNR=22.36,SSIM=0.8092) (8)CSwin2SR(PSNR=25.33,SSIM=0.8574) + + +(9) Swin2SR (10) CSwin2SR +Fig. 5. Comparison between Swin2SR and CSwin2SR (wheel). + + + +(1) Original + +(2) Compressed (PSNR=27.50, SSIM=0.9011) + +(3) Swin2SR (PSNR=29.72, SSIM=0.9681) + +(4) CSwin2SR (PSNR=31.18, SSIM=0.9726) + + + (5)Original (6)Compressed(PSNR=26.04,SSIM=0.9468) + + +(7)Swi2SR(PSNR=26.87,SSIM=0.9695) (8)CSwin2SR(PSNR=30.73,SSIM=0.9860) + + +(9) Swin2SR (10) CSwin2SR +Fig. 6. Comparison between Swin2SR and CSwin2SR (desktop). +CSwin2SR can efficiently remove pixel shifts in the +reconstruction image of the classical Swin2SR. On DIV2K +test and valid datasets, the increment of average PSNR is +greater than 1dB and the related increment of SSIM is greater +than 0.006. +In our future work, the theory basis of CSwin2SR will be +deeply studied. More SR factors, compression algorithms, +image datasets will also be explored. +ACKNOWLEDGMENT +We would very much like to thank the authors of Swin2SR +for selflessly sharing their open-source codes on Microsoft +GitHub. We also express deep thanks to Google COLAB for +free GPU computing service. +REFERENCES +[1] +D.C. Lepcha, B. Goyal, A. Dogra, and V. Goyal, “Image super- +resolution: A comprehensive review, recent trends, challenges and +applications,” Information Fusion, vol. 91, pp. 230-260, Mar 2023. +[2] +W.K. Huang, F.B. Zhou, T. Zou, P.W. Lu, Y.H. Xue, J.J. Liang, and +Y.K. Dong, “Alternating positive and negative feedback control model +based on catastrophe theories,” Mathematics, vol. 9, no. 22, pp. 1-19, +Nov 2021. +[3] +M.V. Conde, U.J. Choi, M. Burchi, and R. Timofte, “SwinV2 +transformer for compressed image super-resolution and restoration,” +Proceedings of the European Conference on Computer Vision (ECCV), +Advances in Image Manipulation (AIM) workshop, pp. 1-19, Tel Aviv, +Israel, Oct 2022. +[4] +Z.H. Wang; J. Chen, and S. Hoi, “Deep learning for image super- +resolution: a survey,” IEEE Trans. on Pattern Analysis and Machine +Intelligence, 43(10) , pp.3365-3387, Oct 2021. +[5] +V.C. Marcos, M.D. Steven, M. Matteo, L. Ales, and P.P. Eduardo, +“Model-based image signal processors via learnable dictionaries,” +Proceedings of the AAAI Conference on Artificial Intelligence, vol. 36, +no. 1, pp. 481–489, Vancouver, Canada, Jun 2022. +[6] +Z. Liu, Y.T. Lin, Y. Cao, H. Hu, Y.X. Wei, Z. Zhang, S. Lin, and B.N. +Guo, “Swin transformer: hierarchical vision transformer using shifted +windows,” Proceedings of IEEE/CVF International Conference on +Computer Vision (ICCV), pp. 9992-10002, Virtual, Oct 2021. +[7] +Z. Liu, H. Hu, Y.T. Lin, Z.L. Yao, Z.D. Xie, Y.X. Wei, J. Ning, Y. Cao, +Z. Zhang, L. Dong, F.R. Wei, and B.N. Guo, “Swin transformer v2: +scaling up capacity and resolution,” Processings of IEEE/CVF +Conference on Computer Vision and Pattern Recognition (CVPR), pp. +11999-12009, New Orleans, USA, Jun 2022. +[8] +J.Y. Liang; J.Z. Cao; G.L. Sun; K. Zhang; L.V. Gool; R.D. Timofte, +“SwinIR: image restoration using swin transformer,” Proceedings of +IEEE/CVF International Conference on Computer Vision Workshops +(ICCVW), pp. 1833-1844, Virtual, Oct 2021. +[9] X.B. Liu, S.Q. Chen, L.P. Song, M. Wozniak, and S. Liu, “Self- +attention negative feedback network for real-time image super- +resolution,” Journal of King Saud University - Computer and +Information Sciences (B), vol. 34, no. 8, pp. 6179-6186, Aug 2022. +[10] E. Agustsson and R. Timofte, “NTIRE 2017 Challenge on Single +Image Super-Resolution: Dataset and Study,” Proceeding of IEEE +Conference on Computer Vision and Pattern Recognition (CVPR) +Workshops, pp. 1-10, Honolulu, USA, Jul 2017. + + + + + + + + + + + + + +00 \ No newline at end of file diff --git a/f9FAT4oBgHgl3EQf8R46/content/tmp_files/load_file.txt b/f9FAT4oBgHgl3EQf8R46/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..28206aea0216e98fd4bf3add92c08db8f55ddbb6 --- /dev/null +++ b/f9FAT4oBgHgl3EQf8R46/content/tmp_files/load_file.txt @@ -0,0 +1,362 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf,len=361 +page_content='CSwin2SR: Circular Swin2SR for Compressed Image Super-Resolution Honggui Li College of Information Engineering Yanghzou University Yangzhou, China hgli@yzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='cn Maria Trocan LISITE Research Laboratory ISEP Paris, France maria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='trocan@isep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content="fr Dimitri Galayko Laboratoire d'Informatique de Paris 6 Sorbonne University Paris, France dimitri." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='galayko@sorbonne-universite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='fr Mohamad Sawan College of Engineerig Westlake University Hangzhou, China sawan@westlake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='cn Abstract—Closed-loop negative feedback mechanism is extensively utilized in automatic control systems and brings about extraordinary dynamic and static performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' In order to further improve the reconstruction capability of current methods of compressed image super-resolution, a circular Swin2SR (CSwin2SR) approach is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The CSwin2SR contains a serial Swin2SR for initial super-resolution reestablishment and circular Swin2SR for enhanced super- resolution reestablishment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Simulated experimental results show that the proposed CSwin2SR dramatically outperforms the classical Swin2SR in the capacity of super-resolution recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' On DIV2K test and valid datasets, the average increment of PSNR is greater than 1dB and the related average increment of SSIM is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Keywords—closed-loop, negative feedback, Swin2SR, CSwin2SR, compressed image, super-resolution I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' INTRODUCTION With the exponential growth of realistic and virtual image data, it is necessary to compress the original image for the purpose of highly-efficient image storage, transmission, processing, analysis, understanding and some emergent applications, such as augment reality and metaverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' In order to gain a high ratio of image compression, lossy image compression methods are preferentially considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' However, lossy image compression will unavoidably lead to the distortion to the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Hence, there is a urgent demand for compressed image restoration, including compressed image denoising, deblurring, deblocking, inpainting, artifact removing and super-resolution (SR) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Classical compressed image SR methods utilized an open- close architecture to obtain a high-quality image from a low- quality image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Closed-loop negative feedback framework is broadly used in automatic control systems and attains outstanding static and dynamic performance [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' For the sake of further lifting the capability of existing open-loop compressed image SR approaches, a closed-loop structure is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Shifted windows V2 transformer for compressed image super-resolution and restoration (Swin2SR) is one of the state- of-the-art algorithms for compressed image SR [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Swin2SR is also one of the top solutions at the advances in image manipulation challenge on SR of compressed image and video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' This paper attempts to enhance the image recovery performance of Swin2SR by forming a closed-loop architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The new algorithm is named as circular Swin2SR (CSwin2SR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' To the best of our knowledge, there are no previous work on closed-loop compressed image SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The main contributions of this paper are listed as follows: (1) a serial Swin2SR architecture is proposed by adding down- sampling and compression units into the traditional Swin2SR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' (2) a circular Swin2SR architecture is proposed by building a closed-loop negative feedback mechanism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' (3) dramatic performance improvement of CSwin2SR is gained compared with the conventional Swin2SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The remainder of this paper is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Section 2 is related work, section 3 is theoretical basis, section 4 is simulation experiments, and section 5 is final conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' RELATED WORK Compressed image SR is one of the challenge branches of image restoration [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' It can be classified into two categories: model-based method and learning-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Model- based method wields hand-crafted priors and learning-based method wields data-driven priors [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Currently, learning- based method is the mainstream direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Learning-based method makes use of convolutional neural network (CNN) and vision transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Compared with classical CNN, emerging vision transformer has the advantage of long-range attention and achieves better image SR performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Shifted windows (Swin) transformer is a hierarchical general-purpose baseline for computer vision [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Swin transformer acquires greater efficiency by confining self- attention to non-overlapping local windows while permitting cross-window linking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' SwinV2 transformer reaches excellent performance in image classification, object detection and semantic segmentation by scaling up capacity and resolution [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' SwinV2 transformer puts forward some innovations: a residual post normalization algorithm and a scaled cosine attention algorithm to strengthen the stability of large vision transformer models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' a log-spaced continuous position bias algorithm to effectively transfer vision transformer models at low-resolution images and windows to their high-resolution versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Swin transformer image restoration (SwinIR) is the pioneering research on image SR via Swin transformer [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' SwinIR consists of three subparts: shallow subnetwork, deep subnetwork and reconstruction subnetwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Swin2SR is the upgradation of SwinIR through SwinV2 transformer [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Compressed image SR can also be divided into two categories: known-compression method and unknown- compression method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' For the known-compression method, the image compression algorithm and its parameters are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' For instance, lossy JPEG image compression algorithm is employed and its parameter of compression quality is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' For unknown-compression method, the compression algorithm or its parameters is indeterminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' It is more difficult to resolve unknown-compression method than known-compression method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' This paper focuses on the known-compression method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The proposed method fully takes advantage of the determinate compression algorithm and its parameters to establish a circular structure for upgrading the SR reconstruction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' It should be mentioned that local self-attention negative- feedback module has already been adopted for image SR [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' However, this module does not build a closed-loop architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' THEORY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Improved Open-Loop Swin2SR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Architecture of improved open-loop Swin2SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The architecture of the improved open-loop Swin2SR is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' It comprises three serial units: down- sampling (DS), compression (CP) and super-resolution (SR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The SR unit includes three subunits: shallow feature extraction (SF), deep feature extraction (DF) and reconstruction (RC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The DF subunit consists of multiple residual SwinV2 transformer blocks (RSTB) and a convolution block (CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' xh0 is the original high-resolution image, xd0 is the down-sampling low-resolution image, xc0 is the compressed low-resolution image, and xs0 the reconstruction high-resolution image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Compared with the classical Swin2SR, DS and CP units are extraordinarily added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The introduction of the DS unit is due to the reasonable assumption that a low-resolution image before compression can be regarded as a down-sampling version of a high-resolution image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Although the DS and CP units are implicitly utilized in the classical Swin2SR, they are not obviously analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The DS unit can adopt any traditional down-sampling approaches, such as nearest neighbor and interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The CP unit can employ any existing lossy image compression standards, such as JPEG and WebP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The DS and CP units are designed for training SR unit and evaluating SR recovery performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The improved Swin2SR can de described by following mathematical equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' ( ) ( ) ( ) ( ) ( ) ( ) d0 h0 c0 d0 s0 c0 s0 h0 D d h0 s0 d0 c0 DS , CP , SR SR CP DS , R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' , R = = = = \uf0ce \uf0ce x x x x x x x x x x x x \uf02c (\uf031) where: D is the dimension of high-resolution image and d is the dimension of low-resolution image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Proposed Closed-Loop CSwin2SR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Framework of proposed closed-loop CSwin2SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The framework of the proposed closed-loop CSwin2SR is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' It is composed of two elements: the top half is a serial Swin2SR and the bottom half is a circular Swin2SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The serial Swin2SR obtains the initial high-quality reconstruction xs0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' the circular Swin2SR obtains the enhanced high-quality reconstruction xh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The circular Swin2SR comprises six modules: DS, CP, SR, two summators and multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' It constructs a closed-loop negative feedback system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' xd is the down-sampling version of xh, xc is the compression version of xd, xr the reconstruction version of xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' xe is the error vector and xc is the control vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The circular Swin2SR can be depicted by following mathematical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( \uf05d d h c d s c e s0 s c e h h c h h s0 h D d h s e c d c DS , CP , SR , λ λ SR CP DS , , , R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' , R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='λ 0,1 = = = = − = \uf0ac + \uf0ac + − \uf0ce \uf0ce \uf0ce x x x x x x x x x x x x x x x x x x x x x x x x \uf02c (\uf032) where: λ is a small constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' symbol ‘←’ means iterative updating;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' the initial value of xh can be any choices, such as zero vector, random vector or initial reconstruction xs0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' According to the theory of negative feedback, the error vector is close to zero when the circular Swin2SR system reaches steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Therefore, xs approaches xs0 and xh approaches xh0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Ultimately, the perfect high-resolution reconstruction xh is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The whole procedure can be expressed by the following mathematical formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' ( ) ( ) ( ) ( ) ( ) ( ) e e s0 s s s0 s h s0 h0 h h0 0 SR CP DS , SR CP DS → = − \uf0de → = = \uf0de → x x x x x x x x x x x x \uf02c (\uf033) DS CP SR SF RC DF RSTB CONV xh0 xd0 xc0 xs0 xc0 xs0 DS CP SR xh0 xd0 xc0 xs0 DS CP SR xh xd xc xs λ + − + + xe xc xe xc xh The proposed CSwin2SR algorithm can be described by following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The proposed CSwin2SR algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Experimental Conditions The experimental section is designed to evaluate the reconstruction capability of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The experimental hardware platform is NVIDIA GPUs performing on the cloud and the experimental software platform is Pytorch running on Linux operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The open-source codes and corresponding pretrained models of Swin2SR is employed (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='com/mv-lab/swin2sr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The DIV2K test and valid datasets are utilized and both datasets hold 100 high-resolution colorful images with size form 1320×2040 to 1368×2040 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The lossy image compression algorithm is JPEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The performance metrics include peak signal-to-noise ratio (PSNR) and structural similarity (SSIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' For the purpose of computing PSNR and SSIM, the low-resolution image is up-sampled to the same size as the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The experimental parameters are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' EXPERIMENTAL PARAMETERS Symbol Name Value λ Iterative constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='1 N Total number of iteration 10 q JPEG compression quality 10% dr Downsampling rate 4 sr Super-resolution factor 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Experimental Results Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 2 and 3 summarize the average PSNR, PSNR-Y, SSIM and SSIM-Y on DIV2K test and valid datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' PSNR- Y and SSIM-Y denote only the illuminance component is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The increment of average PSNR and PSNR-Y is larger than 1dB and the related increment of average SSIM and SSIM-Y is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Compared with classical Swin2SR, the proposed CSwin2SR efficiently and hugely improves the capability of super-resolution reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 4, 5 and 6 illustrate exemplar images for comparing the proposed CSwin2SR and the classical Swin2SR, where subfigure (1) is the original high-resolution image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' subfigure (2) is the compressed low-resolution image of JPEG which is zoomed up to the same size as the original image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' subfigure (3) is the high-resolution reconstruction image of classical Swin2SR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' subfigure (4) is the enhanced high-resolution reconstruction image of propose CSwin2SR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' subfigure (5) is the enlarged image patch which is marked with red box in the original image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' subfigure (6) is the enlarged image patch of the compressed image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' subfigure (7) is the enlarged image patch of Swin2SR reconstruction image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' subfigure (8) is the enlarged image patch of CSwin2SR reconstruction image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' subfigure (9) is the image patch of absolute difference between subfigure (7) and subfigure (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' subfigure (10) is the image patch of absolute difference between subfigure (8) and subfigure (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Because it is uneasy for human visual system to discriminate the reconstruction image patches of Swin2SR and CSwin2SR, the image patch of absolute difference is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The absolute difference demonstrates that the proposed CSwin2SR has smaller reconstruction error than the classical Swin2SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' In a word, the proposed CSwin2SR is superior to the classical Swin2SR in the performance of SR reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' According to subfigure (9) and (10) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 4, 5 and 6, the lossy JPEG compression results in horizontal, vertical and diagonal shifts of pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The proposed CSwin2R can more efficiently eliminate the shifts than the traditional Swin2SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' It should be remarkable that the PSNR increment of image patch in subfigure (8) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 4 compared with that of subfigure (7) is greater than 3dB and the related SSIM increment is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The PSNR increment of image patch in subfigure (8) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 5 compared with that of subfigure (7) is close to 3dB and the related SSIM increment is close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The PSNR increment of image patch in subfigure (8) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 6 compared with that of subfigure (7) is greater than 3dB and the related SSIM increment is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' It powerfully demonstrates the outstanding image SR reconstruction capability of the proposed CSwin2SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' It should be further remarkable that the image patches in subfigure (6) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 4 and 5 have higher PSNR than those in subfigure (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Nevertheless, the patches in subfigure (6) hold lower SSIM than those in subfigure (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' EXPERIMENTAL RESULTS ON DIV2K TEST DATASET Method Performance Metrics PSNR(dB) PSNR-Y(dB) SSIM SSIM-Y SwinSR 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='15 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9448 CSwin2SR 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='27 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9510 Increment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='0062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='0062 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' EXPERIMENTAL RESULTS ON DIV2K VALID DATASET Method Performance Metrics PSNR(dB) PSNR-Y(dB) SSIM SSIM-Y SwinSR 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='93 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9427 CSwin2SR 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='10 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9492 Increment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='0065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='0065 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' CONCLUSIONS This paper presents a closed-loop CSwin2R method for compressed image SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' It consist of a serial Swin2SR and a circular Swin2SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The serial Swin2SR is an improvement of classical Swin2SR by supplementing a down-sampling unit and a compression unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The circular Swin2SR is a closed-loop architecture by introducing a negative feedback mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' It is verified by the experimental results that the proposed CSwin2SR is superior to the classical Swin2SR in the capability of image SR reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' The proposed Algorithm: CSwin2SR Input: Xco Initialization: 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='1, n = 1, Xso = SR(Xco), Xh = Xs0 while n <= 10 Xn = Xn + >(Xso-SR(CP(DS(xn) n =n+1 end Output: Xh (1) Original (2) Compressed (PSNR=21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='84, SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='7880) (3) Swin2SR (PSNR=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='11, SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='8828) (4) CSwin2SR (PSNR=24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='07, SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='8982) (5) Original (6)Compressed(PSNR=15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='69,SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='5185) (7)Swin2SR(PSNR=15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='20, SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='5893) (8)CSwin2SR(PSNR=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='27, SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='6110) (9) Swin2SR (10) CSwin2SR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Comparison between Swin2SR and CSwin2SR (zebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' (1) Original (2) Compressed (PSNR=26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='34, SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='8383) (3) Swin2SR (PSNR=28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='52, SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9494) (4) CSwin2SR (PSNR=30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='29, SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9508) (5) Original (6)Compressed(PSNR=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='46,SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='6873) (7)Swin2SR(PSNR=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='36,SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='8092) (8)CSwin2SR(PSNR=25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='33,SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='8574) (9) Swin2SR (10) CSwin2SR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Comparison between Swin2SR and CSwin2SR (wheel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' (1) Original (2) Compressed (PSNR=27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='50, SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9011) (3) Swin2SR (PSNR=29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='72, SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9681) (4) CSwin2SR (PSNR=31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='18, SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9726) (5)Original (6)Compressed(PSNR=26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='04,SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9468) (7)Swi2SR(PSNR=26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='87,SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9695) (8)CSwin2SR(PSNR=30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='73,SSIM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='9860) (9) Swin2SR (10) CSwin2SR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' Comparison between Swin2SR and CSwin2SR (desktop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' CSwin2SR can efficiently remove pixel shifts in the reconstruction image of the classical Swin2SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' On DIV2K test and valid datasets, the increment of average PSNR is greater than 1dB and the related increment of SSIM is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' In our future work, the theory basis of CSwin2SR will be deeply studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' More SR factors, compression algorithms, image datasets will also be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' ACKNOWLEDGMENT We would very much like to thank the authors of Swin2SR for selflessly sharing their open-source codes on Microsoft GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} +page_content=' We also express deep thanks to Google COLAB for free GPU computing service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9FAT4oBgHgl3EQf8R46/content/2301.08749v1.pdf'} 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sha256:fc5aa761f293cd6ca9a977a83f5678db5bd879086c3e25f5913f1dfcbcaec1bb +size 228121 diff --git a/lNAyT4oBgHgl3EQfyflP/content/tmp_files/2301.00684v1.pdf.txt b/lNAyT4oBgHgl3EQfyflP/content/tmp_files/2301.00684v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f12d4d1241f829c7bea04bd5e17b69321c07461 --- /dev/null +++ b/lNAyT4oBgHgl3EQfyflP/content/tmp_files/2301.00684v1.pdf.txt @@ -0,0 +1,312 @@ +arXiv:2301.00684v1 [gr-qc] 27 Dec 2022 +Proceedings of RAGtime #24, 10–14 October, 2022, Opava, Czech Republic +1 +S. Hled´ık and Z. Stuchl´ık, editors, Silesian University in Opava, CZ, pp. 1–5 +Astrophysical black holes embedded in +organized magnetic fields +Case of a nonvanishing electric charge +Vladim´ır Karas +1Astronomical Institute, Czech Academy of Sciences, Boˇcn´ı II 1401, +CZ-14100 Prague, Czech Republic +E-mail: vladimir.karas@asu.cas.cz +ABSTRACT +Large scale magnetic fields pervade the cosmic environment where the as- +trophysical black holes are often embedded and influenced by the mutual +interaction. In this lecture, we outline the appropriate mathematical frame- +work to describe magnetized black holes within General Relativity and we +show several examples how these can be employed in the astrophysical con- +text. In particular, we examine the magnetized black hole metric in terms +of an exact solution of electro-vacuum Einstein-Maxwell equations under +the influence of a non-vanishing electric charge. New effects emerge: the +expulsion of the magnetic flux out of the black-hole horizon depends on the +intensity of the imposed magnetic field. +Keywords: +Black holes – Electromagnetic fields – General relativity +1 +INTRODUCTION +Astrophysical black holes are cosmic objects that can be mathematically described +by a set of Einstein-Maxwell equations (e.g. Romero & Vila 2014 [10]). Various +formulations of the Uniqueness Theorems express in a rigorous way the conditions +under which the black hole solutions exist and they constrain the parameter space +that is necessary to specify different cases (Wald 1984 [13]). It turns out that clas- +sical black holes are described by a small number of such parameters, in particular, +the mass, electric (or magnetic) charge, and angular momentum (spin). Black holes +do not support their own magnetic field except the gravito-magnetically induced +components in the rotating, charged Kerr-Newman metric. +However, astrophysical black holes are embedded in a magnetic field of external +origin, which then interacts with the internal properties of the black hole (Ruffini +& Wilson [11]). In the case of very strong magnetic intensity, the magnetic field +even contributes to the spacetime metric. In the present contribution we examine +interesting properties of such an electrically charged, magnetized, rotating black +NO-ISBN-SET-X © CZ – SU in Opava. All rights reserved. + +2 +V. Karas +hole. +To this end we employ the solution originally derived in 1970s by means +of Ernst magnetization techniques (Ernst & Wild 1976 [3]) and demonstrate its +interesting features in terms of magnetic flux threading different regions of the +black hole horizon or an entire hemisphere (see Biˇc´ak & Hejda 2015 [1], and further +references cited therein). +We limit our discussion to axially symmetric and stationary solutions. These are +vacuum, asymptotically non-flat solutions, where the influence of plasma is ignored +but the effects of strong gravity are taken into account. +2 +MAGNETIZED BLACK HOLES WITH SPIN AND CHARGE +We can write the system of mutually coupled, Einstein-Maxwell partial differential +equations (Chandrasekhar 1983 [2]), +Rµν − 1 +2Rgµν = 8πTµν, +(1) +where the source term Tµν is of purely electromagnetic origin, +T αβ ≡ T αβ +EMG = 1 +4π +� +F αµF β +µ − 1 +4F µνFµνgαβ +� +, +(2) +T µν;ν = −F µαjα, +F µν ;ν = 4πjµ, +⋆F µν;ν = 4πMµ, +(3) +and ⋆Fµν ≡ 1 +2εµνρσFρσ. We will consider the spacetime solutions for the metric that +satisfies electro-vacuum case with a regular event horizon under the constraints of +axial symmetry and stationarity, +ds2 = f −1 � +e2γ � +dz2 + dρ2� ++ ρ2 dφ2� +− f ( dt − ω dφ)2 , +(4) +with f, ω, and γ being functions of z and ρ only. In the weak electromagnetic +field approximation, the electromagnetic (test) field is supposed to reside in the +background of a rotating black hole, e.g., Kerr metric or a weakly charged Kerr +metric (e.g. Wald 1984 [13], Gal’tsov 1986 [4]). +As an example, in an asymptotically flat spacetime, the axial Killing vector ∂φ +generates a uniform magnetic field, whereas the field vanishes asymptotically for +the time-like Killing vector ∂t. These two solutions are known as the Wald’s field +(Wald 1974 [12]): +F = 1 +2B0 +� +d˜ξ + 2J +M dξ +� +. +(5) +Magnetic flux surfaces are defined, +4πΦM = +� +S +F = const, +(6) +Magnetic and electric (Lorentz) forces are then given by +m ˙u = qm⋆F.u, +m ˙u = qeF.u, +(7) + +Charged black holes embedded in organized magnetic fields +3 +and the magnetic field lines (in the axisymmetric case) are determined by +dr +dθ = Br +Bθ +, +(8) +in a perfect analogy with classical electromagnetism. We will employ the above- +given quantities in our discussion further below. +Magnetic (electric) lines of force are defined by the direction of Lorentz force that +acts on electric (magnetic) charges, +duµ +dτ ∝ ⋆F µ +ν uν, +duµ +dτ ∝ F µ +ν uν. +(9) +In an axially symmetric system, the equation for magnetic lines takes a lucid form, +dr +dθ = −Fθφ +Frφ +, +dr +dφ = Fθφ +Frθ +, +(10) +that is again in correspondence with eq. (8). +Let us now turn our attention to the case of strong magnetic field, where we +cannot ignore its influence on the spacetime metric. The latter is not necessarilly +flat in the asymptotical spatial region far from the black hole (Ernst & Wild 1976 +[3]; Karas & Vokrouhlick´y 1990 [9]). +Magnetized Kerr-Newman black hole metric can be expressed in the form (Garc´ıa +D´ıaz 1985 [5]) +ds2 = |Λ|2Σ +� +∆−1 dr2 + dθ2 − ∆A−1 dt2� ++|Λ|−2Σ−1A sin2 θ ( dφ − ω dt)2 , +(11) +Σ = r2 + a2 cos2 θ, ∆ = r2 − 2Mr + a2 + e2, A = (r2 + a2)2 − ∆a2 sin2 θ are +functions from the Kerr-Newman metric. The outer horizon is located at radius +r≡r+ = 1+(1−a2−e2)1/2, like in an unmagnetized case, and the horizon existence +is restricted to the range of parameters a2 + e2 ≤ 1. Let us emphasise that, in the +magnetized case, the traditional Kerr-Newman parameters a and e are not identical +with the black hole total spin and electric charge, as we will see further below. +Moreover, because of asymptotically non-flat nature of the spacetime, the Komar- +type angular momentum and electric charge (as well as the black hole mass) have +to be defined by integration over the horizon sphere rather than at radial infinity. +The magnetization function Λ = 1 + βΦ − 1 +4β2E reads, in terms of the Ernst +potentials Φ(r, θ) and E(r, θ), +ΣΦ = ear sin2 θ − ℑe +� +r2 + a2� +cos θ, +(12) +ΣE = −A sin2 θ − e2 � +a2 + r2 cos2 θ +� ++2ℑa +� +Σ +� +3 − cos2 θ +� ++ a2 sin4 θ − re2 sin2 θ +� +cos θ. +(13) +The corresponding components of the electromagnetic field can be written con- +veniently with respect to orthonormal LNRF components, +H(r) + iE(r) = A−1/2 sin−1θ Φ′ +,θ, +(14) +H(θ) + iE(θ) = − (∆/A)1/2 sin−1θ Φ′ +,r, +(15) + +4 +V. Karas +−1.5 +−1 +−0.5 +0 +0.5 +1 +1.5 +−4 +−3 +−2 +−1 +0 +1 +2 +3 +4 +F +Q +β=0 +β=0 +3 +3 +Figure 1. The “butterfly diagram” shows the magnetic flux of magnetized Kerr-Newman +black hole with a2 + e2 = 1 as a function of the total electric charge Q. Solid curves +correspond to a constant value of the dimensionless magnetization parameter β = BM +(β = 0 is the case of an unmagnetized Kerr-Newman black hole). The area of the plot +with ultra-strong magnetization is bounded by β = 1 (red curve) and emphasized by +yellow colour in the plot. The lines of constant ratio of a/e and varying β are also plotted +(dashed; the cases of a/e = ±0.85 and 0 are shown); some distinctive combinations of the +parameters a, e are emphasized by colour points. +where Φ′(r, θ) = Λ−1 � +Φ − 1 +2βE +� +. The total electric charge QH is +QH = −|Λ0|2 ℑm Φ′ (r+, 0) , +(16) +and the magnetic flux Φm(θ) across a cap placed in an axisymmetric position on +the horizon is +Φm = 2π|Λ0|2 ℜe Φ′ � +r+, ¯θ +���� +θ +¯θ=0, +(17) +where Λ0 = Λ(θ = 0). +Let us note that the span of the azimuthal coordinate in the magnetized solu- +tion must be rescaled by the multiplication factor Λ0 in order to avoid a conical +singularity on the symmetry axis (Hiscock 1981 [6]): +Λ0 = +� +1 + 3 +2β2e2 + 2β3ae + β4 � 1 +16e4 + a2��1/2 . +(18) +This rescaling procedure effectively leads to the increase of the horizon surface area, +and thereby also magnetic flux across the horizon (Karas 1988 [7]). + +Charged black holes embedded in organized magnetic fields +5 +In Figure 1, the magnetic flux across the entire black hole hemisphere in Kerr- +Newman strongly magnetized black hole solution, F = Φm(θ = π/2), is shown as +a function of electric charge on the horizon, Q = QH (additional details in Karas +& Bud´ınov´a 2000 [8]). +Let us note that cases of intersection of the β = const +curves with F = 0 and non-zero total charge, Q ̸= 0 correspond to the vanishing +total angular momentum of the black hole, J = 0. This property is rather different +from the behaviour of weakly magnetized black holes with only test magnetic field +imposed on them. +On the other hand, this exact solution does not allow us to +study the effects of mis-alignment of the magnetic field with respect to the rotation +axis, which is so far possible only in the test-field approximation or by numerical +techniques. +ACKNOWLEDGEMENTS +The author acknowledges continued support from the Czech Science Foundation +EXPRO grant titled “Accreting black holes in the new era of X-ray polarimetry +missions”, No. 21-06825X. +REFERENCES +[1] +Biˇc´ak J., Hejda F. (2015), “Near-horizon description of extremal magnetized sta- +tionary black holes and Meissner effect”, Physical Review D, 92, id.104006 +[2] +Chandrasekhar S. (1983), The Mathematical Theory of Black Holes (Oxford: Oxford +University Press) +[3] +Ernst F. J., and Wild W. J. (1976), “Kerr black holes in a magnetic universe”, J. +Math. Phys., 12, 1845 +[4] +Gal’tsov D. V. (1986), Particles and Fields around Black Holes (Moscow: Moscow +University Press) +[5] +Garc´ıa D´ıaz A. (1985), “Magnetic generalization of the Kerr-Newman metric”, J. +Math. Phys., 26, 155 +[6] +Hiscock W. A. (1981), “On black holes in magnetic universes”, J. Math. Phys., 22, +1828 +[7] +Karas V. (1988), “Magnetic fluxes across black holes. Exact models”, Bulletin of the +Astronomical Institute of Czechoslovakia, 39, 30 +[8] +Karas V., Bud´ınov´a Z. (2000), “Magnetic fluxes across black holes in a strong mag- +netic field regime”, Physica Scripta, 61, 253 +[9] +Karas V., Vokrouhlick´y D. (1990), “On interpretation of the magnetized Kerr- +Newman black hole”, J. Math. Phys., 32, 714 +[10] +Romero G. E., Vila G. S. (2014), Introduction to Black Hole Astrophysics, Lecture +Notes in Physics, vol. 876 (Berlin: Springer) +[11] +Ruffini R., Wilson J. R. (1975), “Relativistic magnetohydrodynamical effects of +plasma accreting into a black hole”, Physical Review D, 12, 2959 +[12] +Wald R. M. (1974), “Black hole in a uniform magnetic field”, Physical Review D, +10, 1680 +[13] +Wald R. M. (1984), General Relativity (Chicago: University of Chicago Press) + + diff --git a/lNAyT4oBgHgl3EQfyflP/content/tmp_files/load_file.txt b/lNAyT4oBgHgl3EQfyflP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..06d4f620d6e984fa775f3cf5865890aec3fcf8d1 --- /dev/null +++ b/lNAyT4oBgHgl3EQfyflP/content/tmp_files/load_file.txt @@ -0,0 +1,121 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf,len=120 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='00684v1 [gr-qc] 27 Dec 2022 Proceedings of RAGtime #24, 10–14 October, 2022, Opava, Czech Republic 1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Hled´ık and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Stuchl´ık, editors, Silesian University in Opava, CZ, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' 1–5 Astrophysical black holes embedded in organized magnetic fields Case of a nonvanishing electric charge Vladim´ır Karas 1Astronomical Institute, Czech Academy of Sciences, Boˇcn´ı II 1401, CZ-14100 Prague, Czech Republic E-mail: vladimir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='karas@asu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='cz ABSTRACT Large scale magnetic fields pervade the cosmic environment where the as- trophysical black holes are often embedded and influenced by the mutual interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' In this lecture, we outline the appropriate mathematical frame- work to describe magnetized black holes within General Relativity and we show several examples how these can be employed in the astrophysical con- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' In particular, we examine the magnetized black hole metric in terms of an exact solution of electro-vacuum Einstein-Maxwell equations under the influence of a non-vanishing electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' New effects emerge: the expulsion of the magnetic flux out of the black-hole horizon depends on the intensity of the imposed magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Keywords: Black holes – Electromagnetic fields – General relativity 1 INTRODUCTION Astrophysical black holes are cosmic objects that can be mathematically described by a set of Einstein-Maxwell equations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Romero & Vila 2014 [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Various formulations of the Uniqueness Theorems express in a rigorous way the conditions under which the black hole solutions exist and they constrain the parameter space that is necessary to specify different cases (Wald 1984 [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' It turns out that clas- sical black holes are described by a small number of such parameters, in particular, the mass, electric (or magnetic) charge, and angular momentum (spin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Black holes do not support their own magnetic field except the gravito-magnetically induced components in the rotating, charged Kerr-Newman metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' However, astrophysical black holes are embedded in a magnetic field of external origin, which then interacts with the internal properties of the black hole (Ruffini & Wilson [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' In the case of very strong magnetic intensity, the magnetic field even contributes to the spacetime metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' In the present contribution we examine interesting properties of such an electrically charged, magnetized, rotating black NO-ISBN-SET-X © CZ – SU in Opava.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' 2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Karas hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' To this end we employ the solution originally derived in 1970s by means of Ernst magnetization techniques (Ernst & Wild 1976 [3]) and demonstrate its interesting features in terms of magnetic flux threading different regions of the black hole horizon or an entire hemisphere (see Biˇc´ak & Hejda 2015 [1], and further references cited therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' We limit our discussion to axially symmetric and stationary solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' These are vacuum, asymptotically non-flat solutions, where the influence of plasma is ignored but the effects of strong gravity are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' 2 MAGNETIZED BLACK HOLES WITH SPIN AND CHARGE We can write the system of mutually coupled, Einstein-Maxwell partial differential equations (Chandrasekhar 1983 [2]), Rµν − 1 2Rgµν = 8πTµν, (1) where the source term Tµν is of purely electromagnetic origin, T αβ ≡ T αβ EMG = 1 4π � F αµF β µ − 1 4F µνFµνgαβ � , (2) T µν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='ν = −F µαjα, F µν ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='ν = 4πjµ, ⋆F µν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='ν = 4πMµ, (3) and ⋆Fµν ≡ 1 2εµνρσFρσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' We will consider the spacetime solutions for the metric that satisfies electro-vacuum case with a regular event horizon under the constraints of axial symmetry and stationarity, ds2 = f −1 � e2γ � dz2 + dρ2� + ρ2 dφ2� − f ( dt − ω dφ)2 , (4) with f, ω, and γ being functions of z and ρ only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' In the weak electromagnetic field approximation, the electromagnetic (test) field is supposed to reside in the background of a rotating black hole, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=', Kerr metric or a weakly charged Kerr metric (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Wald 1984 [13], Gal’tsov 1986 [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' As an example, in an asymptotically flat spacetime, the axial Killing vector ∂φ generates a uniform magnetic field, whereas the field vanishes asymptotically for the time-like Killing vector ∂t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' These two solutions are known as the Wald’s field (Wald 1974 [12]): F = 1 2B0 � d˜ξ + 2J M dξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' (5) Magnetic flux surfaces are defined, 4πΦM = � S F = const, (6) Magnetic and electric (Lorentz) forces are then given by m ˙u = qm⋆F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='u, m ˙u = qeF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='u, (7) Charged black holes embedded in organized magnetic fields 3 and the magnetic field lines (in the axisymmetric case) are determined by dr dθ = Br Bθ , (8) in a perfect analogy with classical electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' We will employ the above- given quantities in our discussion further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Magnetic (electric) lines of force are defined by the direction of Lorentz force that acts on electric (magnetic) charges, duµ dτ ∝ ⋆F µ ν uν, duµ dτ ∝ F µ ν uν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' (9) In an axially symmetric system, the equation for magnetic lines takes a lucid form, dr dθ = −Fθφ Frφ , dr dφ = Fθφ Frθ , (10) that is again in correspondence with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Let us now turn our attention to the case of strong magnetic field, where we cannot ignore its influence on the spacetime metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' The latter is not necessarilly flat in the asymptotical spatial region far from the black hole (Ernst & Wild 1976 [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Karas & Vokrouhlick´y 1990 [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Magnetized Kerr-Newman black hole metric can be expressed in the form (Garc´ıa D´ıaz 1985 [5]) ds2 = |Λ|2Σ � ∆−1 dr2 + dθ2 − ∆A−1 dt2� +|Λ|−2Σ−1A sin2 θ ( dφ − ω dt)2 , (11) Σ = r2 + a2 cos2 θ, ∆ = r2 − 2Mr + a2 + e2, A = (r2 + a2)2 − ∆a2 sin2 θ are functions from the Kerr-Newman metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' The outer horizon is located at radius r≡r+ = 1+(1−a2−e2)1/2, like in an unmagnetized case, and the horizon existence is restricted to the range of parameters a2 + e2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Let us emphasise that, in the magnetized case, the traditional Kerr-Newman parameters a and e are not identical with the black hole total spin and electric charge, as we will see further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Moreover, because of asymptotically non-flat nature of the spacetime, the Komar- type angular momentum and electric charge (as well as the black hole mass) have to be defined by integration over the horizon sphere rather than at radial infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' The magnetization function Λ = 1 + βΦ − 1 4β2E reads, in terms of the Ernst potentials Φ(r, θ) and E(r, θ), ΣΦ = ear sin2 θ − ℑe � r2 + a2� cos θ, (12) ΣE = −A sin2 θ − e2 � a2 + r2 cos2 θ � +2ℑa � Σ � 3 − cos2 θ � + a2 sin4 θ − re2 sin2 θ � cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' (13) The corresponding components of the electromagnetic field can be written con- veniently with respect to orthonormal LNRF components, H(r) + iE(r) = A−1/2 sin−1θ Φ′ ,θ, (14) H(θ) + iE(θ) = − (∆/A)1/2 sin−1θ Φ′ ,r, (15) 4 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Karas −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='5 −4 −3 −2 −1 0 1 2 3 4 F Q β=0 β=0 3 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' The “butterfly diagram” shows the magnetic flux of magnetized Kerr-Newman black hole with a2 + e2 = 1 as a function of the total electric charge Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Solid curves correspond to a constant value of the dimensionless magnetization parameter β = BM (β = 0 is the case of an unmagnetized Kerr-Newman black hole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' The area of the plot with ultra-strong magnetization is bounded by β = 1 (red curve) and emphasized by yellow colour in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' The lines of constant ratio of a/e and varying β are also plotted (dashed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' the cases of a/e = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content='85 and 0 are shown);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' some distinctive combinations of the parameters a, e are emphasized by colour points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' where Φ′(r, θ) = Λ−1 � Φ − 1 2βE � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' The total electric charge QH is QH = −|Λ0|2 ℑm Φ′ (r+, 0) , (16) and the magnetic flux Φm(θ) across a cap placed in an axisymmetric position on the horizon is Φm = 2π|Λ0|2 ℜe Φ′ � r+, ¯θ ���� θ ¯θ=0, (17) where Λ0 = Λ(θ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Let us note that the span of the azimuthal coordinate in the magnetized solu- tion must be rescaled by the multiplication factor Λ0 in order to avoid a conical singularity on the symmetry axis (Hiscock 1981 [6]): Λ0 = � 1 + 3 2β2e2 + 2β3ae + β4 � 1 16e4 + a2��1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' (18) This rescaling procedure effectively leads to the increase of the horizon surface area, and thereby also magnetic flux across the horizon (Karas 1988 [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Charged black holes embedded in organized magnetic fields 5 In Figure 1, the magnetic flux across the entire black hole hemisphere in Kerr- Newman strongly magnetized black hole solution, F = Φm(θ = π/2), is shown as a function of electric charge on the horizon, Q = QH (additional details in Karas & Bud´ınov´a 2000 [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' Let us note that cases of intersection of the β = const curves with F = 0 and non-zero total charge, Q ̸= 0 correspond to the vanishing total angular momentum of the black hole, J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' This property is rather different from the behaviour of weakly magnetized black holes with only test magnetic field imposed on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' On the other hand, this exact solution does not allow us to study the effects of mis-alignment of the magnetic field with respect to the rotation axis, which is so far possible only in the test-field approximation or by numerical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The author acknowledges continued support from the Czech Science Foundation EXPRO grant titled “Accreting black holes in the new era of X-ray polarimetry 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfyflP/content/2301.00684v1.pdf'} diff --git a/mNE2T4oBgHgl3EQfJAb4/content/tmp_files/2301.03688v1.pdf.txt b/mNE2T4oBgHgl3EQfJAb4/content/tmp_files/2301.03688v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad3c59962621cdcf7715bd6dbfdb8650f9deb525 --- /dev/null +++ b/mNE2T4oBgHgl3EQfJAb4/content/tmp_files/2301.03688v1.pdf.txt @@ -0,0 +1,3199 @@ +arXiv:2301.03688v1 [math.AP] 9 Jan 2023 +SIGN-CHANGING SOLUTIONS FOR THE SINH–POISSON +EQUATION WITH ROBIN BOUNDARY CONDITION +PABLO FIGUEROA, LEONELO ITURRIAGA, AND ERWIN TOPP +Abstract. Given ǫ ∈ (0, 1) and λ > 1, we address the existence of +solutions for the Sinh-Poisson equation with Robin boundary value con- +dition +� +∆u + ǫ2(eu − e−u) = 0 +in Ω +∂u +∂ν + λu = 0 +on ∂Ω, +where Ω ⊂ R2 is a bounded smooth domain. We prove two existence +results under a suitable relation between ǫ small and λ large. When Ω +is symmetric with respect to an axis, we prove the existence of a family +of solutions uǫ,λ concentrating at two points with different spin, both +located on the symmetry line and close to the boundary. In the second +result, we assume Ω is not simply connected and we construct sign- +changing solutions concentrating at points located close to the boundary, +each of them on a different connected component of the boundary. +1. Introduction +Let Ω ⊂ R2 a bounded domain with smooth boundary. In this paper we +are interested in study of the singular perturbation problem +(1.1) +� ∆u + ǫ2(eu − e−u) += 0 +in Ω, +Rλu := ∂u +∂ν + λu += 0 +on ∂Ω, +where ν(x) denotes the exterior unit normal on x ∈ ∂Ω, and ǫ, λ are positive +parameters. Our main concern is the construction of sign-changing solutions +when ǫ is small, and simultaneously, λ is large. +Robin boundary condition (also known as boundary condition of the third +type) can be seen as a combination of Neumann and Dirichlet boundary +condition, in a proportion cast by λ. In [4], Berestycki and Wei study con- +centration phenomena for the least energy solution of equations of Ni-Takagi +type with Robin boundary condition. In fact, they prove the existence of +λ∗ such that, for λ ≥ λ∗, the problem resembles the behavior of the Dirich- +let problem studied (c.f. [31]), meanwhile for λ < λ∗ problem is closer to +Date: January 11, 2023. +2020 Mathematics Subject Classification. 35J25, 35B25, 35B38. +Key words and phrases. Concentrating solutions, sinh-Poisson equation, Robin bound- +ary condition, Lyapunov-Schmidt reduction. +1 + +2 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +the one with Neumann boundary condition (c.f. [30]). In terms of applica- +tions, Robin boundary condition is considered in biological models [16], and +thermal conductivity [10]. +If we formally take the limit λ → ∞ in (1.1), the problem becomes +� +∆u + ǫ2(eu − e−u) = 0, +in Ω, +u = 0, +on ∂Ω. +(1.2) +This equation is tipically referred as sinh-Poisson equation with Dirich- +let boundary condition, and it has connection with the description of two- +dimensional turbulent Euler flows, see Onsager [32, 23, 28, 6, 27] for a phys- +ical discussion of this problem. Roughly speaking, the location of vortices +in the flow can be described through concentration points of the solution +to (1.2). +In this respect, Bartolucci and Pistoia [2] prove that for every m ∈ N, +there exists a solution to problem (1.2) that concentrates around stable +critical points of the Hamiltonian associated to the Dirichlet problem in Ω, +given by +(1.3) +ϕm(ξ1, ..., ξm) = +m +� +i=1 +H(ξi, ξi) + +� +i̸=j +aiajG(ξi, ξj), +(ξ1, ..., ξk) ∈ Ωk, +as ǫ → 0. Here, ai ∈ {−1, 1} for all i = 1, ..., m determine the spin config- +uration of the concentrating points; G is the Green function with Dirichlet +boundary condition +� +−∆xG(x, y) = 8πδy(x) +in Ω, +G(x, y) = 0 +on ∂Ω, +and H is the regular part of the Green function, also referred as the associ- +ated Robin function. Moreover, it is proven in [2] that +uǫ(x) → 8π +k +� +i=1 +aiG(x, ξi) +in C1,σ(¯Ω \ {ξ1, ..., ξm}), as ǫ → 0, +and the solution concentrates each ξi with a sign determined by ai. Concen- +tration points are away to the boundary in consonance with the Dirichlet +condition. +Positive concentrating solutions, concentrating around critical points of a +Hamiltonian function draws back to the early nineties in the seminal work of +Nagasaki and Suzuki [29] for the Liouville equation (namely, when eu − e−u +is replaced by eu in (1.2)) with Dirichlet boundary condition, and it is shown +the effect of the domain determines the existence of concentration, and also +the location of the blow-up points. In the case of the sinh-Poisson equa- +tion, negative concentration points are allowed . In [2], the authors provide +the existence of solutions with two concentration points with different sign, +provided Ω is axially symmetric. This result was later extended by Bartsch, +Pistoia and Weth in [3], where the authors prove the existence of solutions + +SIGN-CHANGING SOLUTIONS +3 +with an arbitrary number of concentrating points, located in the symme- +try axis and alternating sign. For general domains, they prove the result +for 3 and 4 concentrating points, under suitable configuration of the spins. +We also would like to mention the contributions in [14, 15] concerning con- +struction of solutions to equations with nonlinearities of exponential type +in dimension two, in [18, 20] for compact Riemann surfaces, and in [13] for +fractional equations. +Concerning Robin boundary condition, in [12] the authors address the +Liouville equation with Robin boundary condition +(1.4) +� +∆u + ǫ2eu += 0 +inΩ, +Rλu += 0 +on ∂Ω, +and prove the existence of positive solutions with arbitrary number of con- +centration points, located at critical points of the Hamiltonian function +(1.5) +ϕm(ξ) = +m +� +i=1 +Hλ(ξi, ξi) + +m +� +i̸=j +Gλ(ξi, ξj), +ξ = (ξ1, ..., m) ∈ Ωm, +where, this time, Gλ is the Green function +(1.6) +� +−∆Gλ(x, y) += 8πδy(x) +x ∈ Ω +Rλ(Gλ(·, y)) += 0 +on ∂Ω, +and in this case, Hλ is the associated Robin function, defined as +(1.7) +Hλ(x, y) = Gλ(x, y) − Γ(x − y), +x, y ∈ Ω, +where Γ(x−y) = −4 log |x−y| is the fundamental solution for the Laplacian +in the plane. The analysis in [12] (see also [7, 33]) uses in a crucial way the +asymptotic behavior Hλ when λ is large. Accurate asymptotics can be found +in D´avila, Kowalczyk and Montenegro in [11], showing that the behavior of +the map x �→ Hλ(x, x) in the normal direction to the boundary develops a +strict minima at distance of order O(λ−1) to ∂Ω. Thus, critical points for +ϕm in (1.5) can be found for tuples (ξ1, ..., ξm) such that each ξi is sufficiently +close to the boundary, and away each other. In fact, two types of different +solutions can be found, one associated to minima of ϕm, and the other +associated to a critical point of linking-type. +Our approach follows several of the concepts discussed above. +In the +first main result, we rely on axial symmetry just as in [2, 3] to conclude +the existence of sign-changing, two-point concentrating solution in a certain +regime of ǫ small and λ large. +Theorem 1.1. Assume that Ω ⊂ R2 is bounded domain with smooth bound- +ary, and symmetric with respect to the axis x. Denote (a, b) = Ω ∩ R × {0}. +Then, for each α > 16, there exist λ0 > 1, ǫ0 ∈ (0, 1) such that, for every +λ ≥ λ0 and ǫ satisfying ǫλα ≤ ǫ0, problem (1.1) has a solution uǫ,λ that +concentrates with different sign at two points ξi = (ti, 0), i = 1, 2, with +|t1 − a|, |t2 − b| = O(λ−1). + +4 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +Here, by concentration at a point ξ ∈ Ω we mean that for all δ ∈ (0, 1), +supBδ(ξ) |uǫ,λ| → +∞ as ǫ → 0, λ → ∞. Notice that by the invariance of the +problem, if uǫ,λ is solution, −uǫ,λ is also a solution. +The proof of the previous theorem rely on Lyapunov-Schmidt reduction, +and most of the technical arguments are performed over the equation (2.17), +equivalent to (1.1). Our approach in the construction of approximate solu- +tion are rather close to those present in [12, 17, 15]. We use a two-parameter +family of entire solutions of the Liouville equation R2 as the first ansatz, in +junction with smooth correctors that allows us to satisfy the boundary con- +dition. The reduced problem corresponds to that of adjusting variationally +the location of the concentrating points, as critical points of an energy func- +tional associated to the weak formulation of the problem. In a saturation +regime of the parameters, that functional is close to the Hamiltonian ϕm, +which is given by +(1.8) +ϕm(ξ) = +m +� +i=1 +Hλ(ξi, ξi) + +m +� +i̸=j +aiajGλ(ξi, ξj), +ξ = (ξ1, ..., m) ∈ Ωm, +with ai ∈ {−1, 1} for i = 1, .., m (m = 2 in the setting of Theorem 1.1). +Notice that in contrast with (1.5), this time the contribution of the Green +function maybe unbounded from below for tuples (ξ1, ξ2) ∈ Ω × Ω close to +the diagonal if a1a2 = −1, and this makes more difficult to locate concen- +trating points as minima. The relation among ǫ and λ in the theorem allows +us to control the error terms to get the criticality from the minima of the +Robin function showed in [11], which occurs close to boundary. In partic- +ular, the method leads to a relation among ǫ and λ involving logarithmic +corrections that explains the corresponding hypothesis in Theorem 1.1, see +Proposition (5.2). For simplicity, we state our main result in a simpler form, +at the expense of a non sharp estimate. +Concentration near the boundary is in big contrast with previous results +regarding interior concentration, see for instance [29, 5, 24, 26, 1, 19, 17]. +We believe that by methods similar to the ones presented here, solutions +with interior concentration can be constructed for (1.1), since we expect +to have Hλ(·, ξ) → H(·, ξ) as λ → ∞, locally uniform in Ω, where H is the +regular part of the Green function with Dirichlet boundary condition. These +solution are associated to maxima for the Hamiltonian function. We do not +pursue in this direction and focus on concentration near the boundary. +In our second main result, we are able to replace the symmetry assumption +of the previous theorem by a topological one. +Theorem 1.2. Assume that Ω is not simply connected, and let Γ1, .., Γn, +n ∈ N, be the connected components of ∂Ω, and let k ∈ {1, ..., m}. Then, +for each α > 16 there exist λ0 > 1 and ǫ0 ∈ (0, 1) such that for each λ ≥ λ0 +and each ǫ satisfying ǫλα ≤ ǫ0, problem (1.1) has a sign-changing solution +uǫ,λ that concentrates at points ξ1, ..., ξk ∈ Ω, with dist(ξi, Γi) = O(λ−1) for +all i = 1, ..., k, and where Γi ̸= Γj if i ̸= j. + +SIGN-CHANGING SOLUTIONS +5 +This theorem follows the same strategy of the previous result, where as +before the concentration points are also minima to the Hamiltonian func- +tion. In this case, the uniform distance among connected components of the +boundary of the domain allows us to control the contribution of the Green +function in ϕm, no matter the sign of the interacting points is. In fact, in the +notation of the theorem, we have �n +k=1 2k�n +k +� +different solutions (including +the ones concentrating at one point, and the solutions due to the invariance +of the equation). +It is well-known that the use of index/linking arguments have been suc- +cessful for the study of criticality of smooth functionals, and it has been +employed in problems similar to ours, see for example [17, 1, 12] and its +references. It would be interesting to know if solutions of saddle-point type +could be obtained for this problem, but we did not pursue in this direction. +This paper is organized as follows. In Section 2, describing a first ap- +proximation solution to problem (1.1) and estimating the error. Section 3 is +devoted to perform the finite dimensional reduction. In Section 4 we study +the associated nonlinear problem. Section 5 contains the asymptotic expan- +sion of the reduced energy. Finally, in Section 6 we will prove our main +results. +2. Preliminaries and ansatz for solutions +In this section, we denote d(x) = dist(x, ∂Ω) for each x ∈ Ω. +2.1. Preliminaries about Green and Robin function. We start pro- +viding some estimates for the Green function with Robin boundary condition +Gλ in (1.6), and its regular part Hλ in (1.7). +Lemma 2.1. Let ξ ∈ Ω. +• For each δ > 0 small, there exists C depending on δ and Ω such that +if d(ξ) ≥ δ we have +∥Hλ(·, ξ)∥∞ ≤ C. +• There exists δ ∈ (0, 1) small and CΩ > 0 large depending on the +smoothness of Ω, such that, for all λ large in terms of δ, if (λ log λ)−1 ≤ +d(ξ) ≤ δ, then +(2.1) +− CΩ + 2 log |x − ξ∗| ≤ Hλ(x, ξ) ≤ CΩ, +where ξ∗ ∈ Ωc is the reflection of ξ with respect to the tangent to ∂Ω +supported at the projection of ξ. +• In particular, for any 0 < δ there exists Cδ > 0 such that +(2.2) +∥Gλ(·, ξ)∥L∞(Ω\Bδ(ξ)) ≤ Cδ. +Proof. Let ξ ∈ Ω close to the boundary. Up to rotation and translation, we +can assume that the projection of ξ onto the boundary is the origin, and +ν(0) = −e2. We denote ξ∗ = (0, −d(ξ)). + +6 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +We divide the analysis depending on the distance of ξ to the boundary. +Let ˜δ ∈ (0, 1) to be specified later on. +By definition of Hλ, for each x ∈ ∂Ω we have +RλHλ(x) = Rλ(−Γ) =|x − ξ|−2(x − ξ) · ν(x) + λ log |x − ξ|. +If d(ξ) ≥ ˜δ, then it is easy to see using maximum principle for Robin +boundary condition (see Lemma 2.1 in [12]) that +∥Hλ(·, ξ)∥∞ ≤ C˜δ. +From here, we assume (λ log λ)−1 ≤ d(ξ) ≤ ˜δ. +For x ∈ ∂Ω such that |x − ξ| ≥ ˜δ we have +|RλHλ(x)| ≤ λCΩ, +where CΩ = log max{1, diam(Ω)} + ˜δ−1. +From here, we concentrate on the points x on the boundary such that +|x − ξ| ≤ ˜δ. +We start with upper bounds for RλHλ. If |x − ξ| ≤ λ−1, then +RλHλ(x) ≤ λ log λ + λ log |x − ξ| ≤ 0, +meanwhile, for λ−1 ≤ |x − ξ| ≤ ˜δ we have +RλHλ(x) ≤ λ + λ log(˜δ) ≤ 0, +provided ˜δ ≤ e−1. +From here, using maximum principle for Robin boundary condition, using +CΩ as supersolution, we conclude the upper bound in (2.1). +For the lower bound, we start recalling that for all |x − ξ| ≤ ˜δ on the +boundary, we have |x − ξ| and |x − ξ∗| are comparable, in the sense that +there exists cΩ ∈ (0, 1) such that +cΩ ≤ |x − ξ| +|x − ξ∗| ≤ c−1 +Ω , +x ∈ ∂Ω ∩ B(ξ, 3˜δ). +This is a consequence of the smoothness of the domain. +If |x − ξ| ≥ λ−θ for θ ∈ (1/2, 1), we have +RλHλ ≥ − |x − ξ|−1 + λ log |x − ξ| +≥ − λθ + λ log |x − ξ∗| + λ log cΩ +≥2λ(log |x − ξ∗| + log cΩ). +Tomamos S = log |x − ξ∗| y +RλS = � +x − ξ∗ · ν(x)|x − ξ∗|−1 + λ log |x − ξ∗| ≤ c−θ +Ω λθ + λ log |x − ξ∗| +≤ λ(c−θ +Ω λθ−1 + log |x − ξ∗|) +≤1 +2λ log |x − ξ∗| + +SIGN-CHANGING SOLUTIONS +7 +If |x − ξ| ≤ λ−θ, we use that x can be writen as x = (x′, ψ(x′)) with +ψ(0) = 0, ψ′(0) = 0, ψ′′ bounded, from which +� +(x − ξ) · ν(x) = +x′ψ′ − ψ + d(ξ) +� +1 + (ψ′)2� +|x′|2 + (ψ − d(ξ))2 ≥ 1 +2, +for all λ large enough. This implies that +RλHλ ≥ +cΩ +2|x − ξ∗| + λ log |x − ξ∗| + λ log cΩ. +Thus, we use the function S(x) = 2 log |x−ξ∗|−2 log cΩ, which, in view of +the estimates above, satisfies RλHλ ≥ RλS on ∂Ω. Hence, we use maximum +principle again, to conclude that Hλ ≥ S in Ω. This leads to the lower bound +in (2.1). The remaining estimates can be easily obtained from (2.1) and the +definition of H, G and Γ. This completes the proof. +□ +2.2. Ansatz for the solution. +Reduced problem. In this section we +will construct an approximation of the solution to problem (1.1). Then we +estimate the error of such approximation in a suitable norm. The basic idea +is to consider a parameter µ > 0 and the functions +(2.3) +wµ(x) = log +8µ2 +(µ2 + |x|2)2 , +x ∈ R2 +which are solutions to the Liouville equation in the whole plane +(2.4) +∆u + eu = 0 +in R2. +An actual solution will have an asymptotic profile as ǫ → 0 and λ → +∞ +which resembles these solutions, properly translated and rescaled in terms +of these parameters. Specifically, we choose our scaling parameter as +(2.5) +ρ := ǫ +λ2 . +Given m ∈ N, we consider {µj}m +j=1 ⊂ (0, +∞), {ξj}m +j=1 ⊂ Ω, and for each +j = 1, ..., m, we denote +wj(x) = wµjρ(x − ξj) + 2 log 1 +ǫ = log +8µ2 +j +(µ2 +jρ2 + |x − ξj|2)2 + 2 log ρ +ǫ , +(2.6) +for all x ∈ R2. It is easy to see that for each j the function wj satisfies +∆wj + ǫ2ewj = 0 +in R2. +(2.7) +In order to satisfy the Robin boundary condition we introduce harmonic +functions Hj satisfying +(2.8) +� +−∆Hj += 0 +in Ω, +Rλ(Hj) += −Rλ(wj) +on ∂Ω. +Our single-point ansatz takes the form +(2.9) +Uj(x) = wj(x) + Hj(x), +x ∈ Ω. + +8 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +Using the explicit form of wj in (2.6) together with (2.8), we can use +maximum principle in Lemma 2.1 in [12] over x �→ Hλ(x, ξj) − Hj(x) to +conclude that +Hj(x) + 2 log ρ +ǫ + log(8µ2 +j) − Hλ(x, ξj) = O +�µ2 +jρ2 +d2 +j +� ++ O +� +µ2 +jρ2 +λd3 +j +� +, +uniformly in ¯Ω. Here and in what follows, we have adopted the notation +d(x) = dist(x, ∂Ω) for x ∈ Ω, and dj = d(ξj). +Thus, in view of the definition of wj and the estimate above, we have the +expansion +Uj(x) = Gλ(x, ξj) − 2 log +� +1 + +µ2 +jρ2 +|x − ξj|2 +� ++ O +�µ2 +jρ2 +d2 +j +� +. +Now we provide the first estimates concerning the ansantz when we con- +sider the expected location of the concentration points {ξj}. +Lemma 2.2. Let {ξj}m +j=1 ⊂ Ω, {µj}m +j=1 ⊂ (0, +∞) satisfying +dj ∈ (δλ−1, δ−1λ−1), +(2.10) +µj ∈ (δ, δ−1), +(2.11) +for some δ ∈ (0, 1). Then, for each j = 1, ..., m we have +Hj(x) = Hλ(x, ξj) − log(8µ2 +j) + 4 log λ + O +� +ρ2λ2� +, +(2.12) +uniformly in ¯Ω; and for each K compact subset of ¯Ω \ {ξj}, there exists +CK > 0 such that +(2.13) +|Uj(x) − Gλ(x, ξj)| ≤ CKρ2 + O(λ2ρ2) +in K, +where the O term is independent of K. +Consider {aj}m +j=1 with aj ∈ {−1, 1} for all j. +We introduce the first +approximation of the problem (1.1) as +U(x) := +m +� +j=1 +ajUj(x). +(2.14) +Following the directions of [17], we study the problem in expanded vari- +ables depending on ρ given in (2.5). +For this, we consider the function +v(y) = u(ρy) for y ∈ Ωρ := {ξ ∈ R2 : ρξ ∈ Ω}, so that u is a solution to +(1.1) if and only if v satisfy +(2.15) +� +∆v + ρ2ǫ2(ev − e−v) += 0 +in Ωρ, +Rλρv += 0 +on ∂Ωρ. +We also adopt the notation ξ′ +j = 1 +ρξj, Vj(y) = Uj(ρy) and V (y) = U(ρy) +for y ∈ Ωρ and we look for solutions to (2.15) in the form v = V + φ with +φ : Ωρ → R small in an adequate norm. + +SIGN-CHANGING SOLUTIONS +9 +Thus, considering v = V + φ, problem (2.15) can be equivalently formu- +lated as +(2.16) +� +Lφ += −[R + Λφ + N(φ)] +in Ωρ, +Rλρφ += 0 +on ∂Ωρ, +where +Lφ = ∆φ + Wφ, +with +W = +m +� +j=1 +8µ2 +j +(µ2 +j + |y − ξ′ +j|2)2 , +(2.17) +Λφ = +� +(ǫρ)2(eV + e−V ) − W +� +φ, +(2.18) +N(φ) = (ǫρ)2 � +eV (eφ − φ − 1) − e−V (e−φ + φ − 1) +� +, +(2.19) +R(y) = ∆V + (ǫρ)2(eV − e−V ). +(2.20) +2.3. First estimates for problem (2.16). In what follows, we recall rel- +evant estimates concerning the Robin function Hλ we collect from [11] for +the two-dimensional case. Define +h(θ) = −4 log(2θ) + 8 +� ∞ +0 +e−t log (2θ + t) dt, +(2.21) +v(θ) = −2θ − 4θ +� ∞ +0 +e−2θs +(1 + s)2 ds. +(2.22) +It is possible to see that h : (0, ∞) → R has a unique nodegenerate +minimum θ0 ∈ (0, ∞). +The Robin function obeys the expansion +(2.23) +Hλ(x, x) = −4 log λ + h(λd(x)) + λ−1κ(˜x)v(λd(x)) + O(λ−1−α) +for each x such that a1 ≤ λd(x) ≤ a2 for some constants 0 < a1 < a2, and +all λ > 0 large enough. Here, α ∈ (0, 1), ˜x is the projection of x onto the +boundary, κ(˜x) is the mean curvature of ∂Ω at ˜x, see Lemma 2.1 in [11]. +For reasons that will be made clear later in the next section, for measur- +able functions h : Ωρ → R we introduce the following weighted norm +(2.24) +∥h∥∗ = sup +y∈Ωρ + + +m +� +j=1 +(1 + |y − ξ′ +j|)−2−σ + ρ2 + + +−1 +|h(y)|, +for 0 < σ < 1 fixed. +Lemma 2.3. Let {ξj}m +j=1 ⊂ Ω such that (2.10) holds for some δ > 0, and +assume further that +(2.25) +|ξi − ξj| ≥ δ +for i ̸= j. +Set {µj}m +j=1 as +log(8µ2 +j) = Hλ(ξj, ξj) + +� +i̸=j +aiGλ(ξi, ξj) + 4 log λ. +(2.26) + +10 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +Then, there exists C > 1 just depending on δ such that C−1 ≤ µj ≤ C for +all j = 1, .., m. +Moreover, we have +(2.27) +∥R∥∗ ≤ Cρλ9 log λ = Cǫλ7 log λ, +and +∥Λφ∥∗ ≤ Cρλ9 log λ∥φ∥L∞(Ωρ), +for all φ ∈ L∞(Ωρ). +Proof: The uniform estimates for µj are a consequence of the expansion of +the Robin function (2.23), together with the estimates for the Green function +in Lemma 2.2. +We estimate R by dividing the analysis in different regions. Note that by +definition of Vi and (2.7) we see that for any i = 1, . . . , m +∆Vi(y) = ρ2∆wi(ρy) = −ρ2ǫ2ewi(ρy) = − +8µ2 +i ρ4 +(µ2 +i ρ2 + |ρy − ξi|2)2 . +Case 1: Assume that |y − ξ′ +j| ≥ δ +ρ for all j = 1, ..., m with δ > 0 small but +independent of ǫ, λ. Then, for each j and for y ∈ Ωρ with |y − ξ′ +j| ≥ δ +ρ we +have from (2.7) that +(2.28) +∆Vj(y) = − +8µ2 +jρ4 +(µ2 +jρ2 + |ρy − ξj|2)2 = O(ρ4), +where the O term is uniform if δ > 0 and µj > 0 are fixed. On the other +hand, using the estimate (2.13) we have +max +a=±1{eaVj(y)} ≤ C max +a=±1{eaGλ(ρy,ξj)}. +From Lemma 2.1 we deduce that eVj is uniformly bounded away ξj. This +together with (2.28) leads us to +(2.29) +|R(y)| = |∆V + ρ2ǫ2g(V )| ≤ C(ρ4 + ρ2ǫ2) +in Ωρ \ +m +� +j=1 +Bδ/ρ(ξ′ +j). +Case 2: Now, assume that δ/(ρλ log λ) ≤ |y − ξ′ +j| ≤ δ/ρ for some j ∈ +{1, . . . , m}. As in (2.28) in the previous case, we see that +|∆Vj(y)| = +8µ2 +jρ4 +(µ2 +jρ2 + δ2[λ log λ]−2)2 = O(ρ4λ4 log4 λ) = O(ρ2ǫ2 log λ). +Using again estimates (2.13), (2.1) and the fact that dj = O(λ−1), we +arrive at +eV (y) = eajGλ(ρy,ξj)+� +l̸=j alGλ(ρy,ξl)+O(ρ2λ2 log2 λ). + +SIGN-CHANGING SOLUTIONS +11 +If aj = 1 then we get that +eV (y) = eHλ(ρy,ξj)+O(1) +|ρy − ξj|4 += O +� +λ4 +ρ4|y − ξ′ +j|4 +� +and if aj = −1 then we find that +eV (y) = |ρy − ξj|4e−Hλ(ρy,ξj)+O(1) = O +� +ρ4λ4|y − ξ′ +j|4� +. +Taking into account that aj = 1 becomes −1 in e−V (y) and aj = −1 becomes +1 in e−V (y), a similar argument to estimate e−V (y) lead us to obtain +|ρ2ǫ2g(V (y))| ≤ Cρ2ǫ2 +� +λ4 +ρ4|y − ξ′ +j|4 + ρ4λ4|y − ξ′ +j|4 +� +, +for y ∈ Bδ/ρ(ξ′ +j) \ Bδ/(ρλ log λ)(ξ′ +j) and for some constant just depending on +δ. This estimate allows us to conclude that +(2.30) +|R(y)| ≤ Cǫλ7 log λ +m +� +j=1 +1 +1 + |y − ξ′ +j|3 , +y ∈ Ωρ ∩ (Bδ/ρ(ξ′ +j) \ Bδ/(λρ log λ)(ξ′ +j)), +in view of +ρ2ǫ2 +� +log λ + +λ4 +ρ4|y − ξ′ +j|4 + ρ4λ4|y − ξ′ +j|4 +� +[1 + |y − ξ′ +j|3] ≤ Cǫλ7 log λ +for y ∈ Ωρ ∩ (Bδ/ρ(ξ′ +j) \ Bδ/(λρ log λ)(ξ′ +j)). +Case 3: Finally, assume that |y −ξ′ +j| ≤ δ/(ρλ log λ) for some j ∈ {1, . . . , m}. +In this case, by the condition over the points ξj we have that |ξj − ξi| ≥ δ +when i ̸= j (we can choose ˜δ > δ if necessary) and therefore we can use the +estimates of Case 1 for the portion of R relative to points indexed by i ̸= j. +Note that we get that for j +(2.31) +∆V (y) = −aj +8µ2 +j +(µ2 +j + |y − ξ′ +j|2)2 + O(ρ4), +in the considered region where the O-term depends only on δ and µj. +Now, we deal with the exponential term. We first assume that aj = 1. +Notice that +ρ2ǫ2g(V ) = ρ2ǫ2(ewj(ρy)eHj(ρy)+� +i̸=j aiVi − e−V ). +By estimates (2.12) we see that +eV (y) = ewj(ρy)eHj(ρy)+� +i̸=j aiVi += ewj(ρy) exp +� +Hλ(ρy, ξj) + +� +i̸=j +aiGλ(ρy, ξi) − log(8µ2 +j) ++ 4 log λ + O +�µ2 +jρ2 +d2 +j +�� + +12 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +At this point, we notice that by a first-order Taylor expansion, for y in +this region we have +(2.32) +Hλ(x, ξj) = Hλ(ξj, ξj) + O(λ log λ |x − ξj|), +and for i ̸= j we have +(2.33) +Gλ(x, ξi) = Gλ(ξj, ξi) + O(λ log λ |x − ξj|), +where the O terms are inependent of ǫ and λ. +Using this and assumption (2.26), we arrive at +ρ2ǫ2eV (y) = ρ2ǫ2ewj(ρy) · exp +� +O(λ log λ |ρy − ξj|) + O +�µ2 +jρ2 +d2 +j +�� += +8µ2 +j +(µ2 +j + |y − ξ′ +j|2)2 +� +1 + O(ρλ log λ|y − ξ′ +j|) + O(ρ2λ2) +� +Using the same computation, it is possible to see that +ρ2ǫ2e−V (y) = ρ2ǫ2 (µ2 +jρ2 + |ρy − ξj|2)2ǫ2 +8µ2 +jρ2 +O(1) += O +� +ρ4ǫ4[1 + |y − ξ′ +j|]4� += O +� +ρ4ǫ4� δ +ρλ +�4� += O(ρ2ǫ2) +From here, we conclude that +(2.34) +ρ2ǫ2g(V ) = +8µ2 +j +(µ2 +j + |y − ξ′ +j|2)2 +� +1 + O(ρλ log λ|y − ξ′ +j|) + O(ρ2λ2) +� ++ O(ρ2ǫ2) +Then, joining the above estimates and (2.31) we get for y ∈ Bδ/(ρλ log λ)(ξ′ +j) +(2.35) +R(y) = +8µ2 +j +(µ2 +j + |y − ξ′ +j|2)2 +� +O(ρλ log λ|y − ξ′ +j|) + O(ρ2λ2) +� ++ O(ρ4 + ρ2ǫ2). +We finish the proof by assuming aj = −1. Similarly as above, we have the +following estimates +ρ2ǫ2g(V ) = ρ2ǫ2(eV − ewj(ρy)eHj(ρy)+� +i̸=j aiVi). +ρ2ǫ2e−V (y) = +8µ2 +j +(µ2 +j + |y − ξ′ +j|2)2 +� +1 + O(ρλ log λ|y − ξ′ +j|) + O(ρ2λ2) +� +. +and +ρ2ǫ2eV (y) = O +� +ρ4ǫ4[1 + |y − ξ′ +j|]4� += O(ρ2ǫ2). +Thus, we get (2.34) and using (2.31) we also obtain (2.35). From the choice +of ρ, the definition of ∥·∥∗ and estimates (2.29), (2.30) and (2.35) we deduce +(2.27). The estimate for Λφ follows the same computations. This completes +the proof. +□ + +SIGN-CHANGING SOLUTIONS +13 +3. Linear problem +In this section we shall reduce the solvability of (2.16) by using the so- +called Lyapunov-Schmidt finite dimensional variational reduction and prove +the main result. In this procedure an important step is the solvability theory +for the linear operator, obtained as the linearization of (2.16) at the approx- +imating solution V , namely, (2.17). Observe that, as ǫ → 0 and λ → +∞, +formally the operator L, around a point ξ′ +j, approaches Lj defined in R2 as +Lj(φ) = ∆φ + +8µ2 +j +(µ2 +j + |y − ξ′ +j|2)2 φ, +j = 1, . . . , m. +We start this section with some notation. +Given vj = vµj the entire +functions solving the equation +∆v + +8µ2 +j +(µ2 +j + |x|2)2 v = 0 +x ∈ R2, +as a consequence of the invariance under dilations and traslations of the +problem (2.4), it is possible to find nontrivial solutions of this equations +and, for each x ∈ R2, are denoted by +zij(x) = ∂ +∂ζi +vj(|x + ζ|) +��� +ζ=0 = +4µjxi +µ2 +j + |x|2 , +i = 1, 2, j = 1, ..., m, +z0j(x) = ∂ +∂s(vj(|sx|) + 2 log(s)) +��� +s=1 = +µ2 +j − |x|2 +µ2 +j + |x|2 , +j = 1, ..., m. +(3.1) +The kernel of Lj in L∞(R2) is non-empty and is spanned by the functions +Zij, i = 0, 1, 2, due to the intrinsic invariances of the problem (2.16), where +Z0j(y) = +µ2 +j − |y − ξ′ +j|2 +µ2 +j + |y − ξ′ +j|2 , +Zij(y) = +4µj(y − ξ′ +j)i +µ2 +j + |y − ξ′ +j|2 , +i = 1, 2, +(3.2) +see [1] for a proof. Consider a large but fixed number R0 > 0 and a radial +smooth cut-off function χ with χ(r) = 1 if r < R0 and χ(r) = 0 if r > R0+1. +Write +χj(y) = χ +� +|y − ξ′ +j| +� +. +(3.3) +Recalling L in (2.17), the main result of this section is the following +Proposition 3.1. There exist λ0 > 0 and ǫ0 > 0 such that for λ ≥ λ0 +and ǫ > 0 satisfying 0 < ρλ < ǫ0, for any points ξ = (ξ1, . . . , ξm) satisfying +(2.10) and for any h ∈ L∞(Ωρ) with ∥h∥∗ < +∞, there is a unique solution + +14 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +φ := Tλ(h) and coefficients cij(ξ) ∈ R, i = 1, 2, j = 1, . . . , m of problem + + + + + + + + + + + + + + + + + +L(φ) = h + +2 +� +i=1 +m +� +j=1 +cijχjZij +in Ωρ +Rρλφ = 0 +on ∂Ωρ +� +Ωρ +χjZijφ = 0 +for i = 1, 2, j = 1, . . . , m. +(3.4) +Moreover, the map ξ �→ φ(ξ) is differentiable with +(3.5) +∥φ∥L∞(Ωρ) ≤ C| log(ρλ)| ∥h∥∗, +|cij| ≤ C∥h∥∗, +and +(3.6) +∥∂(ξl)kφ∥L∞(Ωρ) ≤ C| log(ρλ)|2 ∥h∥∗, +for +l = 1 . . . , m, k = 1, 2. +In order to prove the above result, we require some a priori estimates. We +start with the following result which can be found in Lemma 3.2 in [12]. +Lemma 3.2. There exist λ0 > 0 and ǫ0 > 0 such that for λ ≥ λ0 and +ǫ > 0 satisfying 0 < ρλ < ǫ0, any family of points ξ = (ξ1, . . . , ξm) satisfying +(2.10) and for any solution φ of the problem + + + + + +∆φ + Wφ = h +in Ωρ; +∂φ +∂ν + ρλφ = g +on ∂Ωρ; +(3.7) +satisfying the orthongonality conditions +(3.8) +� +Ωρ +φZijχj = 0 +for i = 0, 1, 2, j = 1, . . . , m, +we have the estimate +(3.9) +∥φ∥L∞(Ωρ) ≤ C(∥h∥∗ + 1 +λρ∥g∥L∞(∂Ωρ)). +The next step is to find an a priori estimate for the solution avoiding the +elements of the kernel due to dilations χjZ0j’s. +Lemma 3.3. There exist λ0 > 0 and ǫ0 > 0 such that for λ ≥ λ0 and +ǫ > 0 satisfying 0 < ρλ < ǫ0, any family of points ξ = (ξ1, . . . , ξm) satisfying +(2.10) and for any solution φ of (3.7) with g = 0, satisfying the orthogonality +conditions +� +Ωρ +φZijχj = 0 +for i = 1, 2, j = 1, . . . , m, +we have +∥φ∥L∞(Ωρ) ≤ C| log(ρλ)| ∥h∥∗. + +SIGN-CHANGING SOLUTIONS +15 +Proof. We will construct functions ˜zj and fix constants bj ∈ R such that the +function +˜φ = φ + +m +� +j=1 +bj˜zj, +satisfies, for certain bj ∈ R, the orthogonality conditions with respect to the +dilations, and subsequenly apply Lemma 3.2. +We divide the proof in several steps: +1.- Construction of the correctors ˜zj: Here we require certain estimates +concerning the Green function with homogeneous Robin boundary condition +in the upper half-space, that is +−∆G(·, y) = δy +in R2 ++; +RaG = 0 +on {x2 = 0}, +where a > 0, and in this case ν = −e2 is the exterior unit normal. As it can +be seen in p. 120 in [22], we have the expression +(3.10) +Ga(x, y) = Γ(x − y) − Γ(x − y∗) − 2 +� +∞ +0 +e−as(x2 + s + y2) +|x + se2 − y∗|2 ds, +and where for y = (y1, y2) ∈ R2 ++, we denote y∗ = (y1, −y2). +Let Fj : Br(ξj) ∩ Ω → R2 ++ be a conformal mapping such that if we denote +ˆξj ∈ ∂Ω is the projection of ξj onto the boundary, then Fj(ˆξj) = 0. +In +addition, DFj(ˆξj) is a rotation (making DFj(ˆξj)ν(ˆξj) = −e2), and is such +that Fj(Br(ξj) ∩ ∂Ω) is an interval in {x : x2 = 0}. Thus, we consider its +expanded version +Fj,ρ(y) = ρ−1Fj(ρy), +y ∈ Br/ρ(ξ′ +j) ∩ Ωρ. +Since Fj is a conformal mapping, we have the existence of a constant +c ∈ (0, 1) (just depending on Ω) such that +(3.11) +c|z1 − z2| ≤ |Fj,ρ(z1) − Fj,ρ(z2)| ≤ c−1|z1 − z2|, +for all z1, z2 ∈ Br/ρ(ξ′ +j) ∩ Ωρ. +Taking into account that dist(ξj, ∂Ω) = O(λ−1), we have +(3.12) +Fj,ρ(ξ′ +j) = ρ−1dje2 + O(ρ−1λ−2). +Let η2j be a cut-off function making η2j = 1 in B r +2ρ (ξ′ +j), η2j = 0 in Bcr +ρ (ξ′ +j), +|Dη2j| ≤ Cρ, |D2η2j| ≤ Cρ2 in R2, and such that ∂η2j +∂ν = 0 in ∂Ωρ. This can +be done using the conformal map Fj and an adequate scaling. +Finally, we consider a cut-off function η1j such that η1j = 1 in BR(ξ′ +j), +η1j = 0 in BR+1(ξ′ +j) and uniform bounds for its first and second-order deriva- +tives. +For y ∈ Ωρ, we define +ˆz2j(y) = η2,j(y) +1 +log(2ρ−1dj)Z0j(y)gj(y), + +16 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +where, for simplicity, we have denoted +(3.13) +gj(y) = Gλρ(Fj,ρ(y), Fj,ρ(ξ′ +j)). +Thus, we define +˜z = +� +j +bj˜zj, +with +˜zj = η1jZ0j + (1 − η1j)ˆz2j, +(3.14) +with bj ∈ R given by the expression +(3.15) +bj +� +Ωρ +χj|Z0j|2 + +� +Ωρ +χjZ0jφ = 0. +2.- Application of the Lemma 3.2: By linearity +L˜φ = h + L˜z in Ωρ; +Rλρ ˜φ = Rλρ˜z on ∂Ωρ, +and by the choice of bj, we are in position to use Lemma 3.2. Then, we see +that +∥˜φ∥∞ ≤ C +� +∥h∥∗ + +� +j +|bj|(∥L˜zj∥∗ + 1 +λρ∥Rλρ˜zj∥L∞(∂Ωρ)) +� +, +from which we get +(3.16) ∥φ∥∞ ≤ C +� +∥h∥∗ + +� +j +|bj|(∥˜zj∥∞ + ∥L˜zj∥∗ + 1 +λρ∥Rλρ˜zj∥L∞(∂Ωρ)) +� +. +Now we estimate each term in the right-hand side above, stating with the +terms concerning ˜zj. +3.- Estimates for gj in (3.13): We start with some estimates concerning the +function gj, consequence of (3.10) and the properties of the conformal map +Fj. +First, by (3.12), for all y ∈ Ωρ with |y − ξ′ +j| ≥ R, we have +����� +� +∞ +0 +e−λρs (Fj,ρ(y))2 + s + (Fj,ρ(ξ′ +j))2 +|Fj,ρ(y) + se2 − Fj,ρ(ξ′ +j)∗|2 ds +����� ≤ +� +∞ +0 +ρe−λρsds +|s + dj + O(λ−2)| ≤ C, +for some C > 0. Using (3.11), (3.12), we have +Γ(Fj,ρ(y) − Fj,ρ(ξ′ +j)) = O(log |y − ξ′ +j|), +and +Γ(Fj,ρ(y)−Fj,ρ(ξ′ +j)∗) = +� log(2ρ−1dj) + O(Rλρ) +if |y − ξ′ +j| ≤ R + 1, +O(log |y − ξ′ +j|) + log(2djρ−1) +if |y − ξ′ +j| > R + 1, +where the O terms are independent of ǫ, λ. Moreover, in the particular case +y ∈ ∂Ωρ, since Fj,ρ(y) belongs to ∂R2 ++, there exists C > 0 such that +(3.17) +���� +gj +log(2djρ−1) +���� +L∞(Ωρ\BR(ξ′ +j)) +≤ C. + +SIGN-CHANGING SOLUTIONS +17 +Additionally, a direct computation leads us to +∇gj(y) =O(|y − ξ′ +j|−1) + O(|Fj,ρ(y) − Fj,ρ(ξ′ +j)∗|−1) ++ O(1) +� +∞ +0 +e−λρs +ds +|se2 − Fj,ρ(ξ′ +j)∗|2 +=O(|y − ξ′ +j|−1) + O(ρλ). +In fact, we have that for |y − ξ′ +j| = R, using that Fj(ˆξj) is a rotation, the +following expansion takes place +(3.18) +∇gj(y) = +y − ξ′ +j +|y − ξ′ +j|2 +� +1 + O(λ−1) +� ++ O(ρλ). +Now we look for the terms involving ˜zj in (3.16). By the previous dis- +cussion, it is easy to see the existence of C > 0 not depending on ǫ, λ such +that +(3.19) +∥˜zj∥∞ ≤ C. +Now we deal with ∥L˜zj∥∗. It is clear that L˜zj = O(ρ3) in BR(ξ′ +j). +Thus, we concentrate on the case |y −ξ′ +j| ≥ R. With the above estimates, +for R ≤ |y − ξ′ +j| ≤ R + 1 (recalling that since F is conformal we have gj is +harmonic there) we have +L˜zj(y) =L +� +η1Z0j +� +1 − +gj +log(2djρ−1) +�� +(y) + Lˆzj(y) +=O(ρ3) + O +� +1 +| log(λρ)| +� +For |y − ξ′ +j| ≥ R + 1, we see that +L˜zj(y) = O +� +log |y − ξ′ +j| +|y − ξ′ +j|3| log(λρ)| +� +. +From which we conclude +(3.20) +∥L˜zj∥∗ ≤ CR2+σ +| log(λρ)|, +for some C > 0. +Now we concentrate on Rλρ˜zj. Using the properties of η2j, we have +(3.21) +Rλρ˜zj = +1 +| log(λρ)|η2j +� +Z0jRλρgj + ∂Z0j +∂ν gj +� +in ∂Ωρ. +Notice that Rλρ˜zj(y) = 0 for |y − ξ′ +j| ≥ r/ρ, so we concentrate on the +analysis of y ∈ ∂Ωρ with |y −ξ′ +j| < r/ρ. For simplicity, we denote this region +as Aj. We have gj is uniformly bounded on ∂Ωρ. In fact, notice that if + +18 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +y ∈ ∂Ωρ, we have |Fj,ρ(y) − Fj,ρ(ξ′ +j)| = |Fj,ρ(y) − Fj,ρ(ξ′ +j)∗| and thus, using +that Fjρ(ξ′ +j)2 ≥ c/(λρ) we have +|gj(y)| ≤ C +� +∞ +0 +e−λρs +|se2 − (Fjρ(ξ′ +j))∗|ds ≤ C. +Using this and the explicit formula for Z0j, we conclude that +Rλρ˜zj = +η2j +| log(λρ)|(O(1)Rλρgj + O(λρ)). +Noticing that +Rλρgj(y) = c +� +− ∂G +∂x2 +(Fjρ(y), Fjρ(ξ′ +j) +� ++ λρG(Fjρ(y), Fjρ(ξ′ +j)) += λρ(c − 1)gj(y), +where c = 1 + O(ρy) is the conformal factor of the map F. Thus, for each +y ∈ ∂Ωρ with |y − ξ′ +j| ≤ r/ρ, we get +(3.22) +Rλρgj(y) = O(ρy)λρgj(y), +and from here, replacing in (3.21), we conclude that +(3.23) +∥Rλρ˜zj∥∞ ≤ C +λρ +| log(λρ)|. +In particular, using this estimate together with (3.20) and replacing them +into (3.9) we have +(3.24) +∥˜φ∥∞ ≤ C(∥h∥∗ + +1 +| log(λρ)| +� +j +|bj|), +meanwhile, using (3.19), (3.20) and (3.23) and replacing them into (3.16), +we get +(3.25) +∥φ∥∞ ≤ C(∥h∥∗ + +� +j +|bj|), +where in both inequalities the constant C > 0 depends on R, but not on λ +nor ǫ. +4.- Estimate for bj: Using the equation for ˜φ, multiplying by ˜zj and inte- +grating by parts, we conclude that +bj +� +Ωρ +˜zjL˜zj = +� +Ωρ +˜φL˜zj − +� +∂Ωρ +˜φRλρ˜zj + bj +� +∂Ωρ +˜zjRλρ˜zj − +� +Ωρ +h˜zj +=: I1 + I2 + bjI3 + I4. +(3.26) +for each j. Now we proceed to estimate each term. +It is direct to see that +|I4| ≤ C∥h∥∗, + +SIGN-CHANGING SOLUTIONS +19 +and using (3.20) and (3.24), we get +|I1| ≤ C∥˜φ∥∞R−σ∥L˜zj∥∗ ≤ C +R2 +| log(λρ)|(∥h∥∗ + +R2+σ +| log(λρ)| +� +j +|bj|). +Now, for I2 we make +I2 ≤ +C +| log(λρ)|∥˜φ∥∞ +� +∂Ωρ +|Rλρgj| =: +C +| log(λρ)|∥˜φ∥∞ ˜I2. +By (3.22) we have +˜I2 ≤ Cλρ +� +Aj +|gj(y)|ds(y), +and using the explicit formula (3.10), we get that +˜I2 ≤Cλρ +� +Aj +� +∞ +0 +e−λρs|s + Fjρ(ξ′ +j)2| +|Fλρ(y) + se2 − Fjρ(ξ′ +j)∗|2 dsds(y). +From here, using the estimate Fjρ(ξ′ +j) = +1 +λρ(O( 1 +λ), 1 + O( 1 +λ)), by a change +of variables, we can find universal constants C, c > 0 such that +˜I2 ≤Cλρ +� 1/ρ +0 +� +∞ +0 +e−cλρs|s + 1 +λρ| +t2 + (s + 1 +λρ)2 dsdt, +by taking λ suitably large. Thus, by algebraic manipulations, we conclude +that ˜I2 ≤ C for some C > 0 not depending on R, ǫ, λ. From here conclude +that +|I2| ≤ +C +| log(λρ)|(∥h∥∗ + +R2+σ +| log(λρ)| +� +j +|bj|), +and by similar estimates, we conclude +|I3| ≤ +� +∂Ωρ +|˜zjRλρ˜zj|ds(y) = O +� +1 +| log(λρ)|2 +� +. +Collecting the previous estimates and replacing them int (3.26), we con- +clude that +(3.27) +bjI0 = O(1) +� +∥h∥∗ + +1 +| log(λρ)|2 +� +j +|bj| +� +, +where I0 := +� +Ωρ ˜zjL˜zj and O(1) depends on R, but not on ǫ, λ. +We claim that +(3.28) +|I0| ≥ +c +| log(λρ)| +for some c > 0 independent of R, λ and ǫ. If we assume this is true, we +replace this into (3.27) and by taking λρ small enough in terms of R, we +arrive at +|bj| ≤ C| log(λρ)|∥h∥∗, + +20 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +and replacing this into (3.25) we have been proved the lemma. Thus, it +remains to get the +5.- Proof of estimate (3.28): We divide the integral in the regions |y − +ξ′ +j| ≤ R, R < |y − ξ′ +j| ≤ R + 1, and R + 1 < |y − ξ′ +j|. Then, using the +estimates (3.17), the gradient of gj, (3.19), and the fact that for each i ̸= j +we have |y − ξ′ +i| ≥ r/(2ρ) in the support of η2j, we have +(3.29) +I0 = O(R2λ4ρ4) + I01 + O +� +R−3 +1 +| log(λρ)| +� +, +where +I01 = +� +BR+1(ξ′ +j)\BR(ξ′ +j) +L˜z0j ˜z0j. +Writing ˜z0j = η1,j(Z0j − ˆz0j) + ˆz0j valid in the annular region AR = {y : +R < |y − ξ′ +j| < R + 1}, we can write +I01 = +� +AR +� +∆η1j(Z0j − ˆz0j) + 2∇η1j∇(Z0j − ˆz0j) +� +˜z0j ++ +� +AR +η1jLZ0j ˜z0j + +� +AR +(1 − η1j)Lˆz0j ˜z0j. +By similar arguments as in (3.29), we have +I01 = +� +AR +� +∆η1j(Z0j − ˆz0j) + 2∇η1j∇(Z0j − ˆz0j) +� +˜z0j + O +� +R−3 +1 +| log(λρ)| +� +=:I02 + O +� +R−3 +1 +| log(λρ)| +� +. +Then, for I02, we integrate by parts and using that (1−gj/ log(2djρ−1)) = +O( log(R) +| log(λρ)|) in AR, we can get +I02 = − +� +|y−ξ′ +j|=R +˜z0j∇(Z0j − ˆz0j)ν +− +� +AR +η1j(∆(Z0j − ˆz0j)˜z0j + ∇(Z0j − ˆz0j)∇˜z0j) +− +� +AR +∇η1j∇˜z0j(Z0j − ˆz0j) +=: − I03 + O +� +R−2 +| log(λρ)| +� ++ O +� log2(R) +| log(λρ)|2 +� ++ O +�R2 log2(R) +| log(λρ)|2 +� +. + +SIGN-CHANGING SOLUTIONS +21 +For I03 we have +−I03 = − +� +|y−ξ′ +j|=R +˜z0j(1 − gj/ log(2djρ−1))∇Z0jν ++ +1 +log(2djρ−1) +� +|y−ξ′ +j|=R +Z2 +0j∇gjν +=O +�R−2 log(R) +log(λρ) +� ++ +Z2 +0j(R) +log(2djρ−1) +� +|y−ξ′ +j|=R +∇gjν, +from which, using the expansion (3.18) we arrive at +I03 = − +2πZ2 +0j(R) +log(2djρ−1)(1 + O(λ−1)) + O +�R−2 log(R) +log(λρ) +� +. +From here, taking R, λ large enough we conclude +|I0| ≥ +π +log(2djρ−1) + O +�R−2 log(R) +log(λρ) +� +, +from which (3.28) follows by fixing R large enough independent of λ, ρ. +□ +Now we are in position to provide the +Proof of Proposition 3.1. Consider the Hilbert space +H = {u ∈ H1(Ωρ) : +� +Ωρ +χjZijφ = 0 for i = 1, 2, j = 1, . . . , m} +endowed with the inner product +⟨u, v⟩H = +� +Ωρ +∇u · ∇v + λρ +� +∂Ωρ +uv. +For each ℓ ∈ H∗, we consider the map R : H∗ → H, given by the Riesz +Representation Theorem, as +⟨Rℓ, ϕ⟩ = ℓ(ϕ), +for all ϕ ∈ H, +and with this, we denote the operators K1, K2 : H → H, given by +K1(f) = R(f), +K2(f) = R(Wf), +f ∈ H, +where we have performed the identification f ∈ L2(Ωρ) with the map ϕ �→ +� +Ωρ fϕ ∈ H∗. In fact, because of this identification we can assume that both +operators are compact. +For h ∈ L∞(Ωρ) ∩ H1(Ωρ), let cij the unique constants such that +˜h := h + +� +ij +cijχjZij ∈ H. +A simple computation tells us that for each i, j we have +− +� +Ωρ hχjZij +� +Ωρ χjZ2 +ij += cij, + +22 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +from which, there exists CR > 0 such that +|cij| ≤ CR∥h∥∗ +The solvability of the problem is equivalent to the existence of a function +φ ∈ H such that +φ − K2(φ) = K1˜h, +1 and by Fredholm alternative, we have a unique solution to this equation +if the homogeneous problem +φ − K2(φ) = 0, +has no nontrivial solutions. +This is the case in view of the estimates of +Lemma 3.3 this is the case, from which the well-posedness of (3.4). By the +estimates for cij, we conclude that the solution φ to this problem satisfies +the estimate (3.5). +For (3.6), by standard arguments we have φkl := ∂(ξl)kφ satisfies +Lφkl = − ∂W +∂(ξl)k +φ + ∂(cklχlZkl) +∂(ξl)k +in Ωρ, +together with homogeneous Robin boundary conditions. In order to use the +estimates in Lemma 3.3 we consider a function with the form +φkl + ˜cklχlZkl, +with ˜ckl such that the orthogonality condition with respect to Zkl is satisfied. +We omit the details. +□ +4. Nonlinear Problem +Proposition 4.1. There exist λ0 > 0 and ǫ0 > 0 such that for all λ ≥ λ0 and +ǫ > 0 with 0 < ρλ9 log λ < ǫ0 and for any points ξ = (ξ1, . . . , ξm) satisfying +(2.10), the problem of finding a function φ and coefficients cij(ξ) ∈ R, i = +1, 2, j = 1, . . . , m satisfying +(4.1) + + + + + + + + + + + + + + + + + + + +L(φ) = −(R + Λ(φ) + N(φ)) + +2 +� +i=1 +m +� +j=1 +cijχjZij +in Ωρ +∂φ +∂ν + ρλφ = 0, +on ∂Ωρ +� +Ωρ +χjZijφ = 0 +for i = 1, 2, j = 1, . . . , m, +has a unique solution φ and scalars cij, i = 1, 2, j = 1, 2, . . . , m satisfying +∥φ∥L∞(Ωρ) ≤ Cρλ9 log λ| log(ρλ)|, +|cij| ≤ Cρλ9 log λ, +where L, R, Λ(φ) and N(φ) are given by (2.17), (2.20), (2.18) and (2.19) +respectively. Moreover, the map ξ′ �→ φ into the space C(¯Ωρ), the derivative + +SIGN-CHANGING SOLUTIONS +23 +Dξ′φ exists and defines a continuous function of ξ′, and there is a constant +C > 0, such that +∥Dξ′φ∥L∞(Ωρ) ≤ Cρλ9 log λ| log(ρλ)|2. +(4.2) +Proof. Observe that in terms of the operator T problem (4.1) becomes +(4.3) +φ = T (−[R + Λ(φ) + N(φ)]) := A(φ), +where T is the continuous linear map defined on the set of all h ∈ L∞(Ωρ) +satisfying ∥h∥∗ < +∞, so that φ = T(h) correspond to the unique solution +of the problem (3.4). For a given number ν > 0, let us consider +Fν = {φ ∈ C(¯Ωρ) : ∥φ∥∞ ≤ νρλ9 log λ| log(ρλ)|} +From the Proposition 3.1, we get +∥A(φ)∥∞ ≤ C| log(ρλ)| [∥R∥∗ + ∥Λ(φ)∥∗ + ∥N(φ)∥∗] . +From Lemma 2.3 it follows the estimate ∥R∥∗ ≤ Cρλ9 log λ. Furthermore, +∥Λ(φ)∥∗ ≤ Cρλ9 log λ∥φ∥∞ +and +∥N(φ)∥∗ ≤ C∥φ∥2 +∞ +Hence, we get for any φ ∈ Fν, +∥A(φ)∥∞ ≤ Cρλ9 log λ| log(ρλ)| +� +1 + νρλ9 log λ| log(ρλ)| + ν2ρλ9 log λ| log(ρλ)|2� +. +Given any φ1, φ2 ∈ Fν, we have that +∥Λ(φ1) − Λ(φ2)∥∗ ≤ Cρλ9 log λ ∥φ1 − φ2∥∞ +and +∥N(φ1) − N(φ2)∥∗ ≤ C(∥φ1∥∞ + ∥φ2∥∞)∥φ1 − φ2∥∞ +≤ Cνρλ9 log λ| log(ρλ)| ∥φ1 − φ2∥∞ +with C independent of ν. Therefore, from the Proposition 3.1 +∥A(φ1) − A(φ2)∥∞ ≤ Cνρλ9 log λ| log(ρλ)|2∥φ1 − φ2∥∞, +so that it follows that for all ǫ sufficiently small and λ sufficiently large A +is a contraction mapping of Fν (for ν large enough), and therefore a unique +fixed point of A exists in Fν. +Let us now discuss the differentiability of φ depending on ξ′, i.e., ξ′ �→ +φ(ξ′) ∈ C(¯Ωρ) is C1. Since R depends continuously (in the ∗-norm) on ξ′, +using the fixed point characterization (4.3), we deduce that the mapping +ξ′ �→ φ is also continuous. Then, formally +∂ξ′ +kl[Λ(φ)] = +� +ρ2ǫ2� +∂ξ′ +kl(eV ) − ∂ξ′ +kl(e−V ) +� +− ∂ξ′ +klW +� +φ ++ [ρ2ǫ2(eV + e−V ) − W]∂ξ′ +kl φ. +and +∂ξ′ +kl[N(φ)] = ρ2ǫ2∂ξ′ +kl(eV )(eφ − φ − 1) + ρ2ǫ2eV [eφ − 1]∂ξ′ +kl φ +− ρ2ǫ2∂ξ′ +kl(e−V )(e−φ + φ − 1) + ρ2ǫ2e−V [e−φ − 1]∂ξ′ +kl φ. + +24 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +From the definition of V it follows that +∂ξ′ +klV (y) = ak +4(y − ξ′ +k)l +µ2 +k + |y − ξ′ +k|2 − +m +� +i=1 +ai +2∂ξ′ +kl(µ2 +i ) +µ2 +i + |y − ξ′ +i|2 +ρ∂2lHλ(ρy, ρξ′ +k)+O(ρ2λ2). +Hence, ∥ρ2ǫ2∂ξ′ +kl(eV )∥∗ and ∥ρ2ǫ2∂ξ′ +kl(e−V )∥∗ are uniformly bounded. Then, +we conclude that +∥∂ξ′ +kl[Λ(φ)]∥∗ ≤ +���ρ2ǫ2� +∂ξ′ +kl(eV ) − ∂ξ′ +kl(e−V ) +� +− ∂ξ′ +klW +��� +∗∥φ∥∞ ++ [∥ρ2ǫ2(eV + e−V ) − W∥∗∥∂ξ′ +kl φ∥∞. +and +∥∂ξ′ +kl[N(φ)]∥∗ ≤ C∥φ∥2 +∞ + C∥φ∥∞ ∥∂ξ′ +klφ∥∞ +≤ Cνρλ9 log λ| log(ρλ)| +� +νρλ9 log λ| log(ρλ)| + ∥∂ξ′ +klφ∥∞ +� +. +Also, observe that we have +∂ξ′ +klφ = −(∂ξ′ +klT) (R + Λ(φ) + N(φ)) − T +� +∂ξ′ +kl [R + Λ(φ) + N(φ)] +� +. +So, using (3.5) and (3.6), we get +∥∂ξ′ +klφ∥∞ ≤ C| log(ρλ)| +� +| log(ρλ)| (∥R∥∗ + ∥Λ(φ)∥∗ + ∥N(φ)∥∗) ++ ∥∂ξ′ +klR∥∗ + ∥∂ξ′ +kl[Λ(φ)]∥∗ + ∥∂ξ′ +kl[N(φ)]∥∗ +� +. +Let us estimate ∥∂ξ′ +klR∥∗. We know that +∂ξ′ +klR(y) = ∆∂ξ′ +klV (y) + ρ2ǫ2(eV + e−V )∂ξ′ +klV (y). +From similar computations to deduce Lemma 2.3 it follows that for any +l = 1, 2: +• if |y − ξ′ +j| > δ +ρ for all j = 1, . . . , m then +∂ξ′ +klR(y) = O +� +ρλ log λ[ρ4 + ρ2ǫ2] +� +, +• if +δ +ρλ log λ ≤ |y − ξ′ +j| ≤ δ +ρ for some j ∈ {1, . . . , m} then +∂ξ′ +klR(y) = ρλ2 log2 λ O +� +ρ2ǫ2� +log4 λ + +λ4 +ρ4|y − ξ′ +j|4 + ρ4λ4|y − ξ′ +j|4�� +and +• if |y − ξ′ +j| ≤ +δ +ρλ log λ for some j ∈ {1, . . . , m} then +∂ξ′ +klR(y) = +8µ2 +j +(µ2 +j + |y − ξ′ +j|2)2 O(ρλ log λ[1 + |y − ξ′ +j|]) ++ +4δjk(y − ξ′ +k)l + 2∂ξ′ +kl(µ2 +j) +µ2 +j + |y − ξ′ +j|2 +O(ρ2ǫ2) + O(ρ3ǫ2λ log λ + ρ4). + +SIGN-CHANGING SOLUTIONS +25 +Therefore, from the definition of *-norm we conclude that +∥∂ξ′ +klR∥∗ ≤ C(ρλ log λ + ρ2λ11 log3 λ). +Similar computations as above and those used to deduce Lemma 2.3 lead us +to find the estimate +��ρ2ǫ2(eV − e−V )∂ξ′ +klV − ∂ξ′ +klW +�� +∗ ≤ C(ρλ log λ + ρ2λ11 log3 λ). +Hence, we find the following estimate +∥∂ξ′ +klφ∥∞ ≤ C +� +ρλ9 log λ| log(ρλ)|2 + ρλ9 log λ| log(ρλ)|2∥∂ξ′ +klφ∥∞ +� +Thus, we conclude (4.2). Note that ∂ξ′ +klµj = O(ρλ log λ). +The above computations can be made rigorous by using the implicit func- +tion theorem and the fixed point representation (4.3) which guarantees C1 +regularity in ξ′. +□ +5. Variational reduction and energy computations +In view of Proposition 4.1 we obtain a solution to (2.15) with the form +V + φ if we are able to get that +(5.1) +cij(ξ) = 0, +i = 1, 2, j = 1, . . . , m. +This problem is equivalent to look for critical points of the following +functional +Fǫ,λ(ξ) = Jǫ,λ(U + ˜φ), +where U is the approximation defined in (2.14) and ˜φ(x) = φ +�x +ρ +� +with φ +the solution to (2.16). The energy function Jǫ,λ is given by +(5.2) +Jǫ,λ(u) = 1 +2 +� +Ω +|∇u|2dx − ǫ2 +� +Ω +� +eu + e−u� +dx + λ +2 +� +∂Ω +u2dσ, +u ∈ H1(Ω). +Notice that critical points for Jǫ,λ are weak solutions for (1.1) compare +with [2]. We have the following sufficient condition to have (5.1). +Lemma 5.1. There exists λ0 > 0 and ǫ0 > 0 such that for any λ ≥ λ0 and +ǫ > 0 so that 0 < ρλ9 < ǫ0, if ξ ∈ Ωm is a critical point of Fǫ,λ satisfying +(2.10) then u = U(ξ)+ ˜φ(ξ) is a critical point of Jǫ,λ, that is, if DξFǫ,λ(ξ) = 0 +then ξ satisfies system (5.1), i.e., u is a solution to (1.1). +Proof. Define the energy functional Iǫ,λ associated to problem (2.15), namely, +Iǫ,λ(v) = 1 +2 +� +Ωρ +|∇v|2 − ρ2ǫ2 +� +Ωρ +(ev + e−v) dy + ρλ +2 +� +∂Ωρ +v2 dσ(y). +Let us differentiate the function Fǫ,λ(ξ) with respect to ξ. Since +Iǫ,λ(V (ξ′) + φ(ξ′)) = Jǫ,λ(U(ξ) + ˜φ(ξ)), + +26 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +we can differentiate directly Iǫ,λ(V + φ) (under the integral sign), so that +integrating by parts +∂ξklFǫ,λ(ξ) = 1 +ρDIǫ,λ(V + φ) +� +∂ξ′ +klV + ∂ξ′ +klφ +� += − 1 +ρ +2 +� +i=1 +m +� +j=1 +cij +� +Ωǫ +χjZij +� +∂ξ′ +klV + ∂ξ′ +klφ +� +. +From the results of the previous section, this expression defines a continuous +function of ξ′, and hence of ξ. Let us assume that DξFǫ,λ(ξ) = 0. Then, +from the latter equality +2 +� +i=1 +m +� +j=1 +cij +� +Ωǫ +χjZij +� +∂ξ′ +klV + ∂ξ′ +klφ +� += 0, +k = 1, 2, l = 1, . . . , m. +Using (4.2) and ∂ξ′ +klV = 4 ak +µk Zkl + O +� +ρλ log λ +� +, where O(ρλ log λ) is in the +L∞ norm, it follows that +2 +� +i=1 +m +� +j=1 +cij +� +Ωǫ +χjZij [Zkl + o(1)] = 0, +k = 1, 2, l = 1, . . . , m. +with o(1) small in the sense of the L∞ norm as ρλ9 log λ → 0. The above +system is diagonal dominant and we thus get cij = 0 for i = 1, 2, j = +1, . . . , m. +□ +Next result states an expansion of Fǫ,λ in terms of ϕm. In order to have +a more clearly the relation between ǫ ∈ (0, 1) and λ > 1 such that they +satisfying ǫλ16 ≤ ǫ0 for some ǫ0 < 1, and recalling ρ = ǫλ−2, we see that +(5.3) +0 ≥ log(ρλ) ≥ log ǫ − log ǫ−1/16 ≥ 2 log ǫ, +from which we state the estimates in terms of the leading expression at the +logarithm. +Proposition 5.2. The following expansions holds +Fǫ,λ(ξ) = −16πm + 8πm log 8 − 16πm log(ρλ2) − 4πϕm(ξ) + θǫ,λ(ξ), +in C1-sense, where +(5.4) +ϕm(ξ) = +m +� +j=1 + +Hλ(ξj, ξj) + +m +� +i=1,i̸=j +aiajGλ(ξi, ξj) + + , +|θǫ,λ(ξ)| = O +� +ǫ2λ14| log ǫ|3� +, +and +|∇θǫ,λ(ξ)| = O +� +ǫλ16 log4 ǫ +� +, +uniformly on points ξ = (ξ1, . . . , ξm) ∈ Ωm satisfying the constraints (2.10), +as ǫ → 0 and λ → +∞. + +SIGN-CHANGING SOLUTIONS +27 +Proof. First, we shall expand the energy functional Jǫ,λ evaluated in the +ansatz U, namely, we give an asymptotic estimate of Jǫ,λ(U). +Claim 1. The following expansion does hold +Jǫ,λ(U) = −16πm + 8πm log 8 − 16πm log(ρλ2) − 4πϕm(ξ) + O(ρλ log λ) +Proof. First, we will evaluate the quadratic and boundary parts of energy +evaluated at U, that is, integrating by parts +1 +2 +� +Ω +|∇U|2 dx + λ +2 +� +∂Ω +U 2 dσ = −1 +2 +� +Ω +U∆U dx = −1 +2 +m +� +j=1 +aj +� +Ω +U∆Uj dx, +since on ∂Ω +∂U +∂ν + λU = 0. +Using the equation (2.7) of Uj = wj + Hj (recall Hj is harmonic), we have +� +Ω +U(−∆Uj) dx = +� +Ω +ǫ2ewj(x)U(x) dx += aj +� +Ω +ǫ2ewj(wj + Hj) + +� +i̸=j +ai +� +Ω +ǫ2ewj(wi + Hi). +Then, we expand as follows +� +Ω +ǫ2ewj(wj + Hj) = +� +B(ξj, +dj +2 ) +ǫ2ewj(wj + Hj) + +� +Ω\B(ξj, +dj +2 ) +ǫ2ewj(wj + Hj). +Using (2.12) and x − ξj = µjρy, we obtain that +� +B(ξj, +dj +2 ) +ǫ2ewj(wj + Hj) += +� +B(ξj, +dj +2 ) +8µ2 +jρ2 +(µ2 +jρ2 + |x − ξj|2)2 +� +log +1 +(µ2 +jρ2 + |x − ξj|2)2 + Hλ(x, ξj) ++ O +� +µ2 +jρ2 +d2 +j +� � += +� +B(0, +dj +2µj ρ ) +8 +(1 + |y|2)2 +� +− 2 log(1 + |y|2) − 4 log(µjρ) + Hλ(ξj + µjρy, ξj) ++ O +� +µ2 +jρ2 +d2 +j +� � += − 16π − 32π log(µjρ) + 8πHλ(ξj, ξj) + O(ρλ log λ). +Since µj is uniformly bounded and away from zero, and since dj = O(λ−1), +we see that +� +B(0, +dj +2µj ρ) +8 +(1 + |y|2)2 log(1 + |y|2) dy = 8π + O +�µ2 +jρ2 +d2 +j +�� log µjρ +dj +�� +� +, + +28 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +and +� +B(0, +dj +2µj ρ ) +8 +(1 + |y|2)2 dy = 8π + O +�µ2 +jρ2 +d2 +j +� +and by using (2.32), we get Hλ(ξj +µjρy, ξj) = Hλ(ξj, ξj)+O(µjρλ log λ|y|) +in B(0, +dj +2µjρ) so that +� +B(0, +dj +2µj ρ ) +8 +(1 + |y|2)2 Hλ(ξj + µjρy, ξj) dy = 8πHλ(ξj, ξj) + O(ρλ log λ). +Also, +� +Ω\B(ξj, +dj +2 ) +ǫ2ewj(wj + Hj) = +� +Ω\B(ξj, +dj +2 ) +ǫ2ewj +� +Gλ(x, ξj) + O +� +µ2 +jρ2 +d2 +j +�� += O(ρ2λ2 log λ) +in view of +ǫ2ewj = +8µ2 +jρ2 +(µ2 +jρ2 + |x − ξj|2)2 = O +�µ2 +jρ2 +d2 +j +� +and Gλ(x, ξj) = O(log λ) for |x − ξj| > dj +2 . Therefore, we obtain that +� +Ω +ǫ2ewj(wj + Hj) = −16π − 32π log(µjρ) + 8πHλ(ξj, ξj) + O(ρλ log λ) +(5.5) +Now, for i ̸= j we have that +� +Ω +ǫ2ewj(wi + Hi) = +� +B(ξj, +dj +2 ) +ǫ2ewj(wi + Hi) + +� +Ω\B(ξj, +dj +2 ) +ǫ2ewj(wi + Hi). +Then, by using (2.12), we get that +� +B(ξj, +dj +2 ) +ǫ2ewj(wi + Hi) = 8πGλ(ξi, ξj) + O(ρλ log λ) +� +Ω\B(ξj, +dj +2 ) +ǫ2ewj(wi + Hi) += +� +B(ξi, di +2 ) +ǫ2ewj +� +log +1 +(µ2 +i ρ2 + |x − ξi|)2 + Hλ(x, ξi) + O +�µ2 +i ρ2 +d2 +i +�� +dx ++ +� +Ω\[B(ξj, +dj +2 )∪B(ξi, di +2 )] +ǫ2ewj +� +Gλ(x, ξi) + O +�µ2 +i ρ2 +d2 +i +�� +dx += O +� +ρ2 +�| log ρ| +λ2 ++ | log(ρλ)| +λ2 +�� ++ O +�ρ2 +λ2 log λ + ρ4 +λ4 +� += O +�ρ2 +λ2 | log(ρλ)| +� + +SIGN-CHANGING SOLUTIONS +29 +in view of +� +B(ξi, di +2 ) +8µ2 +jρ2 +(µ2 +jρ2 + |x − ξj|2)2 +� +log +1 +(µ2 +i ρ2 + |x − ξi|)2 + Hλ(x, ξi) + O +�µ2 +i ρ2 +d2 +i +�� +dx += O +� +ρ2 +� +B(ξi, di +2 ) +| log(µ2 +i ρ2 + |x − ξi|)| dx + ρ2 +� +B(ξi, di +2 ) +|Hλ(x, ξi)| dx + ρ4 +� += O +� +ρ2� +d2 +i | log ρ| + ρ2�di +ρ +�2 +log +�di +ρ +��� ++ O(ρ2[d2 +i log λ + d3 +i λ log λ]) +and +� +Ω\[B(ξj, +dj +2 )∪B(ξi, di +2 )] +8µ2 +jρ2 +(µ2 +jρ2 + |x − ξj|2)2 +� +Gλ(x, ξi) + O +�µ2 +i ρ2 +d2 +i +�� +dx += O +� +ρ2 +d2 +j +log λ + +ρ4 +d2 +jd2 +i +� +Thus, we obtain that +(5.6) +� +Ω +ǫ2ewj(wi + Hi) = 8πGλ(ξi, ξj) + O(ρλ log λ). +Taking into account (2.26), (5.5) and (5.6), we find that +1 +2 +� +Ω +|∇U|2 dx + λ +2 +� +Ω +U 2 dσ = 1 +2 +m +� +j=1 + + +� +Ω +ǫ2ewjUj + aj +m +� +i=1,i̸=j +ai +� +Ω +ǫ2ewjUi + + += 1 +2 +m +� +j=1 + +−16π − 32π log(µjρ) + 8πHλ(ξj, ξj) + aj +m +� +i=1,i̸=j +ai8πGλ(ξi, ξj) + + + O(ρλ log λ) +(5.7) +On the other hand, we have +� +Ω +ǫ2(eU+e−U) dx = +m +� +j=1 +� +B(ξj, +dj +log λ ) +ǫ2(eU+e−U) dx+ +� +Ω\∪m +j=1B(ξj, +dj +log λ ) +ǫ2(eU+e−U) dx. +Observe that +� +Ω\∪m +j=1B(ξj,δ) +ǫ2(eU + e−U) dx = +� +Ω\∪m +j=1B(ξj,δ) +ǫ2� +e +�m +i=1 aiGλ(x,ξi)+O(ρ2λ2) ++ e− �m +i=1 aiGλ(x,ξi)+O(ρ2λ2)� +dx += O(ǫ2), +by using that �m +i=1 aiGλ(x, ξi) = O(1) in Ω \ ∪m +j=1B(ξj, δ). So, we get that +� +Ω\∪m +j=1B(ξj, +dj +log λ) +ǫ2(eU+e−U) dx = +m +� +j=1 +� +B(ξj,δ)\B(ξj, +dj +log λ ) +ǫ2(eU+e−U) dx+O(ǫ2). + +30 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +Now, assume that aj = 1 so that by (2.1) we obtain +� +B(ξj,δ)\B(ξj, +dj +log λ) +ǫ2eU = +� +B(ξj,δ)\B(ξj, +dj +log λ) +ǫ2eGλ(x,ξj)+� +l̸=j alGλ(x,ξl)+O(ρ2λ2) dx += +� +B(ξj,δ)\B(ξj, +dj +log λ) +ǫ2 eHλ(x,ξj)+O(1) +|x − ξj|4 +dx += O +� +ǫ2λ4 +� +B(ξj,δ)\B(ξj, +dj +log λ) +|x − ξj|−4 dx +� += O(ǫ2λ6 log2 λ) +and +� +B(ξj,δ)\B(ξj, +dj +log λ) +ǫ2e−U = +� +B(ξj,δ)\B(ξj, +dj +log λ) +ǫ2e−Gλ(x,ξj)−� +l̸=j alGλ(x,ξl)+O(ρ2λ2) dx += +� +B(ξj,δ)\B(ξj, +dj +log λ) +ǫ2|x − ξj|4e−Hλ(x,ξj)+O(1)dx += O +� +ǫ2λ4 +� +B(ξj,δ)\B(ξj, +dj +log λ ) +|x − ξj|4 dx +� += O(ǫ2λ4). +Similarly, for aj = −1 we find that +� +B(ξj,δ)\B(ξj, +dj +log λ ) +ǫ2(eU + e−U) = O(ǫ2λ4 + ǫ2λ6 log2 λ). +Now, assume that aj = 1, so that by Lemma 2.2 and taking x − ξj = µjρy +we get +� +B(ξj, +dj +log λ ) +ǫ2eU dx = +� +B(ξj, +dj +log λ ) +8µ2 +jρ2 +(µ2 +jρ2 + |x − ξj|2)2 exp +� +Hj(x) + +� +l̸=j +Ul(x) +� +dx += +� +B(ξj, +dj +log λ ) +8eHλ(x,ξj)−log(8µ2 +j)+4 log λ+� +l̸=j alGλ(x,ξl)+O(ρ2λ2) +µ2 +jρ2 +� +1 + +� |x−ξj| +µjρ +�2�2 +dx += +� +B(0, +dj +µj ρ log λ ) +8eH(ξj+µjρy,ξj)−log(8µ2 +j )+4 log λ+� +l̸=j alG(ξj+µjρy,ξl)+O(ρ2λ2) +(1 + |y|2)2 +dy += +� +B(0, +dj +µj ρ log λ ) +8 +(1 + |y|2)2 +� +1 + O(ρλ log λ|y| + ρ2λ2) +� +dy += 8π + O(ρλ log λ), + +SIGN-CHANGING SOLUTIONS +31 +and +� +B(ξj, +dj +log λ ) +ǫ2e−U dx = ǫ2 +� +B(ξj, +dj +log λ ) +� +8µ2 +jρ2 +(µ2 +jρ2 + |x − ξj|2)2ǫ2 +�−1 +exp +� +− Hj(x) − +� +l̸=j +Ul(x) +� +dx += +� +B(ξj, +dj +log λ ) +ǫ4 +8µ2 +jρ2 +� +µ2 +jρ2 + |x − ξj|2�2 +× e−Hλ(x,ξj)+log(8µ2 +j)−4 log λ−� +l̸=j alGλ(x,ξl)+O(ρ2λ2) dx += +� +B(ξj, +dj +log λ ) +ǫ4 +8µ2 +jρ2 +� +µ2 +jρ2 + |x − ξj|2�2 � +1 + O(λ log λ|x − ξj| + ρ2λ2) +� +dx += O +� +ρǫ +log4 λ +� +In case aj = −1, using previous ideas we find that +� +B(ξj, +dj +log λ ) +ǫ2eU dx = O +� +ρǫ +log4 λ +� +and +� +B(ξj, +dj +log λ) +ǫ2e−U dx = 8π + O(ρλ log λ). +Therefore, we conclude that +� +Ω +ǫ2(eU + e−U)dx = 8πm + O +� +ρλ log λ + +ρǫ +log4 λ + ǫ2λ6 log2 λ +� +. +From the choice of µj’s in (2.26), we obtain that +Jǫ,λ(U) = −16πm + 8πm log 8 − 16πm log(ρλ2) − 4πϕm(ξ) + O(ρλ log λ) +(5.8) +where ϕm is given by (5.4), if ǫλ7 log λ| log( ǫ +λ)|2 is small enough. +□ +Claim 2. The following expansion does hold +∂(ξl)k[Jǫ,λ(U)] = −4π∂(ξl)kϕm(ξ) + O(ρλ log2 λ) +for l = 1, . . . , m and k = 1, 2. +Proof. First, observe that +∂(ξl)k +� +Jǫ,λ(U) +� += DJǫ,λ(U) +� +∂(ξl)kU +� += − +� +Ω +� +∆U + ǫ2(eU − e−U) +� +∂(ξl)kU. +Now, we have that +� +Ω +∂(ξl)kU(−∆U) = +m +� +j=1 +aj +� +Ω +∂(ξl)kU(−∆Uj) = +m +� +j=1 +aj +� +Ω +ǫ2ewj(x)∂(ξl)kU. + +32 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +Using similar arguments as above and taking into account a suitable expan- +sion for ∂(ξl)kU (see ∂(ξ′ +l)kV in the proof of Proposition 4.1), we conclude +that +� +Ω +∂(ξl)kU(−∆U) = − 16π +m +� +j=1 +∂(ξl)kµj +µj ++ 8π∂2kHλ(ξl, ξl) ++ 8π +m +� +j=1 +j̸=l +ajal∂2kGλ(ξj, ξl) + O(ρλ log2 λ). +On the other hand, from similar arguments as above it follows that +ǫ2 +� +Ω +(eU − e−U)∂(ξl)kU = ǫ2 +m +� +j=1 +aj +� +Ω +(eU − e−U)∂(ξl)kUj = O(ρλ log2 λ). +Therefore, by using the choice of µj in (2.26) the claim follows . +This +completes the proof. +□ +Claim 3. The following expansion does hold +Fǫ,λ(ξ) = Jǫ,λ(U) + θ∗ +ǫ,λ(ξ), +where +|θ∗ +ǫ,λ(ξ)| = O(ρ2λ18 log2 λ| log(ρλ)|), +and +|∇θ∗ +ǫ,λ(ξ)| = O(ρλ18 log2 λ| log(ρλ)|2), +as ρλ19 → 0, uniformly on points ξ = (ξ1, . . . , ξm) ∈ Ωm satisfying the +constraints (2.10). +Proof. Since we have, Iǫ,λ(V ) = Jǫ,λ(U) and +Iǫ,λ(V (ξ′) + φ(ξ′)) = Jǫ,λ(U(ξ) + ˜φ(ξ)), +we write +Jǫ,λ(U + ˜φ) − Jǫ,λ(U) = Iǫ,λ(V + φ) − Iǫ,λ(V ) := A. +Let us estimate A first. Taking into account that DIǫ,λ(V + φ)[φ] = 0, a +Taylor expansion and (4.1) gives us +A = − +� 1 +0 +D2Iǫ,λ(V + tφ)[φ]2 t dt, += − +� 1 +0 +�� +Ωρ +[R + N(φ)] φ +− +� +Ωρ +ρ2ǫ2 � +eV (etφ − 1) + e−V (e−tφ − 1) +� +φ2 +� +t dt. +(5.9) +Therefore, we get +Iǫ,λ(V + φ) − Iǫ,λ(V ) = O(ρ2λ18 log2 λ| log(ρλ)|), + +SIGN-CHANGING SOLUTIONS +33 +since ∥R∥∗ ≤ Cρλ9 log λ, ∥N(φ)∥∗ ≤ C∥φ∥2 +∞ and ∥φ∥∞ ≤ Cρλ9 log λ| log(ρλ)|. +Let us differentiate with respect to ξ′. We use representation (5.9) and dif- +ferentiate directly under the integral sign, thus obtaining, for each k = 1, 2, +l = 1, . . . , m. +Using Lemma 4.1 and the computations in the proof, we +conclude that for k = 1, 2, l = 1, . . . , m +∂ξ′ +kl [Iǫ,λ(V + φ) − Iǫ,λ(V )] = O(ρ2λ18 log2 λ| log(ρλ)|2). +Now, taking ˜θǫ,λ(ξ′) = θ∗ +ǫ,λ(ρξ′) with θ∗ +ǫ,λ(ξ) = Fǫ,λ(ξ) − Jǫ,λ(U), we have +shown that +|˜θǫ,λ| + +1 +| log(ρλ)||∇ξ′ ˜θǫ,λ| = O(ρ2λ18 log2 λ| log(ρλ)|), +as ρλ9 log λ| log(ρλ)|2 → 0. The continuity in ξ of all these expressions is +inherited from that of φ and its derivatives in ξ in the L∞ norm. +□ +Therefore, from (5.3), previous claims and ∇θ∗ +ǫ,λ(ξ) = 1 +ρ∇˜θǫ,λ( ξ +ρ) we con- +clude the proof of Proposition 5.2. +□ +6. Proof of main results +Recall θ0 ∈ (0, ∞) is the unique minima of h in (2.21) in (0, +∞). We +denote +S∗ = {x ∈ Ω : d(x) = θ0}. +Concerning h, it is easy to see that +h(θ) = 4 log θ + O(1) +as θ → +∞, +h(θ) = −4 log θ + O(1) +as θ → 0+. +Furthermore, from (2.23) it follows that there is C′ = C′(K) such that +|Hλ(x, x) − h(λd(x)) + 4 log λ| ≤ C′ +λ , +for all x ∈ Ω satisfying K−1 +λ +≤ dist(x, ∂Ω) ≤ K +λ . +6.1. Symmetric case. Here we follow closely the arguments in [3]. For +this, we assume the domain Ω is such that Ω ∩ R × {0} ̸= ∅, and that it is +symmetric with respect to the reflection at R×{0}. In this setting, we have +Gλ in (1.6) is also symmetric with respect to this reflection in the following +sense: for x = (x1, x2) ∈ Ω, let ˜x = (x1, −x2), then +(6.1) +G(x, y) = G(˜x, ˜y) +for all x ̸= y. +In fact, defining ˜Gλ(x, y) = Gλ(˜x, ˜y), we have +−∆x ˜Gλ(x, y) = −∆xGλ(˜x, ˜y) = 8πδ˜y(˜x) = 8πδy(x), +meanwhile, for each x ∈ ∂Ω, using that ν(˜x) = (ν(x)1, −ν(x)2) we have +Rλ ˜Gλ(x, y) = ∇xGλ(˜x, ˜y) · (ν(x)1, −ν(x)2) + λGλ(˜x, ˜y) = RλG(˜x, ˜y) = 0, +from which, using the uniqueness of the Green function, we conclude the +symmetry property. The symmetry condition is inherited by Hλ as Hλ(˜x) = + +34 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +Hλ(x). For a m-tupe ξ = (ξ1, ..., ξm), we denote ˜ξ = (˜ξ1, ..., ˜ξm), and µξ,j = +µ˜ξ,j for µ as in (2.26), and ϕm(ξ) = ϕm(˜ξ), where ϕ is defined in (5.4). +We stress the notation by writing wξ,j(x), Uξ,j(x) as the functions in- +troduced in (2.6), (2.9) with the particular choice of ξ. Then, by related +invariance of the equation, it is possible to see that w˜ξ,j(˜x) = wξ,j(x), +and that x �→ U˜ξ,j(˜x) solves the same equation than Uξ,j(x), from which +Uξ,j(x) = U˜ξ,j(˜x) and therefore Rξ(x) = R˜ξ(˜x). +Now, if we denote c(ξ) = (cij(ξ))ij and (c(ξ), φξ) as the unique solution +for (4.1) given in Proposition 4.1, we consider the function ˜φ(x) = φ˜ξ(˜x), it +is possible to prove that ˜φ satisfies the same equation than φξ, from which +they are equal by uniqueness. +Then, by definition of F and using the previous symmetry properties, +we have F(˜ξ) = F(ξ) from which we have θ(ξ) = θ(˜ξ). +Thus, we have +that F is a C1 and symmetric with respect to x perturbation of the func- +tion ϕ. +This implies that F has critical points whenever the function +ξ �→ ϕ((ξ11, 0), (ξ12, 0), . . . , (ξ1m, 0)) has stable critical points. +Now we are ready to provide the +Proof of Theorem 1.1: Assume Ω is simply connected, contains the origin +and it is symmetric with respect to the x-axis. By symmetry and the ex- +pansion of the energy given in Proposition 5.2, it suffices to prove that there +is a (nondegenerate) critical point to the function +(t1, t2) �→ ϕ2((t1, 0), (t2, 0)), +t1, t2 ∈ (a, b), +where we identify the set {ξ ∈ Ω : ξ2 = 0} with the interval (a, b). With +a slight abuse of notation, we denote this function by ϕ2(t1, t2). We are +interested in sign-changing solutions, from which ϕ2 takes the form +ϕ2(t1, t2) = Hλ((t1, 0), (t1, 0)) + Hλ((t2, 0), (t2, 0)) − 2Gλ((t1, 0), (t2, 0)). +Using the expansion (2.23), we have +ϕ2(t1, t2) = −8 log λ+h(λd(t1, 0))+h(λd(t2, 0))−2Gλ((t1, 0), (t2, 0))+O(λ−1), +where O(λ−1) does not depend on (t1, t2). +Fix δ = (b − a)/4 and for K > 1 to be determined, consider the set +Ω0 = {(t1, t2) ∈ (a, b)2 : λd((ti, 0)) ∈ (K−1, K), i = 1, 2; |t1 − t2| > δ}. +Let us stress that our symmetry assumption implies that (t1, t2) ∈ ∂Ω0 if +and only if λd((ti, 0)) ∈ {K−1, K} for some i ∈ {1, 2}. First, we choose +(ξ∗ +1, ξ∗ +2) ∈ Ω0 with ξ∗ +i ∈ S∗, namely, λd(ξ∗ +i ) = θ0, for i = 1, 2 so that, by the +positivity of the Green’s function and (2.23) +ϕ2(ξ∗ +1, ξ∗ +2) = Hλ(ξ∗ +1, ξ∗ +1) + Hλ(ξ∗ +2, ξ∗ +2) − 2Gλ(ξ∗ +1, ξ∗ +2) +≤ Hλ(ξ∗ +1, ξ∗ +1) + Hλ(ξ∗ +2, ξ∗ +2) ≤ −8 log λ + 2h(θ0) + C′ +λ + +SIGN-CHANGING SOLUTIONS +35 +Let Cδ > 0 such that |G(x, y)| ≤ Cδ for all x, y ∈ Ω with |x − y| ≥ δ and +d(x), d(y) ≥ (λ log λ)−1. If λd((ti, 0)) = K−1 for some i (say i = 1), we have +that (ξ1, ξ2) ∈ ∂Ω0 and by (2.2), we can write +ϕ2(ξ1, ξ2) ≥ −8 log λ + h(K−1) + h(θ0) − Cδ − C′ +λ . +Now, we fix K large just depending on h(θ0) and Cδ(Ω) such that +h(K−1) > Cδ + h(θ0) + 2, +which is valid for all λ large enough such that (λ log λ)−1 ≤ K−1λ−1. Hence, +choosing λ larger, if necessary, we also get that 2C′ +λ < 1. The same estimate +can be found if λd(ti, 0) = K. From here, we deduce that for any (ξ1, ξ2) ∈ +∂Ω0 +ϕ2(ξ1, ξ2) ≥ −8 log λ + 2h(θ0) − h(θ0) + h(K) − Cδ(Ω) − C′ +λ +≥ −8 log λ + 2h(θ0) + C′ +λ + 1 +> ϕ2(ξ∗ +1, ξ∗ +2). +This implies that inf∂Ω0 ϕ2 > ϕ2(ξ∗ +1, ξ∗ +2), from which there is an interior min- +ima of ϕ2 in Ω0, which is stable under symmetric approximations. Therefore, +by using Proposition 5.2, there is an interior minima of Fǫ,λ in Ω0 for ǫ > 0 +small enough and λ > 0 large as above. This concludes the proof. +□ +6.2. Not simply connected case. Taking advantage of previous estimates +we are now ready to +Proof of Theorem 1.2: Assume that Ω is not simply connected with n holes +n ≥ 1, so that ∂Ω = ∪n+1 +i=1 Γi with Γi’s smooth closed curves satisfying +Γi ∩ Γj = ∅ for all i ̸= j. Fix δ = a +4, with +a = min{dist(Γi, Γj) : i ̸= j, i, j = 1, . . . , n + 1} +and for K > 1 to be determined, consider the set +ΩK = {(ξ1, . . . , ξm) ∈ Ωm | λd(ξi) ∈ (K−1, K), i = 1, 2; |ξi − ξj| > δ}. +Let us stress that our assumption m ≤ n + 1 implies that ξ ∈ ∂ΩK if and +only if λd(ξi) ∈ {K−1, K} for some i ∈ {1, . . . , m} with ξ = (ξ1, . . . , ξm). +For simplicity we shall assume that +{1, . . . , m} = I1 ∪ I2, +and +i ∈ Ik ⇐⇒ ai = (−1)k, +k = 1, 2, +so that |Ik| = mk, k = 1, 2, m1 + m2 = m and +m +� +i=1 +m +� +j=1 +i̸=j +aiajGλ(ξi, ξj) = +� +i,j∈I1 +i̸=j +Gλ(ξi, ξj)+ +� +i,j∈I2 +i̸=j +Gλ(ξi, ξj)−2 +� +i∈I1 +� +j∈I2 +Gλ(ξi, ξj). +In other words, we will find a sign-changing solution uε,λ to (1.1) having a +positive bubble centered at ξi with i ∈ I1 and a negative bubble centered at + +36 +P. FIGUEROA, L. ITURRIAGA, AND E. TOPP +ξj with j ∈ I2. Recall that for some Cδ > 0 fixed we have that |G(x, y)| ≤ Cδ +for all x, y ∈ Ω with |x − y| ≥ δ and d(x), d(y) ≥ (λ log λ)−1. Now, we +choose (ξ∗ +1, . . . , ξ∗ +m) ∈ ΩK with ξi ∈ S∗ for all i, namely, λd(ξ∗ +i ) = θ0, for +all i = 1, . . . , m so that, by the positivity of the Green’s function we obtain +that +ϕm(ξ∗ +1, . . . , ξ∗ +m) = +m +� +i=1 +Hλ(ξ∗ +i , ξ∗ +i ) + +� +i,j∈I1 +i̸=j +Gλ(ξi, ξj) + +� +i,j∈I2 +i̸=j +Gλ(ξi, ξj) +≤ m +� +− 4 log λ + h(θ0) + C′ +λ +� ++ +� +m1(m1 − 1) + m2(m2 − 1) +� +Cδ += −4m log λ + mh(θ0) + mC′ +λ ++ +� +m1(m1 − 1) + m2(m2 − 1) +� +Cδ. +On the other hand, if ξ ∈ ∂Ω, namely, λd(ξi) = K−1 for some i (say i = 1) +by (2.2), we can write +ϕm(ξ1, . . . , ξm) ≥ +m +� +i=1 +Hλ(ξi, ξi) − 2 +� +i∈I1 +� +j∈I2 +Gλ(ξi, ξj) +≥ (m − 1) +� +− 4 log λ + h(θ0) − C′ +λ +� +− 4 log λ + h(K−1) +− C′ +λ − 2m1m2Cδ +≥ −4m log λ + h(K−1) + (m − 1)h(θ0) − 2m1m2Cδ − mC′ +λ . +We fix K large just depending on h(θ0) and Cδ = C(Ω) such that +h(K−1) > +� +m1(m1 − 1) + m2(m2 − 1) + 2m1m2 +� +Cδ + h(θ0) + 2, +which is valid for all λ large enough such that (λ log λ)−1 ≤ K−1λ−1. Hence, +choosing λ larger, if necessary, we get that 2C′ +λ < 1. The same estimate can be +found if λd(ξi) = K. From here, we deduce that for any (ξ1, . . . , ξm) ∈ ∂ΩK +ϕm(ξ1, . . . , ξm) ≥ −4m log λ + h(K−1) + (m − 1)h(θ0) − 2m1m2Cδ − mC1 +λ +≥ −4m log λ + mh(θ0) + mC1 +λ ++ +� +m1(m1 − 1) + m2(m2 − 1) +� +Cδ + 1 +> ϕm(ξ∗ +1, . . . , ξ∗ +m). +This implies that inf∂ΩK ϕm > ϕm(ξ∗ +1, . . . , ξ∗ +m), from which there is a minima +of ϕm in ΩK, which is stable under small C1 perturbations. Therefore, by +using Proposition 5.2, there is an interior minima of Fǫ,λ in ΩK for ǫ > 0 +small enough and λ > 0 large as above. This concludes the proof. +□ + +SIGN-CHANGING SOLUTIONS +37 +Acknowledgements. +P. F. was partially supported by Fondecyt grant +1201884. +L. I. was partially supported by Fondecyt grants 1211766 and +1221365. E. T. was partially supported by Foncecyt grant 1201897. +References +[1] S. Baraket and F. Pacard, Construction of singular limits for a semilinear elliptic +equation in dimension 2, Calc. Var. Partial Differential Equations 6 (1)(1998), 1-38. +[2] D. Bartolucci and A. Pistoia, Existence and qualitative properties of concentrating +solutions for the sinh-Poisson equation, IMA J. Appl. Math. 72 (2007), no. 6, 706– +729. +[3] T. Bartsch, A. Pistoia and T. Weth, N-Vortex equilibria for ideal fluids in bounded +planar domains and new nodal solutions of the sinh-Poisson and the Lane-Emden- +Fowler equations. Commun. Math. Phys. 297 (2010), no. 3, 653–686 (2010). +[4] H. Berestycki and J. Wei, On singular perturbation problems with Robin boundary +condition, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (1) (2003) 199–230. +[5] H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of +−∆u = V (x)eu in two dimensions, Comm. Partial Differential Equations 16 (8-9) +(1991) 1223–1253. +[6] A. J. Chorin, Vorticity and Turbulence, Appl. Math. 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Wei, On the location and profile of spike-layer solutions to singularly +perturbed semilinear Dirichlet problems. Comm. Pure Appl. Math. 48 (1995), no. 7, +731–768. +[32] L. Onsager, Statistical hydrodynamics, Nuovo Cimento (9) 6 (1949), no. 2, 279–287. +[33] Y. Zhang, L. Shi, Concentrating solutions for a planar elliptic problem with large +nonlinear exponent and Robin boundary condition, Adv. Nonlinear Anal. (2019) 8: +1252–1285 +P. Figueroa +Instituto de Ciencias F´ısicas y Matem´aticas, Facultad de Ciencias, Univer- +sidad Austral de Chile, Campus Isla Teja, Valdivia, Chile. +Email address: pablo.figueroa@uach.cl +L. Iturriaga +Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa +Casilla V-110, Avda. Espa˜na, 1680 – Valpara´ıso, Chile. +Email address: leonelo.iturriaga@usm.cl +E. Topp +Departamento de Matem´atica y C.C., Universidad de Santiago de Chile, +Casilla 307, Santiago, Chile. +Email address: erwin.topp@usach.cl + diff --git a/mNE2T4oBgHgl3EQfJAb4/content/tmp_files/load_file.txt b/mNE2T4oBgHgl3EQfJAb4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4c632b6062ab3a99b20c3c3fa73b32ea5b9b157b --- /dev/null +++ b/mNE2T4oBgHgl3EQfJAb4/content/tmp_files/load_file.txt @@ -0,0 +1,1220 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf,len=1219 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='03688v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='AP] 9 Jan 2023 SIGN-CHANGING SOLUTIONS FOR THE SINH–POISSON EQUATION WITH ROBIN BOUNDARY CONDITION PABLO FIGUEROA, LEONELO ITURRIAGA, AND ERWIN TOPP Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Given ǫ ∈ (0, 1) and λ > 1, we address the existence of solutions for the Sinh-Poisson equation with Robin boundary value con- dition � ∆u + ǫ2(eu − e−u) = 0 in Ω ∂u ∂ν + λu = 0 on ∂Ω, where Ω ⊂ R2 is a bounded smooth domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We prove two existence results under a suitable relation between ǫ small and λ large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' When Ω is symmetric with respect to an axis, we prove the existence of a family of solutions uǫ,λ concentrating at two points with different spin, both located on the symmetry line and close to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In the second result, we assume Ω is not simply connected and we construct sign- changing solutions concentrating at points located close to the boundary, each of them on a different connected component of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Introduction Let Ω ⊂ R2 a bounded domain with smooth boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In this paper we are interested in study of the singular perturbation problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) � ∆u + ǫ2(eu − e−u) = 0 in Ω, Rλu := ∂u ∂ν + λu = 0 on ∂Ω, where ν(x) denotes the exterior unit normal on x ∈ ∂Ω, and ǫ, λ are positive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Our main concern is the construction of sign-changing solutions when ǫ is small, and simultaneously, λ is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Robin boundary condition (also known as boundary condition of the third type) can be seen as a combination of Neumann and Dirichlet boundary condition, in a proportion cast by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In [4], Berestycki and Wei study con- centration phenomena for the least energy solution of equations of Ni-Takagi type with Robin boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In fact, they prove the existence of λ∗ such that, for λ ≥ λ∗, the problem resembles the behavior of the Dirich- let problem studied (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' [31]), meanwhile for λ < λ∗ problem is closer to Date: January 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 35J25, 35B25, 35B38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Concentrating solutions, sinh-Poisson equation, Robin bound- ary condition, Lyapunov-Schmidt reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 1 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP the one with Neumann boundary condition (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In terms of applica- tions, Robin boundary condition is considered in biological models [16], and thermal conductivity [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' If we formally take the limit λ → ∞ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1), the problem becomes � ∆u + ǫ2(eu − e−u) = 0, in Ω, u = 0, on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2) This equation is tipically referred as sinh-Poisson equation with Dirich- let boundary condition, and it has connection with the description of two- dimensional turbulent Euler flows, see Onsager [32, 23, 28, 6, 27] for a phys- ical discussion of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Roughly speaking, the location of vortices in the flow can be described through concentration points of the solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In this respect, Bartolucci and Pistoia [2] prove that for every m ∈ N, there exists a solution to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2) that concentrates around stable critical points of the Hamiltonian associated to the Dirichlet problem in Ω, given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3) ϕm(ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', ξm) = m � i=1 H(ξi, ξi) + � i̸=j aiajG(ξi, ξj), (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', ξk) ∈ Ωk, as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Here, ai ∈ {−1, 1} for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', m determine the spin config- uration of the concentrating points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' G is the Green function with Dirichlet boundary condition � −∆xG(x, y) = 8πδy(x) in Ω, G(x, y) = 0 on ∂Ω, and H is the regular part of the Green function, also referred as the associ- ated Robin function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Moreover, it is proven in [2] that uǫ(x) → 8π k � i=1 aiG(x, ξi) in C1,σ(¯Ω \\ {ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', ξm}), as ǫ → 0, and the solution concentrates each ξi with a sign determined by ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Concen- tration points are away to the boundary in consonance with the Dirichlet condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Positive concentrating solutions, concentrating around critical points of a Hamiltonian function draws back to the early nineties in the seminal work of Nagasaki and Suzuki [29] for the Liouville equation (namely, when eu − e−u is replaced by eu in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2)) with Dirichlet boundary condition, and it is shown the effect of the domain determines the existence of concentration, and also the location of the blow-up points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In the case of the sinh-Poisson equa- tion, negative concentration points are allowed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In [2], the authors provide the existence of solutions with two concentration points with different sign, provided Ω is axially symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This result was later extended by Bartsch, Pistoia and Weth in [3], where the authors prove the existence of solutions SIGN-CHANGING SOLUTIONS 3 with an arbitrary number of concentrating points, located in the symme- try axis and alternating sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For general domains, they prove the result for 3 and 4 concentrating points, under suitable configuration of the spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We also would like to mention the contributions in [14, 15] concerning con- struction of solutions to equations with nonlinearities of exponential type in dimension two, in [18, 20] for compact Riemann surfaces, and in [13] for fractional equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Concerning Robin boundary condition, in [12] the authors address the Liouville equation with Robin boundary condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='4) � ∆u + ǫ2eu = 0 inΩ, Rλu = 0 on ∂Ω, and prove the existence of positive solutions with arbitrary number of con- centration points, located at critical points of the Hamiltonian function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='5) ϕm(ξ) = m � i=1 Hλ(ξi, ξi) + m � i̸=j Gλ(ξi, ξj), ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', m) ∈ Ωm, where, this time, Gλ is the Green function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='6) � −∆Gλ(x, y) = 8πδy(x) x ∈ Ω Rλ(Gλ(·, y)) = 0 on ∂Ω, and in this case, Hλ is the associated Robin function, defined as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='7) Hλ(x, y) = Gλ(x, y) − Γ(x − y), x, y ∈ Ω, where Γ(x−y) = −4 log |x−y| is the fundamental solution for the Laplacian in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The analysis in [12] (see also [7, 33]) uses in a crucial way the asymptotic behavior Hλ when λ is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Accurate asymptotics can be found in D´avila, Kowalczyk and Montenegro in [11], showing that the behavior of the map x �→ Hλ(x, x) in the normal direction to the boundary develops a strict minima at distance of order O(λ−1) to ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Thus, critical points for ϕm in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='5) can be found for tuples (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', ξm) such that each ξi is sufficiently close to the boundary, and away each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In fact, two types of different solutions can be found, one associated to minima of ϕm, and the other associated to a critical point of linking-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Our approach follows several of the concepts discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In the first main result, we rely on axial symmetry just as in [2, 3] to conclude the existence of sign-changing, two-point concentrating solution in a certain regime of ǫ small and λ large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Assume that Ω ⊂ R2 is bounded domain with smooth bound- ary, and symmetric with respect to the axis x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Denote (a, b) = Ω ∩ R × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, for each α > 16, there exist λ0 > 1, ǫ0 ∈ (0, 1) such that, for every λ ≥ λ0 and ǫ satisfying ǫλα ≤ ǫ0, problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) has a solution uǫ,λ that concentrates with different sign at two points ξi = (ti, 0), i = 1, 2, with |t1 − a|, |t2 − b| = O(λ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP Here, by concentration at a point ξ ∈ Ω we mean that for all δ ∈ (0, 1), supBδ(ξ) |uǫ,λ| → +∞ as ǫ → 0, λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Notice that by the invariance of the problem, if uǫ,λ is solution, −uǫ,λ is also a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The proof of the previous theorem rely on Lyapunov-Schmidt reduction, and most of the technical arguments are performed over the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='17), equivalent to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Our approach in the construction of approximate solu- tion are rather close to those present in [12, 17, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We use a two-parameter family of entire solutions of the Liouville equation R2 as the first ansatz, in junction with smooth correctors that allows us to satisfy the boundary con- dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The reduced problem corresponds to that of adjusting variationally the location of the concentrating points, as critical points of an energy func- tional associated to the weak formulation of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In a saturation regime of the parameters, that functional is close to the Hamiltonian ϕm, which is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='8) ϕm(ξ) = m � i=1 Hλ(ξi, ξi) + m � i̸=j aiajGλ(ξi, ξj), ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', m) ∈ Ωm, with ai ∈ {−1, 1} for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='., m (m = 2 in the setting of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Notice that in contrast with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='5), this time the contribution of the Green function maybe unbounded from below for tuples (ξ1, ξ2) ∈ Ω × Ω close to the diagonal if a1a2 = −1, and this makes more difficult to locate concen- trating points as minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The relation among ǫ and λ in the theorem allows us to control the error terms to get the criticality from the minima of the Robin function showed in [11], which occurs close to boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In partic- ular, the method leads to a relation among ǫ and λ involving logarithmic corrections that explains the corresponding hypothesis in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1, see Proposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For simplicity, we state our main result in a simpler form, at the expense of a non sharp estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Concentration near the boundary is in big contrast with previous results regarding interior concentration, see for instance [29, 5, 24, 26, 1, 19, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We believe that by methods similar to the ones presented here, solutions with interior concentration can be constructed for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1), since we expect to have Hλ(·, ξ) → H(·, ξ) as λ → ∞, locally uniform in Ω, where H is the regular part of the Green function with Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' These solution are associated to maxima for the Hamiltonian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We do not pursue in this direction and focus on concentration near the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In our second main result, we are able to replace the symmetry assumption of the previous theorem by a topological one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Assume that Ω is not simply connected, and let Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='., Γn, n ∈ N, be the connected components of ∂Ω, and let k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, for each α > 16 there exist λ0 > 1 and ǫ0 ∈ (0, 1) such that for each λ ≥ λ0 and each ǫ satisfying ǫλα ≤ ǫ0, problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) has a sign-changing solution uǫ,λ that concentrates at points ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', ξk ∈ Ω, with dist(ξi, Γi) = O(λ−1) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', k, and where Γi ̸= Γj if i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' SIGN-CHANGING SOLUTIONS 5 This theorem follows the same strategy of the previous result, where as before the concentration points are also minima to the Hamiltonian func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In this case, the uniform distance among connected components of the boundary of the domain allows us to control the contribution of the Green function in ϕm, no matter the sign of the interacting points is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In fact, in the notation of the theorem, we have �n k=1 2k�n k � different solutions (including the ones concentrating at one point, and the solutions due to the invariance of the equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' It is well-known that the use of index/linking arguments have been suc- cessful for the study of criticality of smooth functionals, and it has been employed in problems similar to ours, see for example [17, 1, 12] and its references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' It would be interesting to know if solutions of saddle-point type could be obtained for this problem, but we did not pursue in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In Section 2, describing a first ap- proximation solution to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) and estimating the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Section 3 is devoted to perform the finite dimensional reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In Section 4 we study the associated nonlinear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Section 5 contains the asymptotic expan- sion of the reduced energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Finally, in Section 6 we will prove our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Preliminaries and ansatz for solutions In this section, we denote d(x) = dist(x, ∂Ω) for each x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Preliminaries about Green and Robin function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We start pro- viding some estimates for the Green function with Robin boundary condition Gλ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='6), and its regular part Hλ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let ξ ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For each δ > 0 small, there exists C depending on δ and Ω such that if d(ξ) ≥ δ we have ∥Hλ(·, ξ)∥∞ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' There exists δ ∈ (0, 1) small and CΩ > 0 large depending on the smoothness of Ω, such that, for all λ large in terms of δ, if (λ log λ)−1 ≤ d(ξ) ≤ δ, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) − CΩ + 2 log |x − ξ∗| ≤ Hλ(x, ξ) ≤ CΩ, where ξ∗ ∈ Ωc is the reflection of ξ with respect to the tangent to ∂Ω supported at the projection of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In particular, for any 0 < δ there exists Cδ > 0 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2) ∥Gλ(·, ξ)∥L∞(Ω\\Bδ(ξ)) ≤ Cδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let ξ ∈ Ω close to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Up to rotation and translation, we can assume that the projection of ξ onto the boundary is the origin, and ν(0) = −e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We denote ξ∗ = (0, −d(ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP We divide the analysis depending on the distance of ξ to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let ˜δ ∈ (0, 1) to be specified later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' By definition of Hλ, for each x ∈ ∂Ω we have RλHλ(x) = Rλ(−Γ) =|x − ξ|−2(x − ξ) · ν(x) + λ log |x − ξ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' If d(ξ) ≥ ˜δ, then it is easy to see using maximum principle for Robin boundary condition (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1 in [12]) that ∥Hλ(·, ξ)∥∞ ≤ C˜δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From here, we assume (λ log λ)−1 ≤ d(ξ) ≤ ˜δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For x ∈ ∂Ω such that |x − ξ| ≥ ˜δ we have |RλHλ(x)| ≤ λCΩ, where CΩ = log max{1, diam(Ω)} + ˜δ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From here, we concentrate on the points x on the boundary such that |x − ξ| ≤ ˜δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We start with upper bounds for RλHλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' If |x − ξ| ≤ λ−1, then RλHλ(x) ≤ λ log λ + λ log |x − ξ| ≤ 0, meanwhile, for λ−1 ≤ |x − ξ| ≤ ˜δ we have RλHλ(x) ≤ λ + λ log(˜δ) ≤ 0, provided ˜δ ≤ e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From here, using maximum principle for Robin boundary condition, using CΩ as supersolution, we conclude the upper bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For the lower bound, we start recalling that for all |x − ξ| ≤ ˜δ on the boundary, we have |x − ξ| and |x − ξ∗| are comparable, in the sense that there exists cΩ ∈ (0, 1) such that cΩ ≤ |x − ξ| |x − ξ∗| ≤ c−1 Ω , x ∈ ∂Ω ∩ B(ξ, 3˜δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This is a consequence of the smoothness of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' If |x − ξ| ≥ λ−θ for θ ∈ (1/2, 1), we have RλHλ ≥ − |x − ξ|−1 + λ log |x − ξ| ≥ − λθ + λ log |x − ξ∗| + λ log cΩ ≥2λ(log |x − ξ∗| + log cΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Tomamos S = log |x − ξ∗| y RλS = � x − ξ∗ · ν(x)|x − ξ∗|−1 + λ log |x − ξ∗| ≤ c−θ Ω λθ + λ log |x − ξ∗| ≤ λ(c−θ Ω λθ−1 + log |x − ξ∗|) ≤1 2λ log |x − ξ∗| SIGN-CHANGING SOLUTIONS 7 If |x − ξ| ≤ λ−θ, we use that x can be writen as x = (x′, ψ(x′)) with ψ(0) = 0, ψ′(0) = 0, ψ′′ bounded, from which � (x − ξ) · ν(x) = x′ψ′ − ψ + d(ξ) � 1 + (ψ′)2� |x′|2 + (ψ − d(ξ))2 ≥ 1 2, for all λ large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This implies that RλHλ ≥ cΩ 2|x − ξ∗| + λ log |x − ξ∗| + λ log cΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Thus, we use the function S(x) = 2 log |x−ξ∗|−2 log cΩ, which, in view of the estimates above, satisfies RλHλ ≥ RλS on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Hence, we use maximum principle again, to conclude that Hλ ≥ S in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This leads to the lower bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The remaining estimates can be easily obtained from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) and the definition of H, G and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Ansatz for the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Reduced problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In this section we will construct an approximation of the solution to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then we estimate the error of such approximation in a suitable norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The basic idea is to consider a parameter µ > 0 and the functions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3) wµ(x) = log 8µ2 (µ2 + |x|2)2 , x ∈ R2 which are solutions to the Liouville equation in the whole plane (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='4) ∆u + eu = 0 in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' An actual solution will have an asymptotic profile as ǫ → 0 and λ → +∞ which resembles these solutions, properly translated and rescaled in terms of these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Specifically, we choose our scaling parameter as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='5) ρ := ǫ λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Given m ∈ N, we consider {µj}m j=1 ⊂ (0, +∞), {ξj}m j=1 ⊂ Ω, and for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', m, we denote wj(x) = wµjρ(x − ξj) + 2 log 1 ǫ = log 8µ2 j (µ2 jρ2 + |x − ξj|2)2 + 2 log ρ ǫ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='6) for all x ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' It is easy to see that for each j the function wj satisfies ∆wj + ǫ2ewj = 0 in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='7) In order to satisfy the Robin boundary condition we introduce harmonic functions Hj satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='8) � −∆Hj = 0 in Ω, Rλ(Hj) = −Rλ(wj) on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Our single-point ansatz takes the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='9) Uj(x) = wj(x) + Hj(x), x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 8 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP Using the explicit form of wj in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='6) together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='8), we can use maximum principle in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1 in [12] over x �→ Hλ(x, ξj) − Hj(x) to conclude that Hj(x) + 2 log ρ ǫ + log(8µ2 j) − Hλ(x, ξj) = O �µ2 jρ2 d2 j � + O � µ2 jρ2 λd3 j � , uniformly in ¯Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Here and in what follows, we have adopted the notation d(x) = dist(x, ∂Ω) for x ∈ Ω, and dj = d(ξj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Thus, in view of the definition of wj and the estimate above, we have the expansion Uj(x) = Gλ(x, ξj) − 2 log � 1 + µ2 jρ2 |x − ξj|2 � + O �µ2 jρ2 d2 j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now we provide the first estimates concerning the ansantz when we con- sider the expected location of the concentration points {ξj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let {ξj}m j=1 ⊂ Ω, {µj}m j=1 ⊂ (0, +∞) satisfying dj ∈ (δλ−1, δ−1λ−1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='10) µj ∈ (δ, δ−1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='11) for some δ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', m we have Hj(x) = Hλ(x, ξj) − log(8µ2 j) + 4 log λ + O � ρ2λ2� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='12) uniformly in ¯Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' and for each K compact subset of ¯Ω \\ {ξj}, there exists CK > 0 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='13) |Uj(x) − Gλ(x, ξj)| ≤ CKρ2 + O(λ2ρ2) in K, where the O term is independent of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Consider {aj}m j=1 with aj ∈ {−1, 1} for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We introduce the first approximation of the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) as U(x) := m � j=1 ajUj(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='14) Following the directions of [17], we study the problem in expanded vari- ables depending on ρ given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For this, we consider the function v(y) = u(ρy) for y ∈ Ωρ := {ξ ∈ R2 : ρξ ∈ Ω}, so that u is a solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) if and only if v satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='15) � ∆v + ρ2ǫ2(ev − e−v) = 0 in Ωρ, Rλρv = 0 on ∂Ωρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We also adopt the notation ξ′ j = 1 ρξj, Vj(y) = Uj(ρy) and V (y) = U(ρy) for y ∈ Ωρ and we look for solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='15) in the form v = V + φ with φ : Ωρ → R small in an adequate norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' SIGN-CHANGING SOLUTIONS 9 Thus, considering v = V + φ, problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='15) can be equivalently formu- lated as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='16) � Lφ = −[R + Λφ + N(φ)] in Ωρ, Rλρφ = 0 on ∂Ωρ, where Lφ = ∆φ + Wφ, with W = m � j=1 8µ2 j (µ2 j + |y − ξ′ j|2)2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='17) Λφ = � (ǫρ)2(eV + e−V ) − W � φ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='18) N(φ) = (ǫρ)2 � eV (eφ − φ − 1) − e−V (e−φ + φ − 1) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='19) R(y) = ∆V + (ǫρ)2(eV − e−V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='20) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' First estimates for problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In what follows, we recall rel- evant estimates concerning the Robin function Hλ we collect from [11] for the two-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Define h(θ) = −4 log(2θ) + 8 � ∞ 0 e−t log (2θ + t) dt, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='21) v(θ) = −2θ − 4θ � ∞ 0 e−2θs (1 + s)2 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='22) It is possible to see that h : (0, ∞) → R has a unique nodegenerate minimum θ0 ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The Robin function obeys the expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='23) Hλ(x, x) = −4 log λ + h(λd(x)) + λ−1κ(˜x)v(λd(x)) + O(λ−1−α) for each x such that a1 ≤ λd(x) ≤ a2 for some constants 0 < a1 < a2, and all λ > 0 large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Here, α ∈ (0, 1), ˜x is the projection of x onto the boundary, κ(˜x) is the mean curvature of ∂Ω at ˜x, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1 in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For reasons that will be made clear later in the next section, for measur- able functions h : Ωρ → R we introduce the following weighted norm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='24) ∥h∥∗ = sup y∈Ωρ \uf8eb \uf8ed m � j=1 (1 + |y − ξ′ j|)−2−σ + ρ2 \uf8f6 \uf8f8 −1 |h(y)|, for 0 < σ < 1 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let {ξj}m j=1 ⊂ Ω such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='10) holds for some δ > 0, and assume further that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='25) |ξi − ξj| ≥ δ for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Set {µj}m j=1 as log(8µ2 j) = Hλ(ξj, ξj) + � i̸=j aiGλ(ξi, ξj) + 4 log λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='26) 10 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP Then, there exists C > 1 just depending on δ such that C−1 ≤ µj ≤ C for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='., m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Moreover, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='27) ∥R∥∗ ≤ Cρλ9 log λ = Cǫλ7 log λ, and ∥Λφ∥∗ ≤ Cρλ9 log λ∥φ∥L∞(Ωρ), for all φ ∈ L∞(Ωρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Proof: The uniform estimates for µj are a consequence of the expansion of the Robin function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='23), together with the estimates for the Green function in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We estimate R by dividing the analysis in different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Note that by definition of Vi and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='7) we see that for any i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m ∆Vi(y) = ρ2∆wi(ρy) = −ρ2ǫ2ewi(ρy) = − 8µ2 i ρ4 (µ2 i ρ2 + |ρy − ξi|2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Case 1: Assume that |y − ξ′ j| ≥ δ ρ for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', m with δ > 0 small but independent of ǫ, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, for each j and for y ∈ Ωρ with |y − ξ′ j| ≥ δ ρ we have from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='7) that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='28) ∆Vj(y) = − 8µ2 jρ4 (µ2 jρ2 + |ρy − ξj|2)2 = O(ρ4), where the O term is uniform if δ > 0 and µj > 0 are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' On the other hand, using the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='13) we have max a=±1{eaVj(y)} ≤ C max a=±1{eaGλ(ρy,ξj)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1 we deduce that eVj is uniformly bounded away ξj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='28) leads us to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='29) |R(y)| = |∆V + ρ2ǫ2g(V )| ≤ C(ρ4 + ρ2ǫ2) in Ωρ \\ m � j=1 Bδ/ρ(ξ′ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Case 2: Now, assume that δ/(ρλ log λ) ≤ |y − ξ′ j| ≤ δ/ρ for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' As in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='28) in the previous case, we see that |∆Vj(y)| = 8µ2 jρ4 (µ2 jρ2 + δ2[λ log λ]−2)2 = O(ρ4λ4 log4 λ) = O(ρ2ǫ2 log λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Using again estimates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='13), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) and the fact that dj = O(λ−1), we arrive at eV (y) = eajGλ(ρy,ξj)+� l̸=j alGλ(ρy,ξl)+O(ρ2λ2 log2 λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' SIGN-CHANGING SOLUTIONS 11 If aj = 1 then we get that eV (y) = eHλ(ρy,ξj)+O(1) |ρy − ξj|4 = O � λ4 ρ4|y − ξ′ j|4 � and if aj = −1 then we find that eV (y) = |ρy − ξj|4e−Hλ(ρy,ξj)+O(1) = O � ρ4λ4|y − ξ′ j|4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Taking into account that aj = 1 becomes −1 in e−V (y) and aj = −1 becomes 1 in e−V (y), a similar argument to estimate e−V (y) lead us to obtain |ρ2ǫ2g(V (y))| ≤ Cρ2ǫ2 � λ4 ρ4|y − ξ′ j|4 + ρ4λ4|y − ξ′ j|4 � , for y ∈ Bδ/ρ(ξ′ j) \\ Bδ/(ρλ log λ)(ξ′ j) and for some constant just depending on δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This estimate allows us to conclude that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='30) |R(y)| ≤ Cǫλ7 log λ m � j=1 1 1 + |y − ξ′ j|3 , y ∈ Ωρ ∩ (Bδ/ρ(ξ′ j) \\ Bδ/(λρ log λ)(ξ′ j)), in view of ρ2ǫ2 � log λ + λ4 ρ4|y − ξ′ j|4 + ρ4λ4|y − ξ′ j|4 � [1 + |y − ξ′ j|3] ≤ Cǫλ7 log λ for y ∈ Ωρ ∩ (Bδ/ρ(ξ′ j) \\ Bδ/(λρ log λ)(ξ′ j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Case 3: Finally, assume that |y −ξ′ j| ≤ δ/(ρλ log λ) for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In this case, by the condition over the points ξj we have that |ξj − ξi| ≥ δ when i ̸= j (we can choose ˜δ > δ if necessary) and therefore we can use the estimates of Case 1 for the portion of R relative to points indexed by i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Note that we get that for j (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='31) ∆V (y) = −aj 8µ2 j (µ2 j + |y − ξ′ j|2)2 + O(ρ4), in the considered region where the O-term depends only on δ and µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now, we deal with the exponential term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We first assume that aj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Notice that ρ2ǫ2g(V ) = ρ2ǫ2(ewj(ρy)eHj(ρy)+� i̸=j aiVi − e−V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' By estimates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='12) we see that eV (y) = ewj(ρy)eHj(ρy)+� i̸=j aiVi = ewj(ρy) exp � Hλ(ρy, ξj) + � i̸=j aiGλ(ρy, ξi) − log(8µ2 j) + 4 log λ + O �µ2 jρ2 d2 j �� 12 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP At this point, we notice that by a first-order Taylor expansion, for y in this region we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='32) Hλ(x, ξj) = Hλ(ξj, ξj) + O(λ log λ |x − ξj|), and for i ̸= j we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='33) Gλ(x, ξi) = Gλ(ξj, ξi) + O(λ log λ |x − ξj|), where the O terms are inependent of ǫ and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Using this and assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='26), we arrive at ρ2ǫ2eV (y) = ρ2ǫ2ewj(ρy) · exp � O(λ log λ |ρy − ξj|) + O �µ2 jρ2 d2 j �� = 8µ2 j (µ2 j + |y − ξ′ j|2)2 � 1 + O(ρλ log λ|y − ξ′ j|) + O(ρ2λ2) � Using the same computation, it is possible to see that ρ2ǫ2e−V (y) = ρ2ǫ2 (µ2 jρ2 + |ρy − ξj|2)2ǫ2 8µ2 jρ2 O(1) = O � ρ4ǫ4[1 + |y − ξ′ j|]4� = O � ρ4ǫ4� δ ρλ �4� = O(ρ2ǫ2) From here, we conclude that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='34) ρ2ǫ2g(V ) = 8µ2 j (µ2 j + |y − ξ′ j|2)2 � 1 + O(ρλ log λ|y − ξ′ j|) + O(ρ2λ2) � + O(ρ2ǫ2) Then, joining the above estimates and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='31) we get for y ∈ Bδ/(ρλ log λ)(ξ′ j) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='35) R(y) = 8µ2 j (µ2 j + |y − ξ′ j|2)2 � O(ρλ log λ|y − ξ′ j|) + O(ρ2λ2) � + O(ρ4 + ρ2ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We finish the proof by assuming aj = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Similarly as above, we have the following estimates ρ2ǫ2g(V ) = ρ2ǫ2(eV − ewj(ρy)eHj(ρy)+� i̸=j aiVi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ρ2ǫ2e−V (y) = 8µ2 j (µ2 j + |y − ξ′ j|2)2 � 1 + O(ρλ log λ|y − ξ′ j|) + O(ρ2λ2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' and ρ2ǫ2eV (y) = O � ρ4ǫ4[1 + |y − ξ′ j|]4� = O(ρ2ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Thus, we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='34) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='31) we also obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From the choice of ρ, the definition of ∥·∥∗ and estimates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='29), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='30) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='35) we deduce (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The estimate for Λφ follows the same computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' □ SIGN-CHANGING SOLUTIONS 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Linear problem In this section we shall reduce the solvability of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='16) by using the so- called Lyapunov-Schmidt finite dimensional variational reduction and prove the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In this procedure an important step is the solvability theory for the linear operator, obtained as the linearization of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='16) at the approx- imating solution V , namely, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Observe that, as ǫ → 0 and λ → +∞, formally the operator L, around a point ξ′ j, approaches Lj defined in R2 as Lj(φ) = ∆φ + 8µ2 j (µ2 j + |y − ξ′ j|2)2 φ, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We start this section with some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Given vj = vµj the entire functions solving the equation ∆v + 8µ2 j (µ2 j + |x|2)2 v = 0 x ∈ R2, as a consequence of the invariance under dilations and traslations of the problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='4), it is possible to find nontrivial solutions of this equations and, for each x ∈ R2, are denoted by zij(x) = ∂ ∂ζi vj(|x + ζ|) ��� ζ=0 = 4µjxi µ2 j + |x|2 , i = 1, 2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', m, z0j(x) = ∂ ∂s(vj(|sx|) + 2 log(s)) ��� s=1 = µ2 j − |x|2 µ2 j + |x|2 , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) The kernel of Lj in L∞(R2) is non-empty and is spanned by the functions Zij, i = 0, 1, 2, due to the intrinsic invariances of the problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='16), where Z0j(y) = µ2 j − |y − ξ′ j|2 µ2 j + |y − ξ′ j|2 , Zij(y) = 4µj(y − ξ′ j)i µ2 j + |y − ξ′ j|2 , i = 1, 2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2) see [1] for a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Consider a large but fixed number R0 > 0 and a radial smooth cut-off function χ with χ(r) = 1 if r < R0 and χ(r) = 0 if r > R0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Write χj(y) = χ � |y − ξ′ j| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3) Recalling L in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='17), the main result of this section is the following Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' There exist λ0 > 0 and ǫ0 > 0 such that for λ ≥ λ0 and ǫ > 0 satisfying 0 < ρλ < ǫ0, for any points ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξm) satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='10) and for any h ∈ L∞(Ωρ) with ∥h∥∗ < +∞, there is a unique solution 14 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP φ := Tλ(h) and coefficients cij(ξ) ∈ R, i = 1, 2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m of problem \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 L(φ) = h + 2 � i=1 m � j=1 cijχjZij in Ωρ Rρλφ = 0 on ∂Ωρ � Ωρ χjZijφ = 0 for i = 1, 2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='4) Moreover, the map ξ �→ φ(ξ) is differentiable with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='5) ∥φ∥L∞(Ωρ) ≤ C| log(ρλ)| ∥h∥∗, |cij| ≤ C∥h∥∗, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='6) ∥∂(ξl)kφ∥L∞(Ωρ) ≤ C| log(ρλ)|2 ∥h∥∗, for l = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m, k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In order to prove the above result, we require some a priori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We start with the following result which can be found in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2 in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' There exist λ0 > 0 and ǫ0 > 0 such that for λ ≥ λ0 and ǫ > 0 satisfying 0 < ρλ < ǫ0, any family of points ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξm) satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='10) and for any solution φ of the problem \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ∆φ + Wφ = h in Ωρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ∂φ ∂ν + ρλφ = g on ∂Ωρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='7) satisfying the orthongonality conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='8) � Ωρ φZijχj = 0 for i = 0, 1, 2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m, we have the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='9) ∥φ∥L∞(Ωρ) ≤ C(∥h∥∗ + 1 λρ∥g∥L∞(∂Ωρ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The next step is to find an a priori estimate for the solution avoiding the elements of the kernel due to dilations χjZ0j’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' There exist λ0 > 0 and ǫ0 > 0 such that for λ ≥ λ0 and ǫ > 0 satisfying 0 < ρλ < ǫ0, any family of points ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξm) satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='10) and for any solution φ of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='7) with g = 0, satisfying the orthogonality conditions � Ωρ φZijχj = 0 for i = 1, 2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m, we have ∥φ∥L∞(Ωρ) ≤ C| log(ρλ)| ∥h∥∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' SIGN-CHANGING SOLUTIONS 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We will construct functions ˜zj and fix constants bj ∈ R such that the function ˜φ = φ + m � j=1 bj˜zj, satisfies, for certain bj ∈ R, the orthogonality conditions with respect to the dilations, and subsequenly apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We divide the proof in several steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='- Construction of the correctors ˜zj: Here we require certain estimates concerning the Green function with homogeneous Robin boundary condition in the upper half-space, that is −∆G(·, y) = δy in R2 +;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' RaG = 0 on {x2 = 0}, where a > 0, and in this case ν = −e2 is the exterior unit normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' As it can be seen in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 120 in [22], we have the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='10) Ga(x, y) = Γ(x − y) − Γ(x − y∗) − 2 � +∞ 0 e−as(x2 + s + y2) |x + se2 − y∗|2 ds, and where for y = (y1, y2) ∈ R2 +, we denote y∗ = (y1, −y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let Fj : Br(ξj) ∩ Ω → R2 + be a conformal mapping such that if we denote ˆξj ∈ ∂Ω is the projection of ξj onto the boundary, then Fj(ˆξj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In addition, DFj(ˆξj) is a rotation (making DFj(ˆξj)ν(ˆξj) = −e2), and is such that Fj(Br(ξj) ∩ ∂Ω) is an interval in {x : x2 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Thus, we consider its expanded version Fj,ρ(y) = ρ−1Fj(ρy), y ∈ Br/ρ(ξ′ j) ∩ Ωρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Since Fj is a conformal mapping, we have the existence of a constant c ∈ (0, 1) (just depending on Ω) such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='11) c|z1 − z2| ≤ |Fj,ρ(z1) − Fj,ρ(z2)| ≤ c−1|z1 − z2|, for all z1, z2 ∈ Br/ρ(ξ′ j) ∩ Ωρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Taking into account that dist(ξj, ∂Ω) = O(λ−1), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='12) Fj,ρ(ξ′ j) = ρ−1dje2 + O(ρ−1λ−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let η2j be a cut-off function making η2j = 1 in B r 2ρ (ξ′ j), η2j = 0 in Bcr ρ (ξ′ j), |Dη2j| ≤ Cρ, |D2η2j| ≤ Cρ2 in R2, and such that ∂η2j ∂ν = 0 in ∂Ωρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This can be done using the conformal map Fj and an adequate scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Finally, we consider a cut-off function η1j such that η1j = 1 in BR(ξ′ j), η1j = 0 in BR+1(ξ′ j) and uniform bounds for its first and second-order deriva- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For y ∈ Ωρ, we define ˆz2j(y) = η2,j(y) 1 log(2ρ−1dj)Z0j(y)gj(y), 16 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP where, for simplicity, we have denoted (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='13) gj(y) = Gλρ(Fj,ρ(y), Fj,ρ(ξ′ j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Thus, we define ˜z = � j bj˜zj, with ˜zj = η1jZ0j + (1 − η1j)ˆz2j, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='14) with bj ∈ R given by the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='15) bj � Ωρ χj|Z0j|2 + � Ωρ χjZ0jφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='- Application of the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2: By linearity L˜φ = h + L˜z in Ωρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Rλρ ˜φ = Rλρ˜z on ∂Ωρ, and by the choice of bj, we are in position to use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, we see that ∥˜φ∥∞ ≤ C � ∥h∥∗ + � j |bj|(∥L˜zj∥∗ + 1 λρ∥Rλρ˜zj∥L∞(∂Ωρ)) � , from which we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='16) ∥φ∥∞ ≤ C � ∥h∥∗ + � j |bj|(∥˜zj∥∞ + ∥L˜zj∥∗ + 1 λρ∥Rλρ˜zj∥L∞(∂Ωρ)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now we estimate each term in the right-hand side above, stating with the terms concerning ˜zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='- Estimates for gj in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='13): We start with some estimates concerning the function gj, consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='10) and the properties of the conformal map Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' First, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='12), for all y ∈ Ωρ with |y − ξ′ j| ≥ R, we have ����� � +∞ 0 e−λρs (Fj,ρ(y))2 + s + (Fj,ρ(ξ′ j))2 |Fj,ρ(y) + se2 − Fj,ρ(ξ′ j)∗|2 ds ����� ≤ � +∞ 0 ρe−λρsds |s + dj + O(λ−2)| ≤ C, for some C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='11), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='12), we have Γ(Fj,ρ(y) − Fj,ρ(ξ′ j)) = O(log |y − ξ′ j|), and Γ(Fj,ρ(y)−Fj,ρ(ξ′ j)∗) = � log(2ρ−1dj) + O(Rλρ) if |y − ξ′ j| ≤ R + 1, O(log |y − ξ′ j|) + log(2djρ−1) if |y − ξ′ j| > R + 1, where the O terms are independent of ǫ, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Moreover, in the particular case y ∈ ∂Ωρ, since Fj,ρ(y) belongs to ∂R2 +, there exists C > 0 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='17) ���� gj log(2djρ−1) ���� L∞(Ωρ\\BR(ξ′ j)) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' SIGN-CHANGING SOLUTIONS 17 Additionally, a direct computation leads us to ∇gj(y) =O(|y − ξ′ j|−1) + O(|Fj,ρ(y) − Fj,ρ(ξ′ j)∗|−1) + O(1) � +∞ 0 e−λρs ds |se2 − Fj,ρ(ξ′ j)∗|2 =O(|y − ξ′ j|−1) + O(ρλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In fact, we have that for |y − ξ′ j| = R, using that Fj(ˆξj) is a rotation, the following expansion takes place (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='18) ∇gj(y) = y − ξ′ j |y − ξ′ j|2 � 1 + O(λ−1) � + O(ρλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now we look for the terms involving ˜zj in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' By the previous dis- cussion, it is easy to see the existence of C > 0 not depending on ǫ, λ such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='19) ∥˜zj∥∞ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now we deal with ∥L˜zj∥∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' It is clear that L˜zj = O(ρ3) in BR(ξ′ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Thus, we concentrate on the case |y −ξ′ j| ≥ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' With the above estimates, for R ≤ |y − ξ′ j| ≤ R + 1 (recalling that since F is conformal we have gj is harmonic there) we have L˜zj(y) =L � η1Z0j � 1 − gj log(2djρ−1) �� (y) + Lˆzj(y) =O(ρ3) + O � 1 | log(λρ)| � For |y − ξ′ j| ≥ R + 1, we see that L˜zj(y) = O � log |y − ξ′ j| |y − ξ′ j|3| log(λρ)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From which we conclude (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='20) ∥L˜zj∥∗ ≤ CR2+σ | log(λρ)|, for some C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now we concentrate on Rλρ˜zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Using the properties of η2j, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='21) Rλρ˜zj = 1 | log(λρ)|η2j � Z0jRλρgj + ∂Z0j ∂ν gj � in ∂Ωρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Notice that Rλρ˜zj(y) = 0 for |y − ξ′ j| ≥ r/ρ, so we concentrate on the analysis of y ∈ ∂Ωρ with |y −ξ′ j| < r/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For simplicity, we denote this region as Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We have gj is uniformly bounded on ∂Ωρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In fact, notice that if 18 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP y ∈ ∂Ωρ, we have |Fj,ρ(y) − Fj,ρ(ξ′ j)| = |Fj,ρ(y) − Fj,ρ(ξ′ j)∗| and thus, using that Fjρ(ξ′ j)2 ≥ c/(λρ) we have |gj(y)| ≤ C � +∞ 0 e−λρs |se2 − (Fjρ(ξ′ j))∗|ds ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Using this and the explicit formula for Z0j, we conclude that Rλρ˜zj = η2j | log(λρ)|(O(1)Rλρgj + O(λρ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Noticing that Rλρgj(y) = c � − ∂G ∂x2 (Fjρ(y), Fjρ(ξ′ j) � + λρG(Fjρ(y), Fjρ(ξ′ j)) = λρ(c − 1)gj(y), where c = 1 + O(ρy) is the conformal factor of the map F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Thus, for each y ∈ ∂Ωρ with |y − ξ′ j| ≤ r/ρ, we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='22) Rλρgj(y) = O(ρy)λρgj(y), and from here, replacing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='21), we conclude that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='23) ∥Rλρ˜zj∥∞ ≤ C λρ | log(λρ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In particular, using this estimate together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='20) and replacing them into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='9) we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='24) ∥˜φ∥∞ ≤ C(∥h∥∗ + 1 | log(λρ)| � j |bj|), meanwhile, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='19), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='20) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='23) and replacing them into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='16), we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='25) ∥φ∥∞ ≤ C(∥h∥∗ + � j |bj|), where in both inequalities the constant C > 0 depends on R, but not on λ nor ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='- Estimate for bj: Using the equation for ˜φ, multiplying by ˜zj and inte- grating by parts, we conclude that bj � Ωρ ˜zjL˜zj = � Ωρ ˜φL˜zj − � ∂Ωρ ˜φRλρ˜zj + bj � ∂Ωρ ˜zjRλρ˜zj − � Ωρ h˜zj =: I1 + I2 + bjI3 + I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='26) for each j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now we proceed to estimate each term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' It is direct to see that |I4| ≤ C∥h∥∗, SIGN-CHANGING SOLUTIONS 19 and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='20) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='24), we get |I1| ≤ C∥˜φ∥∞R−σ∥L˜zj∥∗ ≤ C R2 | log(λρ)|(∥h∥∗ + R2+σ | log(λρ)| � j |bj|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now, for I2 we make I2 ≤ C | log(λρ)|∥˜φ∥∞ � ∂Ωρ |Rλρgj| =: C | log(λρ)|∥˜φ∥∞ ˜I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='22) we have ˜I2 ≤ Cλρ � Aj |gj(y)|ds(y), and using the explicit formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='10), we get that ˜I2 ≤Cλρ � Aj � +∞ 0 e−λρs|s + Fjρ(ξ′ j)2| |Fλρ(y) + se2 − Fjρ(ξ′ j)∗|2 dsds(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From here, using the estimate Fjρ(ξ′ j) = 1 λρ(O( 1 λ), 1 + O( 1 λ)), by a change of variables, we can find universal constants C, c > 0 such that ˜I2 ≤Cλρ � 1/ρ 0 � +∞ 0 e−cλρs|s + 1 λρ| t2 + (s + 1 λρ)2 dsdt, by taking λ suitably large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Thus, by algebraic manipulations, we conclude that ˜I2 ≤ C for some C > 0 not depending on R, ǫ, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From here conclude that |I2| ≤ C | log(λρ)|(∥h∥∗ + R2+σ | log(λρ)| � j |bj|), and by similar estimates, we conclude |I3| ≤ � ∂Ωρ |˜zjRλρ˜zj|ds(y) = O � 1 | log(λρ)|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Collecting the previous estimates and replacing them int (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='26), we con- clude that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='27) bjI0 = O(1) � ∥h∥∗ + 1 | log(λρ)|2 � j |bj| � , where I0 := � Ωρ ˜zjL˜zj and O(1) depends on R, but not on ǫ, λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We claim that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='28) |I0| ≥ c | log(λρ)| for some c > 0 independent of R, λ and ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' If we assume this is true, we replace this into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='27) and by taking λρ small enough in terms of R, we arrive at |bj| ≤ C| log(λρ)|∥h∥∗, 20 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP and replacing this into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='25) we have been proved the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Thus, it remains to get the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='- Proof of estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='28): We divide the integral in the regions |y − ξ′ j| ≤ R, R < |y − ξ′ j| ≤ R + 1, and R + 1 < |y − ξ′ j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, using the estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='17), the gradient of gj, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='19), and the fact that for each i ̸= j we have |y − ξ′ i| ≥ r/(2ρ) in the support of η2j, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='29) I0 = O(R2λ4ρ4) + I01 + O � R−3 1 | log(λρ)| � , where I01 = � BR+1(ξ′ j)\\BR(ξ′ j) L˜z0j ˜z0j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Writing ˜z0j = η1,j(Z0j − ˆz0j) + ˆz0j valid in the annular region AR = {y : R < |y − ξ′ j| < R + 1}, we can write I01 = � AR � ∆η1j(Z0j − ˆz0j) + 2∇η1j∇(Z0j − ˆz0j) � ˜z0j + � AR η1jLZ0j ˜z0j + � AR (1 − η1j)Lˆz0j ˜z0j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' By similar arguments as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='29), we have I01 = � AR � ∆η1j(Z0j − ˆz0j) + 2∇η1j∇(Z0j − ˆz0j) � ˜z0j + O � R−3 1 | log(λρ)| � =:I02 + O � R−3 1 | log(λρ)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, for I02, we integrate by parts and using that (1−gj/ log(2djρ−1)) = O( log(R) | log(λρ)|) in AR, we can get I02 = − � |y−ξ′ j|=R ˜z0j∇(Z0j − ˆz0j)ν − � AR η1j(∆(Z0j − ˆz0j)˜z0j + ∇(Z0j − ˆz0j)∇˜z0j) − � AR ∇η1j∇˜z0j(Z0j − ˆz0j) =: − I03 + O � R−2 | log(λρ)| � + O � log2(R) | log(λρ)|2 � + O �R2 log2(R) | log(λρ)|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' SIGN-CHANGING SOLUTIONS 21 For I03 we have −I03 = − � |y−ξ′ j|=R ˜z0j(1 − gj/ log(2djρ−1))∇Z0jν + 1 log(2djρ−1) � |y−ξ′ j|=R Z2 0j∇gjν =O �R−2 log(R) log(λρ) � + Z2 0j(R) log(2djρ−1) � |y−ξ′ j|=R ∇gjν, from which, using the expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='18) we arrive at I03 = − 2πZ2 0j(R) log(2djρ−1)(1 + O(λ−1)) + O �R−2 log(R) log(λρ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From here, taking R, λ large enough we conclude |I0| ≥ π log(2djρ−1) + O �R−2 log(R) log(λρ) � , from which (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='28) follows by fixing R large enough independent of λ, ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' □ Now we are in position to provide the Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Consider the Hilbert space H = {u ∈ H1(Ωρ) : � Ωρ χjZijφ = 0 for i = 1, 2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m} endowed with the inner product ⟨u, v⟩H = � Ωρ ∇u · ∇v + λρ � ∂Ωρ uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For each ℓ ∈ H∗, we consider the map R : H∗ → H, given by the Riesz Representation Theorem, as ⟨Rℓ, ϕ⟩ = ℓ(ϕ), for all ϕ ∈ H, and with this, we denote the operators K1, K2 : H → H, given by K1(f) = R(f), K2(f) = R(Wf), f ∈ H, where we have performed the identification f ∈ L2(Ωρ) with the map ϕ �→ � Ωρ fϕ ∈ H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In fact, because of this identification we can assume that both operators are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For h ∈ L∞(Ωρ) ∩ H1(Ωρ), let cij the unique constants such that ˜h := h + � ij cijχjZij ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' A simple computation tells us that for each i, j we have − � Ωρ hχjZij � Ωρ χjZ2 ij = cij, 22 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP from which, there exists CR > 0 such that |cij| ≤ CR∥h∥∗ The solvability of the problem is equivalent to the existence of a function φ ∈ H such that φ − K2(φ) = K1˜h, 1 and by Fredholm alternative, we have a unique solution to this equation if the homogeneous problem φ − K2(φ) = 0, has no nontrivial solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This is the case in view of the estimates of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3 this is the case, from which the well-posedness of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' By the estimates for cij, we conclude that the solution φ to this problem satisfies the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='6), by standard arguments we have φkl := ∂(ξl)kφ satisfies Lφkl = − ∂W ∂(ξl)k φ + ∂(cklχlZkl) ∂(ξl)k in Ωρ, together with homogeneous Robin boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In order to use the estimates in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3 we consider a function with the form φkl + ˜cklχlZkl, with ˜ckl such that the orthogonality condition with respect to Zkl is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Nonlinear Problem Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' There exist λ0 > 0 and ǫ0 > 0 such that for all λ ≥ λ0 and ǫ > 0 with 0 < ρλ9 log λ < ǫ0 and for any points ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξm) satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='10), the problem of finding a function φ and coefficients cij(ξ) ∈ R, i = 1, 2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 L(φ) = −(R + Λ(φ) + N(φ)) + 2 � i=1 m � j=1 cijχjZij in Ωρ ∂φ ∂ν + ρλφ = 0, on ∂Ωρ � Ωρ χjZijφ = 0 for i = 1, 2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m, has a unique solution φ and scalars cij, i = 1, 2, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m satisfying ∥φ∥L∞(Ωρ) ≤ Cρλ9 log λ| log(ρλ)|, |cij| ≤ Cρλ9 log λ, where L, R, Λ(φ) and N(φ) are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='17), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='20), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='18) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='19) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Moreover, the map ξ′ �→ φ into the space C(¯Ωρ), the derivative SIGN-CHANGING SOLUTIONS 23 Dξ′φ exists and defines a continuous function of ξ′, and there is a constant C > 0, such that ∥Dξ′φ∥L∞(Ωρ) ≤ Cρλ9 log λ| log(ρλ)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Observe that in terms of the operator T problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) becomes (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3) φ = T (−[R + Λ(φ) + N(φ)]) := A(φ), where T is the continuous linear map defined on the set of all h ∈ L∞(Ωρ) satisfying ∥h∥∗ < +∞, so that φ = T(h) correspond to the unique solution of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For a given number ν > 0, let us consider Fν = {φ ∈ C(¯Ωρ) : ∥φ∥∞ ≤ νρλ9 log λ| log(ρλ)|} From the Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1, we get ∥A(φ)∥∞ ≤ C| log(ρλ)| [∥R∥∗ + ∥Λ(φ)∥∗ + ∥N(φ)∥∗] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3 it follows the estimate ∥R∥∗ ≤ Cρλ9 log λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Furthermore, ∥Λ(φ)∥∗ ≤ Cρλ9 log λ∥φ∥∞ and ∥N(φ)∥∗ ≤ C∥φ∥2 ∞ Hence, we get for any φ ∈ Fν, ∥A(φ)∥∞ ≤ Cρλ9 log λ| log(ρλ)| � 1 + νρλ9 log λ| log(ρλ)| + ν2ρλ9 log λ| log(ρλ)|2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Given any φ1, φ2 ∈ Fν, we have that ∥Λ(φ1) − Λ(φ2)∥∗ ≤ Cρλ9 log λ ∥φ1 − φ2∥∞ and ∥N(φ1) − N(φ2)∥∗ ≤ C(∥φ1∥∞ + ∥φ2∥∞)∥φ1 − φ2∥∞ ≤ Cνρλ9 log λ| log(ρλ)| ∥φ1 − φ2∥∞ with C independent of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Therefore, from the Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1 ∥A(φ1) − A(φ2)∥∞ ≤ Cνρλ9 log λ| log(ρλ)|2∥φ1 − φ2∥∞, so that it follows that for all ǫ sufficiently small and λ sufficiently large A is a contraction mapping of Fν (for ν large enough), and therefore a unique fixed point of A exists in Fν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let us now discuss the differentiability of φ depending on ξ′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', ξ′ �→ φ(ξ′) ∈ C(¯Ωρ) is C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Since R depends continuously (in the ∗-norm) on ξ′, using the fixed point characterization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3), we deduce that the mapping ξ′ �→ φ is also continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, formally ∂ξ′ kl[Λ(φ)] = � ρ2ǫ2� ∂ξ′ kl(eV ) − ∂ξ′ kl(e−V ) � − ∂ξ′ klW � φ + [ρ2ǫ2(eV + e−V ) − W]∂ξ′ kl φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' and ∂ξ′ kl[N(φ)] = ρ2ǫ2∂ξ′ kl(eV )(eφ − φ − 1) + ρ2ǫ2eV [eφ − 1]∂ξ′ kl φ − ρ2ǫ2∂ξ′ kl(e−V )(e−φ + φ − 1) + ρ2ǫ2e−V [e−φ − 1]∂ξ′ kl φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 24 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP From the definition of V it follows that ∂ξ′ klV (y) = ak 4(y − ξ′ k)l µ2 k + |y − ξ′ k|2 − m � i=1 ai 2∂ξ′ kl(µ2 i ) µ2 i + |y − ξ′ i|2 +ρ∂2lHλ(ρy, ρξ′ k)+O(ρ2λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Hence, ∥ρ2ǫ2∂ξ′ kl(eV )∥∗ and ∥ρ2ǫ2∂ξ′ kl(e−V )∥∗ are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, we conclude that ∥∂ξ′ kl[Λ(φ)]∥∗ ≤ ���ρ2ǫ2� ∂ξ′ kl(eV ) − ∂ξ′ kl(e−V ) � − ∂ξ′ klW ��� ∗∥φ∥∞ + [∥ρ2ǫ2(eV + e−V ) − W∥∗∥∂ξ′ kl φ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' and ∥∂ξ′ kl[N(φ)]∥∗ ≤ C∥φ∥2 ∞ + C∥φ∥∞ ∥∂ξ′ klφ∥∞ ≤ Cνρλ9 log λ| log(ρλ)| � νρλ9 log λ| log(ρλ)| + ∥∂ξ′ klφ∥∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Also, observe that we have ∂ξ′ klφ = −(∂ξ′ klT) (R + Λ(φ) + N(φ)) − T � ∂ξ′ kl [R + Λ(φ) + N(φ)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' So, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='6), we get ∥∂ξ′ klφ∥∞ ≤ C| log(ρλ)| � | log(ρλ)| (∥R∥∗ + ∥Λ(φ)∥∗ + ∥N(φ)∥∗) + ∥∂ξ′ klR∥∗ + ∥∂ξ′ kl[Λ(φ)]∥∗ + ∥∂ξ′ kl[N(φ)]∥∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let us estimate ∥∂ξ′ klR∥∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We know that ∂ξ′ klR(y) = ∆∂ξ′ klV (y) + ρ2ǫ2(eV + e−V )∂ξ′ klV (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From similar computations to deduce Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3 it follows that for any l = 1, 2: if |y − ξ′ j| > δ ρ for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m then ∂ξ′ klR(y) = O � ρλ log λ[ρ4 + ρ2ǫ2] � , if δ ρλ log λ ≤ |y − ξ′ j| ≤ δ ρ for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m} then ∂ξ′ klR(y) = ρλ2 log2 λ O � ρ2ǫ2� log4 λ + λ4 ρ4|y − ξ′ j|4 + ρ4λ4|y − ξ′ j|4�� and if |y − ξ′ j| ≤ δ ρλ log λ for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m} then ∂ξ′ klR(y) = 8µ2 j (µ2 j + |y − ξ′ j|2)2 O(ρλ log λ[1 + |y − ξ′ j|]) + 4δjk(y − ξ′ k)l + 2∂ξ′ kl(µ2 j) µ2 j + |y − ξ′ j|2 O(ρ2ǫ2) + O(ρ3ǫ2λ log λ + ρ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' SIGN-CHANGING SOLUTIONS 25 Therefore, from the definition of *-norm we conclude that ∥∂ξ′ klR∥∗ ≤ C(ρλ log λ + ρ2λ11 log3 λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Similar computations as above and those used to deduce Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3 lead us to find the estimate ��ρ2ǫ2(eV − e−V )∂ξ′ klV − ∂ξ′ klW �� ∗ ≤ C(ρλ log λ + ρ2λ11 log3 λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Hence, we find the following estimate ∥∂ξ′ klφ∥∞ ≤ C � ρλ9 log λ| log(ρλ)|2 + ρλ9 log λ| log(ρλ)|2∥∂ξ′ klφ∥∞ � Thus, we conclude (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Note that ∂ξ′ klµj = O(ρλ log λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The above computations can be made rigorous by using the implicit func- tion theorem and the fixed point representation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3) which guarantees C1 regularity in ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Variational reduction and energy computations In view of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1 we obtain a solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='15) with the form V + φ if we are able to get that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) cij(ξ) = 0, i = 1, 2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This problem is equivalent to look for critical points of the following functional Fǫ,λ(ξ) = Jǫ,λ(U + ˜φ), where U is the approximation defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='14) and ˜φ(x) = φ �x ρ � with φ the solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The energy function Jǫ,λ is given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2) Jǫ,λ(u) = 1 2 � Ω |∇u|2dx − ǫ2 � Ω � eu + e−u� dx + λ 2 � ∂Ω u2dσ, u ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Notice that critical points for Jǫ,λ are weak solutions for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) compare with [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We have the following sufficient condition to have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' There exists λ0 > 0 and ǫ0 > 0 such that for any λ ≥ λ0 and ǫ > 0 so that 0 < ρλ9 < ǫ0, if ξ ∈ Ωm is a critical point of Fǫ,λ satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='10) then u = U(ξ)+ ˜φ(ξ) is a critical point of Jǫ,λ, that is, if DξFǫ,λ(ξ) = 0 then ξ satisfies system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', u is a solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Define the energy functional Iǫ,λ associated to problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='15), namely, Iǫ,λ(v) = 1 2 � Ωρ |∇v|2 − ρ2ǫ2 � Ωρ (ev + e−v) dy + ρλ 2 � ∂Ωρ v2 dσ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let us differentiate the function Fǫ,λ(ξ) with respect to ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Since Iǫ,λ(V (ξ′) + φ(ξ′)) = Jǫ,λ(U(ξ) + ˜φ(ξ)), 26 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP we can differentiate directly Iǫ,λ(V + φ) (under the integral sign), so that integrating by parts ∂ξklFǫ,λ(ξ) = 1 ρDIǫ,λ(V + φ) � ∂ξ′ klV + ∂ξ′ klφ � = − 1 ρ 2 � i=1 m � j=1 cij � Ωǫ χjZij � ∂ξ′ klV + ∂ξ′ klφ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From the results of the previous section, this expression defines a continuous function of ξ′, and hence of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let us assume that DξFǫ,λ(ξ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, from the latter equality 2 � i=1 m � j=1 cij � Ωǫ χjZij � ∂ξ′ klV + ∂ξ′ klφ � = 0, k = 1, 2, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2) and ∂ξ′ klV = 4 ak µk Zkl + O � ρλ log λ � , where O(ρλ log λ) is in the L∞ norm, it follows that 2 � i=1 m � j=1 cij � Ωǫ χjZij [Zkl + o(1)] = 0, k = 1, 2, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' with o(1) small in the sense of the L∞ norm as ρλ9 log λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The above system is diagonal dominant and we thus get cij = 0 for i = 1, 2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' □ Next result states an expansion of Fǫ,λ in terms of ϕm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In order to have a more clearly the relation between ǫ ∈ (0, 1) and λ > 1 such that they satisfying ǫλ16 ≤ ǫ0 for some ǫ0 < 1, and recalling ρ = ǫλ−2, we see that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3) 0 ≥ log(ρλ) ≥ log ǫ − log ǫ−1/16 ≥ 2 log ǫ, from which we state the estimates in terms of the leading expression at the logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The following expansions holds Fǫ,λ(ξ) = −16πm + 8πm log 8 − 16πm log(ρλ2) − 4πϕm(ξ) + θǫ,λ(ξ), in C1-sense, where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='4) ϕm(ξ) = m � j=1 \uf8ee \uf8f0Hλ(ξj, ξj) + m � i=1,i̸=j aiajGλ(ξi, ξj) \uf8f9 \uf8fb , |θǫ,λ(ξ)| = O � ǫ2λ14| log ǫ|3� , and |∇θǫ,λ(ξ)| = O � ǫλ16 log4 ǫ � , uniformly on points ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξm) ∈ Ωm satisfying the constraints (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='10), as ǫ → 0 and λ → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' SIGN-CHANGING SOLUTIONS 27 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' First, we shall expand the energy functional Jǫ,λ evaluated in the ansatz U, namely, we give an asymptotic estimate of Jǫ,λ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The following expansion does hold Jǫ,λ(U) = −16πm + 8πm log 8 − 16πm log(ρλ2) − 4πϕm(ξ) + O(ρλ log λ) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' First, we will evaluate the quadratic and boundary parts of energy evaluated at U, that is, integrating by parts 1 2 � Ω |∇U|2 dx + λ 2 � ∂Ω U 2 dσ = −1 2 � Ω U∆U dx = −1 2 m � j=1 aj � Ω U∆Uj dx, since on ∂Ω ∂U ∂ν + λU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Using the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='7) of Uj = wj + Hj (recall Hj is harmonic), we have � Ω U(−∆Uj) dx = � Ω ǫ2ewj(x)U(x) dx = aj � Ω ǫ2ewj(wj + Hj) + � i̸=j ai � Ω ǫ2ewj(wi + Hi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, we expand as follows � Ω ǫ2ewj(wj + Hj) = � B(ξj, dj 2 ) ǫ2ewj(wj + Hj) + � Ω\\B(ξj, dj 2 ) ǫ2ewj(wj + Hj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='12) and x − ξj = µjρy, we obtain that � B(ξj, dj 2 ) ǫ2ewj(wj + Hj) = � B(ξj, dj 2 ) 8µ2 jρ2 (µ2 jρ2 + |x − ξj|2)2 � log 1 (µ2 jρ2 + |x − ξj|2)2 + Hλ(x, ξj) + O � µ2 jρ2 d2 j � � = � B(0, dj 2µj ρ ) 8 (1 + |y|2)2 � − 2 log(1 + |y|2) − 4 log(µjρ) + Hλ(ξj + µjρy, ξj) + O � µ2 jρ2 d2 j � � = − 16π − 32π log(µjρ) + 8πHλ(ξj, ξj) + O(ρλ log λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Since µj is uniformly bounded and away from zero, and since dj = O(λ−1), we see that � B(0, dj 2µj ρ) 8 (1 + |y|2)2 log(1 + |y|2) dy = 8π + O �µ2 jρ2 d2 j �� log µjρ dj �� � , 28 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP and � B(0, dj 2µj ρ ) 8 (1 + |y|2)2 dy = 8π + O �µ2 jρ2 d2 j � and by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='32), we get Hλ(ξj +µjρy, ξj) = Hλ(ξj, ξj)+O(µjρλ log λ|y|) in B(0, dj 2µjρ) so that � B(0, dj 2µj ρ ) 8 (1 + |y|2)2 Hλ(ξj + µjρy, ξj) dy = 8πHλ(ξj, ξj) + O(ρλ log λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Also, � Ω\\B(ξj, dj 2 ) ǫ2ewj(wj + Hj) = � Ω\\B(ξj, dj 2 ) ǫ2ewj � Gλ(x, ξj) + O � µ2 jρ2 d2 j �� = O(ρ2λ2 log λ) in view of ǫ2ewj = 8µ2 jρ2 (µ2 jρ2 + |x − ξj|2)2 = O �µ2 jρ2 d2 j � and Gλ(x, ξj) = O(log λ) for |x − ξj| > dj 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Therefore, we obtain that � Ω ǫ2ewj(wj + Hj) = −16π − 32π log(µjρ) + 8πHλ(ξj, ξj) + O(ρλ log λ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='5) Now, for i ̸= j we have that � Ω ǫ2ewj(wi + Hi) = � B(ξj, dj 2 ) ǫ2ewj(wi + Hi) + � Ω\\B(ξj, dj 2 ) ǫ2ewj(wi + Hi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' we get that � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj 2 ) ǫ2ewj(wi + Hi) = 8πGλ(ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ξj) + O(ρλ log λ) � Ω\\B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj 2 ) ǫ2ewj(wi + Hi) = � B(ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' di 2 ) ǫ2ewj � log 1 (µ2 i ρ2 + |x − ξi|)2 + Hλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ξi) + O �µ2 i ρ2 d2 i �� dx + � Ω\\[B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj 2 )∪B(ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' di 2 )] ǫ2ewj � Gλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ξi) + O �µ2 i ρ2 d2 i �� dx = O � ρ2 �| log ρ| λ2 + | log(ρλ)| λ2 �� + O �ρ2 λ2 log λ + ρ4 λ4 � = O �ρ2 λ2 | log(ρλ)| � SIGN-CHANGING SOLUTIONS 29 in view of � B(ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' di 2 ) 8µ2 jρ2 (µ2 jρ2 + |x − ξj|2)2 � log 1 (µ2 i ρ2 + |x − ξi|)2 + Hλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ξi) + O �µ2 i ρ2 d2 i �� dx = O � ρ2 � B(ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' di 2 ) | log(µ2 i ρ2 + |x − ξi|)| dx + ρ2 � B(ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' di 2 ) |Hλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ξi)| dx + ρ4 � = O � ρ2� d2 i | log ρ| + ρ2�di ρ �2 log �di ρ ��� + O(ρ2[d2 i log λ + d3 i λ log λ]) and � Ω\\[B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj 2 )∪B(ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' di 2 )] 8µ2 jρ2 (µ2 jρ2 + |x − ξj|2)2 � Gλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ξi) + O �µ2 i ρ2 d2 i �� dx = O � ρ2 d2 j log λ + ρ4 d2 jd2 i � Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' we obtain that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='6) � Ω ǫ2ewj(wi + Hi) = 8πGλ(ξi, ξj) + O(ρλ log λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Taking into account (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='26), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='6), we find that 1 2 � Ω |∇U|2 dx + λ 2 � Ω U 2 dσ = 1 2 m � j=1 \uf8ee \uf8f0 � Ω ǫ2ewjUj + aj m � i=1,i̸=j ai � Ω ǫ2ewjUi \uf8f9 \uf8fb = 1 2 m � j=1 \uf8ee \uf8f0−16π − 32π log(µjρ) + 8πHλ(ξj, ξj) + aj m � i=1,i̸=j ai8πGλ(ξi, ξj) \uf8f9 \uf8fb + O(ρλ log λ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='7) On the other hand, we have � Ω ǫ2(eU+e−U) dx = m � j=1 � B(ξj, dj log λ ) ǫ2(eU+e−U) dx+ � Ω\\∪m j=1B(ξj, dj log λ ) ǫ2(eU+e−U) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Observe that � Ω\\∪m j=1B(ξj,δ) ǫ2(eU + e−U) dx = � Ω\\∪m j=1B(ξj,δ) ǫ2� e �m i=1 aiGλ(x,ξi)+O(ρ2λ2) + e− �m i=1 aiGλ(x,ξi)+O(ρ2λ2)� dx = O(ǫ2), by using that �m i=1 aiGλ(x, ξi) = O(1) in Ω \\ ∪m j=1B(ξj, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' So, we get that � Ω\\∪m j=1B(ξj, dj log λ) ǫ2(eU+e−U) dx = m � j=1 � B(ξj,δ)\\B(ξj, dj log λ ) ǫ2(eU+e−U) dx+O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 30 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP Now, assume that aj = 1 so that by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) we obtain � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='δ)\\B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ) ǫ2eU = � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='δ)\\B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ) ǫ2eGλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='ξj)+� l̸=j alGλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='ξl)+O(ρ2λ2) dx = � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='δ)\\B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ) ǫ2 eHλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='ξj)+O(1) |x − ξj|4 dx = O � ǫ2λ4 � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='δ)\\B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ) |x − ξj|−4 dx � = O(ǫ2λ6 log2 λ) and � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='δ)\\B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ) ǫ2e−U = � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='δ)\\B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ) ǫ2e−Gλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='ξj)−� l̸=j alGλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='ξl)+O(ρ2λ2) dx = � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='δ)\\B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ) ǫ2|x − ξj|4e−Hλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='ξj)+O(1)dx = O � ǫ2λ4 � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='δ)\\B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ ) |x − ξj|4 dx � = O(ǫ2λ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Similarly, for aj = −1 we find that � B(ξj,δ)\\B(ξj, dj log λ ) ǫ2(eU + e−U) = O(ǫ2λ4 + ǫ2λ6 log2 λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now, assume that aj = 1, so that by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2 and taking x − ξj = µjρy we get � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ ) ǫ2eU dx = � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ ) 8µ2 jρ2 (µ2 jρ2 + |x − ξj|2)2 exp � Hj(x) + � l̸=j Ul(x) � dx = � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ ) 8eHλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='ξj)−log(8µ2 j)+4 log λ+� l̸=j alGλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='ξl)+O(ρ2λ2) µ2 jρ2 � 1 + � |x−ξj| µjρ �2�2 dx = � B(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj µj ρ log λ ) 8eH(ξj+µjρy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='ξj)−log(8µ2 j )+4 log λ+� l̸=j alG(ξj+µjρy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='ξl)+O(ρ2λ2) (1 + |y|2)2 dy = � B(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj µj ρ log λ ) 8 (1 + |y|2)2 � 1 + O(ρλ log λ|y| + ρ2λ2) � dy = 8π + O(ρλ log λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' SIGN-CHANGING SOLUTIONS 31 and � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ ) ǫ2e−U dx = ǫ2 � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ ) � 8µ2 jρ2 (µ2 jρ2 + |x − ξj|2)2ǫ2 �−1 exp � − Hj(x) − � l̸=j Ul(x) � dx = � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ ) ǫ4 8µ2 jρ2 � µ2 jρ2 + |x − ξj|2�2 × e−Hλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='ξj)+log(8µ2 j)−4 log λ−� l̸=j alGλ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='ξl)+O(ρ2λ2) dx = � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ ) ǫ4 8µ2 jρ2 � µ2 jρ2 + |x − ξj|2�2 � 1 + O(λ log λ|x − ξj| + ρ2λ2) � dx = O � ρǫ log4 λ � In case aj = −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' using previous ideas we find that � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ ) ǫ2eU dx = O � ρǫ log4 λ � and � B(ξj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' dj log λ) ǫ2e−U dx = 8π + O(ρλ log λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Therefore, we conclude that � Ω ǫ2(eU + e−U)dx = 8πm + O � ρλ log λ + ρǫ log4 λ + ǫ2λ6 log2 λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From the choice of µj’s in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='26), we obtain that Jǫ,λ(U) = −16πm + 8πm log 8 − 16πm log(ρλ2) − 4πϕm(ξ) + O(ρλ log λ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='8) where ϕm is given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='4), if ǫλ7 log λ| log( ǫ λ)|2 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' □ Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The following expansion does hold ∂(ξl)k[Jǫ,λ(U)] = −4π∂(ξl)kϕm(ξ) + O(ρλ log2 λ) for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m and k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' First, observe that ∂(ξl)k � Jǫ,λ(U) � = DJǫ,λ(U) � ∂(ξl)kU � = − � Ω � ∆U + ǫ2(eU − e−U) � ∂(ξl)kU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now, we have that � Ω ∂(ξl)kU(−∆U) = m � j=1 aj � Ω ∂(ξl)kU(−∆Uj) = m � j=1 aj � Ω ǫ2ewj(x)∂(ξl)kU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 32 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP Using similar arguments as above and taking into account a suitable expan- sion for ∂(ξl)kU (see ∂(ξ′ l)kV in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1), we conclude that � Ω ∂(ξl)kU(−∆U) = − 16π m � j=1 ∂(ξl)kµj µj + 8π∂2kHλ(ξl, ξl) + 8π m � j=1 j̸=l ajal∂2kGλ(ξj, ξl) + O(ρλ log2 λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' On the other hand, from similar arguments as above it follows that ǫ2 � Ω (eU − e−U)∂(ξl)kU = ǫ2 m � j=1 aj � Ω (eU − e−U)∂(ξl)kUj = O(ρλ log2 λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Therefore, by using the choice of µj in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='26) the claim follows .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' □ Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The following expansion does hold Fǫ,λ(ξ) = Jǫ,λ(U) + θ∗ ǫ,λ(ξ), where |θ∗ ǫ,λ(ξ)| = O(ρ2λ18 log2 λ| log(ρλ)|), and |∇θ∗ ǫ,λ(ξ)| = O(ρλ18 log2 λ| log(ρλ)|2), as ρλ19 → 0, uniformly on points ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξm) ∈ Ωm satisfying the constraints (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Since we have, Iǫ,λ(V ) = Jǫ,λ(U) and Iǫ,λ(V (ξ′) + φ(ξ′)) = Jǫ,λ(U(ξ) + ˜φ(ξ)), we write Jǫ,λ(U + ˜φ) − Jǫ,λ(U) = Iǫ,λ(V + φ) − Iǫ,λ(V ) := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let us estimate A first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Taking into account that DIǫ,λ(V + φ)[φ] = 0, a Taylor expansion and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) gives us A = − � 1 0 D2Iǫ,λ(V + tφ)[φ]2 t dt, = − � 1 0 �� Ωρ [R + N(φ)] φ − � Ωρ ρ2ǫ2 � eV (etφ − 1) + e−V (e−tφ − 1) � φ2 � t dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='9) Therefore, we get Iǫ,λ(V + φ) − Iǫ,λ(V ) = O(ρ2λ18 log2 λ| log(ρλ)|), SIGN-CHANGING SOLUTIONS 33 since ∥R∥∗ ≤ Cρλ9 log λ, ∥N(φ)∥∗ ≤ C∥φ∥2 ∞ and ∥φ∥∞ ≤ Cρλ9 log λ| log(ρλ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let us differentiate with respect to ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We use representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='9) and dif- ferentiate directly under the integral sign, thus obtaining, for each k = 1, 2, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1 and the computations in the proof, we conclude that for k = 1, 2, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m ∂ξ′ kl [Iǫ,λ(V + φ) − Iǫ,λ(V )] = O(ρ2λ18 log2 λ| log(ρλ)|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now, taking ˜θǫ,λ(ξ′) = θ∗ ǫ,λ(ρξ′) with θ∗ ǫ,λ(ξ) = Fǫ,λ(ξ) − Jǫ,λ(U), we have shown that |˜θǫ,λ| + 1 | log(ρλ)||∇ξ′ ˜θǫ,λ| = O(ρ2λ18 log2 λ| log(ρλ)|), as ρλ9 log λ| log(ρλ)|2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The continuity in ξ of all these expressions is inherited from that of φ and its derivatives in ξ in the L∞ norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' □ Therefore, from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='3), previous claims and ∇θ∗ ǫ,λ(ξ) = 1 ρ∇˜θǫ,λ( ξ ρ) we con- clude the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Proof of main results Recall θ0 ∈ (0, ∞) is the unique minima of h in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='21) in (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We denote S∗ = {x ∈ Ω : d(x) = θ0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Concerning h, it is easy to see that h(θ) = 4 log θ + O(1) as θ → +∞, h(θ) = −4 log θ + O(1) as θ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Furthermore, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='23) it follows that there is C′ = C′(K) such that |Hλ(x, x) − h(λd(x)) + 4 log λ| ≤ C′ λ , for all x ∈ Ω satisfying K−1 λ ≤ dist(x, ∂Ω) ≤ K λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Here we follow closely the arguments in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For this, we assume the domain Ω is such that Ω ∩ R × {0} ̸= ∅, and that it is symmetric with respect to the reflection at R×{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In this setting, we have Gλ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='6) is also symmetric with respect to this reflection in the following sense: for x = (x1, x2) ∈ Ω, let ˜x = (x1, −x2), then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) G(x, y) = G(˜x, ˜y) for all x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In fact, defining ˜Gλ(x, y) = Gλ(˜x, ˜y), we have −∆x ˜Gλ(x, y) = −∆xGλ(˜x, ˜y) = 8πδ˜y(˜x) = 8πδy(x), meanwhile, for each x ∈ ∂Ω, using that ν(˜x) = (ν(x)1, −ν(x)2) we have Rλ ˜Gλ(x, y) = ∇xGλ(˜x, ˜y) · (ν(x)1, −ν(x)2) + λGλ(˜x, ˜y) = RλG(˜x, ˜y) = 0, from which, using the uniqueness of the Green function, we conclude the symmetry property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The symmetry condition is inherited by Hλ as Hλ(˜x) = 34 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP Hλ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For a m-tupe ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', ξm), we denote ˜ξ = (˜ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=', ˜ξm), and µξ,j = µ˜ξ,j for µ as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='26), and ϕm(ξ) = ϕm(˜ξ), where ϕ is defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We stress the notation by writing wξ,j(x), Uξ,j(x) as the functions in- troduced in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='9) with the particular choice of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, by related invariance of the equation, it is possible to see that w˜ξ,j(˜x) = wξ,j(x), and that x �→ U˜ξ,j(˜x) solves the same equation than Uξ,j(x), from which Uξ,j(x) = U˜ξ,j(˜x) and therefore Rξ(x) = R˜ξ(˜x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now, if we denote c(ξ) = (cij(ξ))ij and (c(ξ), φξ) as the unique solution for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) given in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1, we consider the function ˜φ(x) = φ˜ξ(˜x), it is possible to prove that ˜φ satisfies the same equation than φξ, from which they are equal by uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Then, by definition of F and using the previous symmetry properties, we have F(˜ξ) = F(ξ) from which we have θ(ξ) = θ(˜ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Thus, we have that F is a C1 and symmetric with respect to x perturbation of the func- tion ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This implies that F has critical points whenever the function ξ �→ ϕ((ξ11, 0), (ξ12, 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , (ξ1m, 0)) has stable critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now we are ready to provide the Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1: Assume Ω is simply connected, contains the origin and it is symmetric with respect to the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' By symmetry and the ex- pansion of the energy given in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2, it suffices to prove that there is a (nondegenerate) critical point to the function (t1, t2) �→ ϕ2((t1, 0), (t2, 0)), t1, t2 ∈ (a, b), where we identify the set {ξ ∈ Ω : ξ2 = 0} with the interval (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' With a slight abuse of notation, we denote this function by ϕ2(t1, t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We are interested in sign-changing solutions, from which ϕ2 takes the form ϕ2(t1, t2) = Hλ((t1, 0), (t1, 0)) + Hλ((t2, 0), (t2, 0)) − 2Gλ((t1, 0), (t2, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Using the expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='23), we have ϕ2(t1, t2) = −8 log λ+h(λd(t1, 0))+h(λd(t2, 0))−2Gλ((t1, 0), (t2, 0))+O(λ−1), where O(λ−1) does not depend on (t1, t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Fix δ = (b − a)/4 and for K > 1 to be determined, consider the set Ω0 = {(t1, t2) ∈ (a, b)2 : λd((ti, 0)) ∈ (K−1, K), i = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' |t1 − t2| > δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let us stress that our symmetry assumption implies that (t1, t2) ∈ ∂Ω0 if and only if λd((ti, 0)) ∈ {K−1, K} for some i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' First, we choose (ξ∗ 1, ξ∗ 2) ∈ Ω0 with ξ∗ i ∈ S∗, namely, λd(ξ∗ i ) = θ0, for i = 1, 2 so that, by the positivity of the Green’s function and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='23) ϕ2(ξ∗ 1, ξ∗ 2) = Hλ(ξ∗ 1, ξ∗ 1) + Hλ(ξ∗ 2, ξ∗ 2) − 2Gλ(ξ∗ 1, ξ∗ 2) ≤ Hλ(ξ∗ 1, ξ∗ 1) + Hλ(ξ∗ 2, ξ∗ 2) ≤ −8 log λ + 2h(θ0) + C′ λ SIGN-CHANGING SOLUTIONS 35 Let Cδ > 0 such that |G(x, y)| ≤ Cδ for all x, y ∈ Ω with |x − y| ≥ δ and d(x), d(y) ≥ (λ log λ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' If λd((ti, 0)) = K−1 for some i (say i = 1), we have that (ξ1, ξ2) ∈ ∂Ω0 and by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2), we can write ϕ2(ξ1, ξ2) ≥ −8 log λ + h(K−1) + h(θ0) − Cδ − C′ λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now, we fix K large just depending on h(θ0) and Cδ(Ω) such that h(K−1) > Cδ + h(θ0) + 2, which is valid for all λ large enough such that (λ log λ)−1 ≤ K−1λ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Hence, choosing λ larger, if necessary, we also get that 2C′ λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The same estimate can be found if λd(ti, 0) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From here, we deduce that for any (ξ1, ξ2) ∈ ∂Ω0 ϕ2(ξ1, ξ2) ≥ −8 log λ + 2h(θ0) − h(θ0) + h(K) − Cδ(Ω) − C′ λ ≥ −8 log λ + 2h(θ0) + C′ λ + 1 > ϕ2(ξ∗ 1, ξ∗ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This implies that inf∂Ω0 ϕ2 > ϕ2(ξ∗ 1, ξ∗ 2), from which there is an interior min- ima of ϕ2 in Ω0, which is stable under symmetric approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Therefore, by using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2, there is an interior minima of Fǫ,λ in Ω0 for ǫ > 0 small enough and λ > 0 large as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Not simply connected case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Taking advantage of previous estimates we are now ready to Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2: Assume that Ω is not simply connected with n holes n ≥ 1, so that ∂Ω = ∪n+1 i=1 Γi with Γi’s smooth closed curves satisfying Γi ∩ Γj = ∅ for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Fix δ = a 4, with a = min{dist(Γi, Γj) : i ̸= j, i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , n + 1} and for K > 1 to be determined, consider the set ΩK = {(ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξm) ∈ Ωm | λd(ξi) ∈ (K−1, K), i = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' |ξi − ξj| > δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Let us stress that our assumption m ≤ n + 1 implies that ξ ∈ ∂ΩK if and only if λd(ξi) ∈ {K−1, K} for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m} with ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' For simplicity we shall assume that {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m} = I1 ∪ I2, and i ∈ Ik ⇐⇒ ai = (−1)k, k = 1, 2, so that |Ik| = mk, k = 1, 2, m1 + m2 = m and m � i=1 m � j=1 i̸=j aiajGλ(ξi, ξj) = � i,j∈I1 i̸=j Gλ(ξi, ξj)+ � i,j∈I2 i̸=j Gλ(ξi, ξj)−2 � i∈I1 � j∈I2 Gλ(ξi, ξj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' In other words, we will find a sign-changing solution uε,λ to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='1) having a positive bubble centered at ξi with i ∈ I1 and a negative bubble centered at 36 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' FIGUEROA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' ITURRIAGA, AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' TOPP ξj with j ∈ I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Recall that for some Cδ > 0 fixed we have that |G(x, y)| ≤ Cδ for all x, y ∈ Ω with |x − y| ≥ δ and d(x), d(y) ≥ (λ log λ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Now, we choose (ξ∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξ∗ m) ∈ ΩK with ξi ∈ S∗ for all i, namely, λd(ξ∗ i ) = θ0, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , m so that, by the positivity of the Green’s function we obtain that ϕm(ξ∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξ∗ m) = m � i=1 Hλ(ξ∗ i , ξ∗ i ) + � i,j∈I1 i̸=j Gλ(ξi, ξj) + � i,j∈I2 i̸=j Gλ(ξi, ξj) ≤ m � − 4 log λ + h(θ0) + C′ λ � + � m1(m1 − 1) + m2(m2 − 1) � Cδ = −4m log λ + mh(θ0) + mC′ λ + � m1(m1 − 1) + m2(m2 − 1) � Cδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' On the other hand, if ξ ∈ ∂Ω, namely, λd(ξi) = K−1 for some i (say i = 1) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2), we can write ϕm(ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξm) ≥ m � i=1 Hλ(ξi, ξi) − 2 � i∈I1 � j∈I2 Gλ(ξi, ξj) ≥ (m − 1) � − 4 log λ + h(θ0) − C′ λ � − 4 log λ + h(K−1) − C′ λ − 2m1m2Cδ ≥ −4m log λ + h(K−1) + (m − 1)h(θ0) − 2m1m2Cδ − mC′ λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' We fix K large just depending on h(θ0) and Cδ = C(Ω) such that h(K−1) > � m1(m1 − 1) + m2(m2 − 1) + 2m1m2 � Cδ + h(θ0) + 2, which is valid for all λ large enough such that (λ log λ)−1 ≤ K−1λ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Hence, choosing λ larger, if necessary, we get that 2C′ λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' The same estimate can be found if λd(ξi) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' From here, we deduce that for any (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξm) ∈ ∂ΩK ϕm(ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξm) ≥ −4m log λ + h(K−1) + (m − 1)h(θ0) − 2m1m2Cδ − mC1 λ ≥ −4m log λ + mh(θ0) + mC1 λ + � m1(m1 − 1) + m2(m2 − 1) � Cδ + 1 > ϕm(ξ∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξ∗ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This implies that inf∂ΩK ϕm > ϕm(ξ∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' , ξ∗ m), from which there is a minima of ϕm in ΩK, which is stable under small C1 perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' Therefore, by using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content='2, there is an interior minima of Fǫ,λ in ΩK for ǫ > 0 small enough and λ > 0 large as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' □ SIGN-CHANGING SOLUTIONS 37 Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' was partially supported by Fondecyt grant 1201884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' was partially supported by Fondecyt grants 1211766 and 1221365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE2T4oBgHgl3EQfJAb4/content/2301.03688v1.pdf'} +page_content=' E.' metadata={'source': 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b/q9FST4oBgHgl3EQfPDjc/content/tmp_files/2301.13754v1.pdf.txt @@ -0,0 +1,1885 @@ +CMB signature of non-thermal Dark Matter produced from +self-interacting dark sector +Dilip Kumar Ghosh,1, ∗ Purusottam Ghosh,1, † and Sk Jeesun1, ‡ +1School of Physical Sciences, Indian Association for the Cultivation of Science, +2A & 2B Raja S.C. Mullick Road, Kolkata 700032, India +Abstract +The basic idea of this work is to achieve the observed relic density of a non-thermal dark mat- +ter(DM) and its connection with Cosmic Microwave Background (CMB) via additional relativistic +degrees of freedom which are simultaneously generated during the period TBBN to TCMB from a +long-lived dark sector particle. To realize this phenomena we minimally extend the type-I seesaw +scenario with a Dirac fermion singlet(χ) and a complex scalar singlet (φ) which transform non- +trivially under an unbroken symmetry Z3. χ being the lightest particle in the dark sector acts as a +stable dark matter candidate while the next to lightest state φ operates like a long lived dark scalar +particle. The initial density of φ can be thermally produced through either self-interacting number +changing processes (3φ → 2φ) within dark sector or the standard annihilation to SM particles +(2φ → 2 SM). The late time (after neutrino decoupling) non-thermal decay of φ can produce dark +matter in association with active neutrinos. The presence of extra relativistic neutrino degrees of +freedom at the time of CMB can have a significant impact on ∆Neff. Thus the precise measure- +ment of ∆Neff by current PLANCK 2018 collaboration and future experiments like SPT-3G and +CMB-IV can indirectly probe this non-thermal dark matter scenario which is otherwise completely +secluded due its tiny coupling with the standard model. +∗Electronic address: tpdkg@iacs.res.in +†Electronic address: spspg2655@iacs.res.in +‡Electronic address: skjeesun48@gmail.com +1 +arXiv:2301.13754v1 [hep-ph] 31 Jan 2023 + +I. +INTRODUCTION +The standard model (SM) of particle physics has been extraordinarily victorious in ex- +plaining properties of elementary particles of the universe and their interactions through +strong, electromagnetic and weak forces. The SM seems complete after the discovery of +Higgs-like particle with mass Mh = 125 GeV at the Large Hadron Collider (LHC)[1, 2], +which is responsible for mass generation mechanism through electroweak symmetry break- +ing in the SM. Inspite of the great triumph of the SM, several theoretical and experimental +issues still persist, that demands physics beyond the framework of the Standard Model. +Based on numerous astrophysical and cosmological observations at a wide range of length +scales, it is now well established fact that about 80% of total mass of the universe con- +sists of Dark matter (DM)[3–6] with relic density (ΩDMh2 = 0.120 ± 0.001) [6]. Another +astonishing experimental evidence is the observation of tiny but non-zero neutrino masses +(mν ≲ O(10−10) GeV) and neutrino flavour oscillations [7–12]. To address these issues, vari- +ous theoretical as well as phenomenological ideas have been proposed. The issue of neutrino +masses and their mixing angles can be resolved by the Seesaw mechanisms [13–22]. However, +any direct experimental verification of these ideas are yet to be confirmed. While in the dark +matter sector, weakly interacting massive particles (WIMP) [23–28] is the most popular and +widely studied thermal DM candidate whose interaction strength with SM particles is of the +order of electroweak interactions and via freeze-out mechanism it fits nicely the observed relic +density of the universe. Nevertheless, null measurements from various dark matter detec- +tion experiments [29–41] severely restricts the WIMP freeze-out mechanism and forcing us +to think if the standard WIMP paradigm is just waning or it is already deceased. To bypass +this deadlock, an alternative framework, coined as freeze-in mechanism has been proposed. +In this framework DM is a feebly interacting massive particle (FIMP) whose interactions +with SM plasma is too small ≲ O(10−10) to keep them in thermal bath [42–47]. Rather +FIMPs are produced non-thermally either from decay or annihilation of bath particles in +the early universe. The FIMP freezes in once the temperature of the universe becomes lower +than the FIMP mass and produces DM relic abundance in the correct ball-park as observed +today. Moreover, FIMPs having such a petite coupling with SM particles can easily ac- +commodate various non-observational signature of DM in different detection experiments +like Panda[29], XENON[30], LUX[31]. However, some attempts have been made to test the +2 + +FIMP scenario indirectly using observational data from big bang nucleosynthesis (BBN) or +cosmic microwave background (CMB) [48–54]. Furthermore, non-thermal production of DM +from the decay of heavier dark sector particles have also been studied in literature [55–58]. +Apart from FIMP, strongly interacting massive particle(SIMP) is another alternate +paradigm to explain the DM abundance [59–61] as well as the structure formation of the +universe[62–64]. SIMPs are produced thermally in the early universe by number changing +processes within itself. SIMP scenario requires strong self interaction and very small anni- +hilation rate to SM particles contrary to WIMPs to successfully satisfy the correct DM relic +density [65–67]. +On the other hand Cosmic Microwave Background (CMB) is an ideal probe of the physics +in the early universe. The very precise measurement of anisotropies in the temperature of +photons which dissociate from visible sector in the recombination phase of the thermal +evolution of our universe, leads to the determination of the energy density in that particular +era. From this one can estimate the number of light species in the universe and in the +massless limit this is provided by the relativistic degrees of freedom g∗ [68, 69]. On the +other hand after neutrino decoupling, one recasts the number of light degrees of freedom +associated with neutrino bath as Neff and in the SM it is roughly number of active neutrinos +(Nν = 3). Thus any physics scenarios beyond the SM (BSM) with new light degrees of +freedom with masses O (eV) or less can subscribe to Neff. We have very precise information +of Neff from recent Planck 2018 [6], which suggests Neff at the time of CMB formation to +be NCMB +eff += 2.99+0.34 +−0.33 at 95% confidence level (C.L), whereas in the SM, NSM +eff = 3.045. The +quantity Neff is parameterized as Neff ≡ (ρrad − ργ)/ρν, where, ργ, ρν, and ρrad denote the +photon energy density, active neutrino energy density and total radiation energy density +of the universe respectively [70]. +The deviation from 3, the number of active neutrinos +can be attributed to various non-trivial effects like non-instantaneous neutrino decoupling, +finite temperature QED corrections to the electromagnetic plasma, flavour oscillations of +neutrinos [70–73]. Multiple upcoming experiments like SPT-3G[74], CMB-IV[75] are going +to be extremely sensitive to the presence of any new radiation /light degrees of freedom and +will put stringent bound on ∆Neff = Neff − NSM +eff ≈ 0.06 at 95% confidence level. Various +BSM scenarios that entail additional entropy injection to the neutrino sector can face a tough +challange from the measurement of ∆Neff by both the present and future generation CMB +experiments [76–81]. This precise measurement of ∆Neff has also non-trivial implications +3 + +on various new physics models that produce dark matter in associated with the injection of +additional light degrees of freedom [82–85]. +In this work, we are interested in non thermal production of dark matter from heavier +dark sector, where the dark sector may or may not have sizeable interaction with the SM +bath. To realize this picture we extend the SM by one complex SM gauge singlet scalar +(φ), one gauge singlet Dirac fermion (χ) and 3 right handed neutrinos (RHN)(N1,2,3). The +three RHNs are responsible for neutrino mass generation through well known Type-I seesaw +mechanism[14, 86]. φ and χ are dark sector particles and an additional discrete Z3 symmetry +has been imposed under which they transform non trivially while the rest of the particles +transform trivially. In our analysis lightest dark sector particle χ can play the role of DM +whereas the heavy dark sector particle (φ) is a long lived owing to its very small coupling (≲ +10−12), which will eventually allows φ → χν decay at temperature below neutrino decoupling +temperature (∼ 1 MeV). Non thermal decay of φ is the only source of DM(χ) production +whereas φ freezes out thermally and gains non-zero number density via either of these two +mechanisms : (i) the number changing self interactions (3φ → 2φ), (ii) annihilation to +SM particles (2φ → 2SM). In this work we emphasise on the first scenario where φ has +strong self interactions but very weak interaction with the SM bath. The implication of +this particular scenario has been so far overlooked. Through our detailed numerical analysis +we will highlight the importance of this mechanism in both DM phenomenology and its +footprint on CMB. For the shake of completeness of the analysis, we will also consider the +second process as well to showcase region of parameter space where these two scenarios are +relevant. +It should be noted that in both cases φ particle maintain kinetic equilibrium with SM bath +via the elastic scattering processes and share common temperature with SM bath contrary +to studies that deal with secluded or decoupled dark sector scenarios [87, 88]. If the decay +of φ is happening after neutrino decoupling then it will increase neutrino bath entropy and +contribute to ∆Neff. If the decay is completed before CMB we can trace the signature of +DM from ∆Neff at the time of CMB and find some interesting correlation of freeze in DM +and ∆Neff in our proposed set up. +The rest of the paper is structured as follows: In section II we introduce the model. +The possible dynamics of DM production have been discussed in section III. In section +IV we discuss the light neutrino production from late time decay of φ. The outcome of +4 + +DM relic density together with the contribution to ∆Neff at CMB for both scenarios-I and +II have been discussed in section V. Finally, we summarize our results in section VI. We +show relevant theoretical constraints and limit from the SM Higgs invisible decay width in +Appendix A and Appendix B respectively. Feynman diagrams and corresponding thermal +averaged cross-section for 3φ → 2φ and 2φ → 2SM processes are explicitly demonstrated in +Appendix C and Appendix D respectively. +II. +THE MODEL +In order to explain DM production from dark sector and its cosmological imprints in +CMB, we extend the SM by a complex scalar φ, one Dirac fermion χ and three neutral +Majorana fermions, N1,2,3 which are singlet under the SM gauge group. An additional Z3 +symmetry provides the stability of the lightest dark sector particle, under which the field φ +and χ transform non-trivially i.e. {φ, χ} → {ei 2π +3 φ, ei 2π +3 χ} while all the SM fields including +N1,2,3 transform trivially i.e. {N1,2,3, SM} → {N1,2,3, SM} 1. The lightest dark state, χ acts +as a stable DM candidate which is produced from the late time decay of the other dark sector +particle, φ. The right handed neutrinos (RHN) i.e. N1,2,3 which do not transform under +Z3, will be responsible for light neutrino mass via Type-I seesaw mechanism [14]. All the +BSM fields and their corresponding charge assignments under the extended SM Electroweak +(EW) gauge group are tabulated in table-I. +BSM Fields +SU(2)L U(1)Y +Z3 +Dark scalar (DS) +φ +1 +0 +ω(≡ ei 2π +3 ) +DM +χ +1 +0 +ω(≡ ei 2π +3 ) +RHN +N1,2,3 +1 +0 +1 +TABLE I: Charge assignment of BSM fields under the extended SM EW gauge group, GEW +SM ⊗ Z3. +1 In general any ZN symmetry can serve similar kind of scenario with different self interacting number +changing processes, m φ → 2 φ (m ≥ 3), as well as the standard annihilation to SM particles, 2φ → 2SM. +For example, Z2 will provide 4φ → 2φ interactions which are more phase space suppressed for Mφ ∼ +O(MeV) compare to 3φ → 2φ interactions realised in Z3 symmetry [59]. +5 + +The Lagrangian of this model takes the following form : +L = LK+Y +SM +− V (H) +� +�� +� +SM ++LN + LDS + LDS−H + LDS−ν . +(1) +Here, V (H) represents the SM Higgs potential which is given by +V (H) = −µ2 +H|H|2 + λH|H|4. +(2) +The BSM part encapsulate interactions of heavy RHN sector (LN), dark sector(LDS) as well +as their connection with the SM. The interaction of heavy RHN sector is described by, +LN = +� +i +i ¯Niγµ∂µNi − +� +i,j +1 +2MNij ¯ +N c +i Nj − +� +ℓ,j +Yℓj ¯Lℓ ˜HNj + h.c. +(3) +where i, j = 1, 2, 3 and ℓ = e, µ, τ are lepton flavour indices. Lℓ = (νℓ ℓ)T are left handed +the SM lepton doublet and H is the SM scalar doublet with ˜H = iσ2H∗. The second term in +eq.(3) is the Majorana mass term associated with N1,2,3 and the last term is the Dirac Yukawa +interactions with N1,2,3. +After electroweak symmetry breaking (EWSB), the SM scalar +doublet, H can be expressed in unitary gauge as H = +� +0 +h+v +√ +2 +�T +where v = 246 GeV is +vaccum expectation value (VEV) of SM Higgs. Active neutrino masses can be generated via +Type-I seesaw mechanism followed from eq.(3) as +� +mν +� +3×3 ≈ +� +Y v/ +√ +2 +�� +MN +�−1� +Y Tv/ +√ +2 +� +and the mixing angle between active neutrino and RHN is then θmix ∼ +� +Y v/ +√ +2 +�� +MN +�−1 , +where MN ≈ +� +MN +� +3×3 [14]. +The dark sector of this model consists of a complex scalar (φ) and a Dirac fermion (χ) +with similar transformation property under Z3. The lightest state behave as a stable DM +particle. The Lagrangian of the dark sector is described as follows: +LBSM ⊃ LDS + LDS−H + LDS−ν += +� +|∂µφ|2 − µ2|φ|2 + i¯χγµ∂µχ − MDM ¯χχ − λφ|φ|4 − µφ +3! (φ3 + φ∗3) − yφχχcχφ +� ++ +� +− λφH|H|2|φ|2� ++ +� +− +� +i +yφNi ¯χφNi + h.c. +� +, +(4) +where, i = 1, 2, 3. In the above equation µ is the bare mass term of φ and MDM is the mass +of dark fermion χ. For simplicity, in this work we consider all parameters to be real. In the +dark scalar sector, we assume µ > 0 and λφ > 0 so that ⟨φ⟩ = 0 which implies unbroken Z3 +symmetry. After EWSB the physical mass of φ can be expressed as, +M 2 +φ = µ2 + λφH v2 +2 +. +(5) +6 + +The most important interaction as far as our analysis is concerned, is given by the Yukawa +interaction involving the dark scalar (φ), the DM (χ) and SM neutrinos (ν): +Lint +DS−ν = y1χνφ + h.c. +(6) +This Lagrangian can be realized from the last term in braces in eq.(4) via small mixing +angles(θmix) with RHNs(N1,2,3). The effective Yukawa coupling, y1 can be understood as +� +i yφNiθi +mix, where i = 1, 2, 3. +We choose the dark sector of our model parameters in such a manner that we always get χ +as the lightest dark sector particle. This mass pattern and the underlying discrete symmetry +ensure us that the Dirac fermion (χ) with mass MDM is the DM particle and φ with mass Mφ +is the next to ligtest particle (NLP) in this framework. The DM interacts with the SM bath +only through φ via the Yukawa interaction shown in eq.(6). Thus for a given mass hierarchy +between φ and χ, the life-time of φ is determined by the strength of the Yukawa coupling +y1. For our analysis, we assume NLP (φ) to be a long-lived (τφ > τBBN) particle and for +this to happen one requires a very tiny Yukawa coupling y1 ≲ 10−12 (for Mφ ∼ O(GeV)). +The NLP φ can be thermally produced via the sizable Higgs-portal interaction or through +number changing self interaction processes. The production of the DM in the thermal bath +through scattering process is highly suppressed because of it feeble coupling (y1). However, +it can be produced non-thermally from the decay of long-lived φ as shown in Fig.1. The +FIG. 1: Diagram of DM production with active neutrinos from NLP φ +decay width of φ to DM and a light neutrino is given by, +Γφ→χν = y2 +1 Mφ +16 π +� +1 − M 2 +DM +M 2 +φ +�2 +. +(7) +Besides this, there are two more production channels of the DM χ: (a) N1,2,3 → χφ and (b) +φ → ¯χcχ. The main aim of this work is to connect non-thermal DM and ∆Neff producing +7 + +Xfrom self-interacting dark sector (NLP) which is achievable via the decay φ → χν. But the +presence of those new channels (a &b) will dilute the effect of the late time decay of φ in ∆Neff +and may even completely imperil our non-thermal dark matter scenario by thermalizing the +dark sector. To avoid DM production from RHNs we set MN1,2,3 ≫ TRH so that their number +densities get Boltzmann suppressed(e−MN/T) [85]. Therefore for our discussion, we choose +the following hierarchy +MN1,2,3 ≫ TRH > Mφ > MDM. +(8) +In order to get active neutrino mass of the order ∼ 0.1 eV, we require MN1,2,3 ∼ O(1010) +GeV and θmix ∼ O(10−10) [89] and to satisfy the criteria of eq.8 we set TRH = 103 GeV +which is consistent with the bound obtained from BBN [90]. Following this argument and +masses of relevant particles of this model, in the rest of our analysis we can safely ignore the +production of DM from RHN decay in the computation of Yφ and Yχ. Moreover to suppress +the process (b) we consider yφχ ≪ y1, and this is necessary to exalt φ → χν decay so that φ +can have the maximal contribution to ∆Neff. +Interestingly, active neutrinos(ν) produced from the decay of NLP φ along with DM(χ) +as shown in Fig.1 can have very intriguing consequences in the observation of CMB. We +assume the value of Yukawa coupling (y1) such that φ → χν decay mostly happens be- +tween neutrino decoupling temperature (T < 2 MeV) and CMB formation (T ≈ 0.1eV). +This promptly opens up the possibility of probing the impact of extra neutrino production +from CMB radiation. And this can be achieved if y1 varies in the range (10−12 − 10−15) +and for such a tiny coupling φ becomes a long-lived particle (τφ > τBBN). Thus the afore- +mentioned supplementary active neutrino (ν) injection in our proposed scenario increases +neutrino sector entropy and which in turn contribute significantly to additional neutrino +degrees of freedom or ∆Neff which is very precisely measured at the time of CMB. Thus +any experimental observation on ∆Neff can have very intriguing impact on the dynamics of +dark scalar φ which in turn can influence the dark matter (χ) abundance via φ → χν decay +process, thus affecting two disjoint (FIMP dark matter & Neff) sectors simultaneously. To +explore this phenomenology, we perform a detailed numerical scan over model parameters +to show that the precise measurement of ∆Neff at CMB can indeed restrict certain region +of parameter space of non-thermal DM production which is otherwise remains elusive to +visible sector due to extremely tiny strength of interactions involved in such non-thermal +DM production process. +8 + +While doing our numerical analysis, we use the following model parameters: +{MDM, Mφ, λφH, λφ, µφ, y1}, +(9) +Here, the Higgs portal coupling λφH which decides the interaction between φ and SM, plays +a significant role in deciding φ’s number density through 2φ → 2 SM annihilation and +also in (φ SM → φ SM) elastic scattering processes. On the other hand, the scalar sector +parameters λφ and µφ decide the self interactions of φ which is relevant for the number +changing processes like 3φ → 2φ. And finally the effective Yukawa coupling, y1 dictataes +both DM abundance and additional contribution to Neff. +III. +DYNAMICS OF DARK SECTOR +In this section, we discuss the dynamics of the dark sector that leads to the early time +production of the heavy NLP dark scalar (φ) followed by the late time non-thermal produc- +tion of DM (χ) from the decay of φ. The number density of DM will be generated at some +later epoch (after the neutrino decoupling temperature) of the Universe via the following +two steps : +• Step I: +thermal production +of heavy dark scalar φ at the early time of Universe +(τ < τBBN). +• Step II: non-thermal production of DM, χ from the late time decay of φ (τ > τBBN). +104 +102 +1 +10-2 +10-4 +10-6 +10-8 +10-10 +10-12 +10-14 +10-10 +10-6 +10-4 +10-2 +T(GeV) +Yi +Yϕ +eq. +3ϕ→2ϕ +/ 2ϕ→2SM +ϕ → χ(DM) + ν +Yϕ +YDM (Ωh2≃0.12) +BBN +CMB +10-4 +10-6 +10-8 +10-10 +10-12 +0.00 +0.05 +0.10 +0.15 +0.20 +T(GeV) +ΔNeff +BBN +CMB +Planck 2018(1σ)CMB +FIG. 2: +A cartoon diagram of DM production(left) and the impact in ∆Neff at the time of +CMB(right). +9 + +A cartoon of our proposed setup is shown in Fig.2. +In the left panel, we show the +variation of co-moving density as a function of temperature. +The purple and red solid +lines correspond to the thermal production of φ (Step I) and the non thermal production +of DM (χ) (Step II) respectively. We also show two important temperatures, namely, the +BBN and CMB that play crucial role in our analysis. Active neutrinos produced in the +aforementioned decay of φ make substantial contributions to Neff, which can attract severe +constraints from various observational limits on ∆Neff, as shown in the right panel of Fig.2. +The gray rectangular band is excluded by the Planck 2018 data at 1σ [6]. +Having this +broad picture in mind we now provide details of the thermal production of NLP followed +by non-thermal production of DM in the rest of this section. +Step-I: Thermal production of φ +We consider a scenario in the early universe, when the interaction rate (Γint +φ ) of the NLP +(φ) dominates over the expansion rate (H) of the Universe, (Γint +φ >> H) so that φ remains +in thermal and chemical equilibrium. As the temperature of the universe cools down, the +interaction rate of φ falls below the expansion rate of universe (Γint +φ +< H), thus the system +departs from thermal equilibrium and the number density of φ freezes out. The number +density of φ is mainly provided by the following two types of number changing processes: (i) +3φ ↔ 2φ via φ self interactions (shown in Fig.3(a)) and (ii) 2φ ↔ 2 SM via the SM Higgs +portal interactions (shown in Fig.3(b)). As a result of these two number changing processes, +the NLP (φ) keeps its chemical equilibrium. On the other hand, the kinetic equilibrium +is maintained between φ and the SM bath via elastic scatterings, generically expressed as +φ SM ↔ φ SM which help φ to keep same temperature with SM bath till freeze out takes +place. +The complete dynamics of thermal production of φ can be described by the following Boltz- +mann equation(BEQ): +dYφ +dx += −0.116 g2 +s +√gρ +M 4 +φ +x5 Mpl +� +σv2� +3φ→2φ (Y 3 +φ − Y 2 +φ Y eq +φ ) +−0.264 gs +√gρ +Mφ +x2 Mpl ⟨σv⟩2φ→2SM (Y 2 +φ − Y eq +φ +2) +− +� +45 +4π3 ⟨Γφ→χν⟩ x +M 2 +φ +Mpl +√gρ +Yφ . +(10) +10 + +(a) +(b) +FIG. 3: A cartoon of number changing process of φ: (a) three φ annihilate to two φ (3φ → 2φ). +and (b) two φ annihilate to two SM particles (2φ → 2SM). +Let us first describe various notations used in eq.(10). Yφ(= nφ +s ) is the co-moving number +density of φ where s is the entropy density and x is the dimensionless parameter defined as +x = Mφ +T . Y eq +φ +is the equilibrium co-moving number density of φ. gs(x) and gρ(x) are the ef- +fective relativistic degrees of freedom associated with entropy density and the energy density +respectively and finally Mpl is the Planck mass(Mpl = 1.22 × 1019GeV). The thermal aver- +aged cross-section of 2φ → 2 SM process is denoted by ⟨σv⟩2φ→2SM and for self-interacting +number changing process (3φ → 2φ), it is defined as ⟨σv2⟩3φ→2φ. The first two terms in +eq.(10) lead to non zero density of φ via thermal freeze-out mechanism and it occurs at +x = xtot. +F , where tot in the superscript implies that both number changing processes i.e. +3φ → 2φ and 2φ → 2SM are involved in φ freeze-out process. The last term in eq.(10) pro- +vides the late time (after BBN) decay of φ into DM (χ) and SM neutrinos, resulting the +dilution of number density of φ into χ and ν. +From eq.(10) it is clear that two number changing processes of NLP (φ) as discussed above are +present to keep φ in the thermal bath. However, depending upon the mass and couplings +of NLP, it can be shown very easily that one of those two number changing processes is +infact sufficient for the freeze-out and the final yield of NLP (φ). To justify our argument +quantitatively we define the interaction rate of 3φ → 2φ process as: Γ3φ→2φ = n2 +φ⟨σv2⟩3φ→2φ +and of 2φ → 2SM as: Γ2φ→2SM = nφ⟨σv⟩2φ→2SM. In addition to these number changing +processes, φ SM → φ SM number preserving scattering process is also present to keep φ +in kinetic equilibrium with SM bath. The interaction rate of this process is defined as: +Γ[φ SM→φ SM] = nSM⟨σv⟩[φ SM→φ SM]. Depending on the relative interaction strength between +11 + +0SM +SMtwo number changing processes of φ, we are interested in the following two production modes +of φ: +Scenario I : +Γ[φ SM→φ SM] > Γ3φ→2φ ≫ Γ2φ→2SM , +Scenario II : +Γ[φ SM→φ SM] > Γ2φ→2SM ≫ Γ3φ→2φ . +In the above hierarchy of scattering processes, Γ[φ SM→φ SM] plays a decisive part in main- +taining kinetic equilibrium of φ. During the freeze-out of φ through processes like: nφ → 2φ, +(for n > 2) the rest mass energy of initial state particles can significantly enhance the ki- +netic energy of final state particles, which in turn can heat up the dark sector [66], leading +to an imbalance between the dark sector temperature (Tφ) and SM bath temperature (T). +Thus, in general, to take into account this temperature imbalance one should consider a +new parameter (Tφ) in the evolution equation of NLP number density (Yφ) [85]. However, in +our study we can avoid this paradigm by considering kinetic equilibrium between φ and SM +bath, i.e. by taking Tφ = T at least upto the temperature at which φ freezes out from the +thermal bath. And to achieve this, Γ[φ SM→φ SM] must be larger than interaction rate of the +other processes as well as the expansion rate H of the universe (i.e Γ[φ SM→φ SM]|xF ≳ H(xF)) +[65]. Most importantly this condition must be satisfied in both Scenario I and II. The rel- +evant Feynman diagrams and thermal averaged cross-sections for 3φ → 2φ, 2φ → 2SM and +φ SM → φ SM processes are shown in Appendices C and D. +• Scenario I: In this scenario we consider the interaction rate of 3φ → 2φ number +changing process (Γ3φ→2φ) is significantly higher than 2φ → 2SM process (Γ2φ→2 SM). +Thus 3φ → 2φ process successfully keeps φ in thermal bath for longer duration in +comparison to the process 2φ → 2 SM. +Hence freeze-out of φ is mainly governed +by the 3φ → 2φ process and it occurs at x = x3φ→2φ +F +≈ xtot +F +> x2φ→2SM +F +. +Here +x3φ→2φ +F +(x2φ→2SM +F +) signifies the inverse freeze out temperature of φ when only 3φ → 2φ +(2φ → 2SM) is considered. +In our model, the interaction rate of 3φ → 2φ (2φ → 2SM) process depends +on the couplings λφ, µφ/Mφ (λφH ) and mass Mφ. +To demonstrate the dynamics +(where Γ3φ→2φ ≫ Γ2φ→2 SM), we show the variation of the co-moving number den- +sity Yφ as a function of x(= Mφ/T) in Fig.4(a) for a sample Benchmark point: +{Mφ, µφ/Mφ, λφH, λφ} = {20 GeV, 0.1, 10−2, 1}. The black solid line corresponds +12 + +3ϕ→2ϕ +2ϕ→2SM +3ϕ→2ϕ+2ϕ→2SM +5 +10 +50 +100 +10-12 +10-10 +10-8 +10-6 +10-4 +10-2 +x +Y +mϕ=20GeV,λϕ=1.0,λϕH=10-2 +(a) +λφ H=10 +-4 +λφ H=10-2 +λφ +10−3 +0.01 +0.1 +1 +Mφ (GeV) +10 +20 +30 +40 +50 +(b) +FIG. 4: (a)Thermal freeze-out of φ governed by 3φ → 2φ and (b)Parameter space for scenario-I. +For other details see the text. +to the equilibrium co-moving density of φ (Y eq +φ ) and the blue dashed line corresponds +to the co-moving number density of φ considering contributions from both the num- +ber changing processes: 3φ → 2φ and 2φ → 2 SM in eq.(10). The brown solid line +(red dotted line) depicts the variation of number density of φ when only 3φ → 2φ +(2φ → 2 SM) process is present in eq.(10). The relative contribution of these two +processes in the evolution of Yφ is clearly seen in this figure. If we consider only the +sub-dominant 2φ → 2 SM process, φ freezes-out earlier (red dotted line) due to small +Γ2φ→2 SM, whereas, the dominant 3φ → 2φ process maintains φ in thermal bath for +longer duration (brown solid line). Thus the freeze-out abundance of φ (blue dashed +line) is governed mainly by the dominant 3φ → 2φ process due to larger Γ3φ→2φ for +our choice of model parameters. Therefore, we can safely ignore the second term in +eq.(10) and the modified BEQ takes the following form: +dYφ +dx += −0.116 g2 +s +√gρ +M 4 +φ +x5 Mpl +� +σv2� +3φ→2φ (Y 3 +φ − Y 2 +φ Y eq +φ ) +− +� +45 +4π3 ⟨Γφ→χν⟩ x +M 2 +φ +Mpl +√gρ +Yφ. +(11) +Based on the above argument we can identify the parameter space for scenario-I satis- +fying the criteria: x3φ→2φ +F +> x2φ→2SM +F +. In Fig.4(b) we display the parameter space +for this scenario in Mφ vs. +λφ plane with µφ/Mφ = 0.1 for two different values +λφH = {10−2, 10−4} depicted by the blue and red shaded region respectively. The +13 + +criteria for scenario-I holds only for the region left to individual lines. With an in- +crease in Mφ ,Γ3φ→2φ becomes more mass suppressed compared to Γ2φ→2SM. Hence +for fixed values of λφ, and λφH with increasing Mφ, Γ3φ→2φ falls below Γ2φ→2SM and +scenario-I doesn’t hold anymore for the parameter space right to the colored lines. +With an increase in λφH, Γ2φ→2SM increases and eventually Γ3φ→2φ falls below Γ2φ→2SM +even with lower Mφ. For that reason we see the shaded region moves toward lower Mφ +(towards left) with an increase in λφH. The regions right to the colored lines demand +a different treatment which will be discussed shortly. +Before we conclude this part of our analysis, it is worth noting that the present dark +sector dynamics also allows 4φ → 2φ number changing process involving the same +λφ coupling that is responsible for 3φ → 2φ process. Inspite of the same interaction +strength (λφ), 4φ → 2φ process is more phase space suppressed compared to that of +3φ → 2φ, hence, we neglect it in our numerical calculation of Yφ. +• Scenario II: In this picture we consider Γ2φ→2SM ≫ Γ3φ→2φ, which is contrary to the +previous scenario. In this case freeze-out of φ is dictated by 2φ → 2SM annihilation +process that keeps φ in thermal bath for a longer period compared to 3φ → 2φ process. +Hence the freeze-out of φ occurs at x = xtot +F ≈ x2φ→2SM +F +> x3φ→2φ +F +. +3ϕ→2ϕ +2ϕ→2SM +3ϕ→2ϕ+2ϕ→2SM +5 +10 +50 +100 +10-12 +10-10 +10-8 +10-6 +10-4 +10-2 +x +Y +mϕ=20GeV,λϕ=0.01,λϕH=10-2 +(a) +λφ=0.1 +λφ=0.01 +λφ H +10−3 +0.01 +0.1 +1 +Mφ (GeV) +10 +20 +30 +40 +50 +(b) +FIG. 5: (a)Thermal freeze-out of φ governed by 2φ → 2SM and (b)Parameter space for scenario-II. +For other details see the text. +In Fig.5(a) we report the evolution of Yφ as a function of x = Mφ +T for λφ = 0.01 keeping +other parameters same as in Scenario-I. From this figure it is evident that Yφ is entirely +14 + +decided by 2φ → 2SM number changing processes contrary to the previous scenario +where 3φ → 2φ process was controlling the dynamics. Thus eq.(10) can be simplified +by neglecting the sub-dominant 3φ → 2φ process: +dYφ +dx += −0.264 gs +√gρ +Mφ +x2 Mpl +� +σv2� +2φ→2SM (Y 2 +φ − (Y eq +φ )2) +− +� +45 +4π3 ⟨Γφ→χν⟩ x +M 2 +φ +Mpl +√gρ +Yφ . +(12) +In Fig.5(b) we display parameter space for this scenario in Mφ vs. λφH plane with +µφ/Mφ = 0.1 for two different values λφ = (10−1 & 10−2) depicted by the blue and red +shaded region respectively. For the same reason discussed in the context of scenario-I, +in this case also Γ3φ→2φ decreases with decrease in λφ and finally falls below Γ2φ→2SM. +And this phenomena is true even for lower Mφ. For this reason here also we see that +the shaded region shifts towards lower Mφ (left) with decrease in λφ. +In summary the main observation of this whole subsection are the following: +Scenario I : +Γ3φ→2φ ≫ Γ2φ→2SM +=⇒ xtot +F +≈ +x3φ→2φ +F +> x2φ→2SM +F +, +Scenario II : +Γ3φ→2φ ≪ Γ2φ→2SM +=⇒ xtot +F +≈ +x2φ→2SM +F +> x3φ→2φ +F +. +(13) +It is worth mentioning that scenario-II is more common and has already been studied +in different literature [57, 84], where mother particles are considered to have sizable +annihilation cross-section with SM bath. +In this work our main focus is on scenario-I, +although for the sake of completeness of the analysis we also discuss scenario-II. +Step-II: Non thermal DM production +Following our previous discussion we now focus on the non-thermal production of DM +(χ) from the dilution of φ density described by the last term in the R.H.S of eq.(10). We +solve the following Boltzmann equation to get the evolution of DM(χ) abundance, +dYχ +dx = +� +45 +4π3 ⟨Γ⟩φ→χν +x +M 2 +sc +Mpl +√gρ +Yφ, +(14) +where, Yχ is the co-moving number density of DM χ. In general the solution of Yφ comes +from the BEQ in eq.(11) for scenario-I and in eq.(12) for scenario-II respectively. In the +calculation of Γ(φ → χν) we consider the Yukawa coupling y1 in the range (∼ 10−12 −10−15) +15 + +so that the decay of φ → χ+ν happens in post BBN and pre CMB era. At this stage, we find +it worth discussing one subtle issue regarding the thermal averaged decay width ⟨Γ⟩φ→χν. +As we have pointed out before, that at the time when φ freezes-out, it maintains the same +temperature as the SM bath via the elastic scattering processes. However, this may not be +true at the time of decay(< TBBN) if Γ[φ SM→φ SM] < H at that time. This results the dark +sector to acquire a different temperature T ′ (̸= T) than the thermal bath and this must be +evaluated in order to get ⟨Γ⟩φ→χν. In this work, as we are studying the dark sector dynamics +at low temperature (T ′ ≪ Mφ), and in this limit the thermally averaged decay width can +simply be approximated as ⟨Γ⟩φ→χν (T ′) ≈ Γφ→χν [42, 91], thus reducing the complication +of tracking temperature dependence of the evolution of φ. After solving eq.(14) we get the +complete picture of DM production as shown by red solid line in the left panel of Fig.2. +IV. +LIGHT NEUTRINO PRODUCTION BEFORE CMB +Now we discuss the production of supplementary light neutrinos from the late time decay +of φ and the relevant mechanism of verifying those light degrees of freedom at CMB. As +revealed earlier, neutrinos that are produced after neutrino decoupling (T ≲ 2 MeV) would +inject entropy in the neutrino bath. At the time of CMB, the number of relativistic neutrino +degrees of freedom is expressed as, +N CMB +eff += 8 +7 +�11 +4 +�4/3 ρSM +ν +ργ +����� +T=TCMB +, +(15) +where, ρSM +ν += 3 × 2 × 7 +8 × π2 +30(T SM +ν +)4 and ργ = 2 × π2 +30T 4 are energy densities of neutrino +and photon respectively. Due to the extra neutrino injection from the non-thermal decay +of φ, the energy density of the neutrino bath increases to ρ′ +ν (ρ′ +ν > ρSM +ν ). In this case, the +relativistic neutrino degrees of freedom(N ′ +eff) also differs from the prediction of SM at the +time of CMB. We parameterise this deviation at the time of CMB in the following manner, +∆Neff = +� ρ′ +ν +ρSM +ν +− 1 +� +NSM +eff +����� +T=TCMB +. +(16) +We now solve the following Boltzmann equation to estimate the evolution of ρ′ +ν with tem- +perature, +dρ′ +ν +dx = −4 β ρ′ +ν +x ++ +1 +xH(x) ⟨EΓ⟩φ→χν Yφ s , +(17) +16 + +where the term β indicates the variation of gs(T) with T and is defined as +β(T) = 1 + 1 +3 +T +gs(T) +d gs(T) +dT +. +(18) +where, x is the dimensionless variable as mentioned earlier in context of equation (10). +Yφ is the co-moving number density of φ which is computed by solving eq.(11) or eq.(12) +depending on the scenario we consider. gs(x) is the number of effective degrees of freedom +related to the entropy density and s is the co-moving entropy density. The term ⟨EΓ⟩φ→χν in +eq.(17) is the most crucial ingredient in this analysis which represents the thermal averaged +energy density transferred to neutrino sector and is defined as +⟨EΓ⟩φ→χν = |M|2 +φ→χν +32π +� +M 2 +φ − M 2 +DM +� +M 2 +φ +� +1 − M 2 +DM +M 2 +φ +� +. +The first term in the R.H.S of eq.(17) is responsible for the dilution of ρ′ +ν due to expansion +of the universe while the second term decides the evolution of augmented contribution to ρ′ +ν +from φ decay. The evolution of ρSM +ν +after the decoupling of neutrinos is governed by only the +expansion effect. Thus in the absence of any new source ρSM +ν +can be computed by setting +the second term of the R.H.S of eq.(17) equal to zero and considering only the dilution of +energy density. +V. +RELIC DENSITY AND ∆Neff +So far we have built up the basic framework of the underlying dynamics of dark sector +particles (φ and χ) that provided freeze-in DM as well as yielded extra active light neutrino +that with its possible footprints in ∆Neff. In this section, we perform an exhaustive nu- +merical analysis of Scenario-I and Scenario-II to quantitatively estimate phenomenological +consequences of the late-time decay of φ in the light of current and future measurements +of ∆Neff. For this we first scrutiny the dependence of DM relic density and the ∆Neff on +various model parameters as elaborated in sec-III and sec-IV respectively. +A. +Relic density +To calculate the DM relic density, we numerically solve eq.(14) along with either eq.(11) +(for scenario-I) or eq.(12) (for scenario-II). The solution of the coupled BEQs for each sce- +nario yields Yφ and Yχ as a function of x(= Mφ/T). Using the co-moving density of DM, Yχ +17 + +at x → ∞, one finds out DM relic density as [23]: +Ωχh2 = 2.755 × 108 × +�MDM +GeV +� +× Y today +χ +, +(19) +where Y today +χ += Yχ(x → ∞). The precise determination of Y today +χ +is highly model dependent. +In the following two sub-sections we pin down Y today +χ +for scenario-I and II and corresponding +relic densities. +Scenario-I: As shown before, densities of φ and χ for the scenario-I are mainly driven by +3φ → 2φ number changing process in the dark scalar sector. Based on this number changing +process, we calculate the co-moving abundances of φ and χ and show their evolution with +x(= Mφ/T) in Fig.6. The solid, dashed and dotted lines signify Yφ, Y eq +φ and Yχ respectively. +It can be seen from these figures that the late-time decay of φ (solid lines) produces the +abundance of χ. As the φ → χ + ν decay proceeds, the number density of φ slowly changes +into χ number density and eventually at the end of the decay, the density of φ completely +dilutes to χ number density (Yχ ≡ Yφ at τ ≫ τφ). One can easily understand this from +the fact that φ → χν decay is the only possible decay mode of the NLP (φ) [92]. Thus, in +the generation of Yχ from Yφ, the magnitude of the Yukawa coupling (y1 > 0) has hardly +any role to play, except for setting the lifetime of φ and this provides Yφ +� +x3φ→2φ +F +� +≃ Y today +χ +. +Therefore the relic density of DM given in eq.(19) turns out to be Ωχh2 ∝ MDM×Yφ +� +x3φ→2φ +F +� +. Consequently in order to get fixed Ωχh2, any increase in MDM demands a decrease in +Yφ +� +x3φ→2φ +F +� +and vice versa. We have pointed out before that φ → χ ν decay to happen +between BBN and CMB the value of y1 should lie in the range : {10−12 − 10−15}. As a +sample representative value we set y1 = 10−12 throughout our numerical analysis. To show +the evolution of Yφ and Yχ with temperature, we fix µφ/Mφ = 0.1 and MDM = 400 keV for +both plots. We consider λφH = 10−4 to realize Scenario-I. +In Fig.6(a) we present evolution of densities Yφ (solid line) and Yχ (dotted line) as a +function of dimension less parameter x(= Mφ/T) for two different Mφ and a fixed self- +interaction coupling λφ = 1.0. The dynamics of dark sector particles for Mφ = 1 GeV and +Mφ = 10 GeV are depicted by red and blue colors respectively. With the increase of Mφ, +⟨σv2⟩3φ→2φ encounters phase space and propagator suppression which is also understood from +the expression given in appendix C. As Yφ goes like Yφ ∝ 1/ ⟨σv2⟩3φ→2φ (using analytical +solution [61]), with the smaller ⟨σv2⟩3φ→2φ, the thermal freeze-out of φ happens at earlier +time with higher abundance Yφ and eventually this Yφ is transfered to the Yχ. As a result Yχ +18 + +yΦ =10-12 +MΦ =10 GeV +MΦ=1 GeV +Y +10−9 +10−8 +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +x +10−2 +10−1 +100 +101 +102 +103 +104 +105 +(a) +yΦ =10-12 + MΦ=1 GeV, MN1= 420 KeV +YΦ +eq +λΦ =1.0 +λΦ=0.1 +Y +10−9 +10−8 +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +x +10−2 +10−1 +100 +101 +102 +103 +104 +105 +(b) +FIG. 6: Evolution of co-moving abundances of φ (solid line) and DM(χ) (dotted line) with x(≡ 1/T) +(T in GeV) for scenario-I. In (a)for a fixed λφ = 1.0 with two different values of Mφ and in (b) +for a fixed Mφ = 1 GeV with two different values of λφ are shown. The Higgs portal coupling is +considered here to small, λφH = 10−4 in order to realise the scenario. The other parameters like +y1 = 10−12, MDM = 400 keV and µφ/Mφ = 0.1 are kept same for both plots. +is also higher for higher Mφ. This feature is portrayed in the Fig.6(a) where higher(lower) +value of Mφ leads to the higher(lower) abundance Yχ represented by the red(blue) dotted +line. +To study the role of dark scalar self-coupling,λφ on DM abundance, in Fig.6(b) we show +the variation in Yχ for two different values of λφ = 0.1 (red line) and 1.0 (blue line) keeping +Mφ = 1 GeV. It is obvious, that as the value of λφ increases, the thermal averaged cross- +section ⟨σv2⟩3φ→2φ also increases which eventually reduces the abundance Yφ and finally this +reduced Yφ generates lower Yχ. This is elucidated in Fig.6(b), where a higher(lower) value +of λφ gives a lower(higher) Yχ as it is shown by the blue(red) dotted line. +The other parameter µφ with mass dimension is also responsible for 3φ → 2φ processes as +⟨σv2⟩3φ→2φ ∝ (µφ/Mφ)2 (see Appendix C). With an increase in the ratio µφ/Mφ, the cross- +section will enhance leading to a decrease in Yφ as well as Yχ. For simplicity, we consider +the ratio µφ/Mφ = 0.1 throughout our analysis and is consistent with the theoretical upper +bound on µφ coming from stable vacuum as discussed in Appendix A. +After describing the dependence of relic abundance on different model parameters, we +now present the allowed region of dark sector parameter space from DM observed density, +(ΩDMh2 = 0.120±0.001) given by PLANCK [6]. We perform a numerical scan on the model +19 + +MDM(MeV) +0.1 +1 +10 +Scenario-II +λφ +10−2 +10−1 +100 +Mφ(GeV) +0.1 +1 +10 +100 +FIG. 7: DM relic density satisfied points in Mφ − λφ plane for scenario-I with µφ/Mφ = 0.1, +λφH = 10−4 and y1 = 10−12. The color gradient indicates the range of MDM satisfying the correct +relic density. The shaded region corresponds to the parameter space where scenario-II is dominating +over scenario-I. White regions are just computational artifact associated with the scan. +parameters in the following range +Mφ : {0.1 − 100 GeV}, +λφ : +{0.001 − 1} ; +(20) +to calculate Ωχh2 using eq.(19). We keep other parameters fixed as in Fig.6. and to allow +the on shell decay φ → χ + ν we set Mφ > MDM. +Our scan result is displayed in Fig.7, where we show points satisfying relic density con- +straints in λφ vs. Mφ plane. The grey shaded region corresponds to the parameter space +where scenario-II dominates which demands a different analysis and will be discussed shortly. +The color gradient in the above figure represents DM mass range varying from 0.1 MeV to +10 MeV set by the observed relic density constraint. One can see from this figure that the +higher value of λφ prefers to higher value of MDM. As explained in the context of Fig.6, +with the increase of λφ, Yχ decreases and hence higher value of MDM is required in order +to satisfy the correct relic density as depicted in above Fig.7. For simplicity we restrict our +scan within the specified range of λφ mentioned above. However one can make the scan even +20 + +for higher value of λφ ≳ 1.0 within the perturbativity limit. For those values of λφ even +heavier DM mass,(10 MeV ≲ MDM < Mφ) will be allowed by the relic density constraint. +Scenario-II: We shall now move to the second scenario where the density of φ is mainly +driven by 2φ → 2 SM number changing process. Following our earlier discussion we know +that the density of φ converts into the density of DM +� +Yφ +� +x2φ→2SM +F +� +≃ Y today +χ +� +via the +late time decay of φ. Therefore the relic density of DM given in eq.(19) becomes Ωχh2 ∝ +MDM × Yφ +� +x2φ→2SM +F +� +. Similar to previous scenario Yφ(x2φ→2SM +F +), decreases with increase in +MDM in order to get fixed density and vice versa. One can also analytically express the yield +of φ at freeze-out as: Yφ ∝ 1/⟨σv⟩2φ→2SM [23]. For heavier Mφ, more annihilation processes +of φ to SM pairs kinematically open up and enhance the cross-section. Thus ⟨σv⟩2φ→2SM can +be expressed as � +X=SM⟨σv⟩φφ→XX Θ +� +Mφ − MX +� +where Θ is the Heaviside step function. +For a fixed Mφ, Yφ as well as Yχ decreases as one increases λφH since ⟨σv⟩2φ→2SM ∝ λ2 +φH. +However, here the dependence of Yχ on Mφ is contrary to that of scenario-I. In this case +with the increase in Mφ, the annihilation cross-section, ⟨σv⟩2φ→SM also increases for the +aforementioned reasons and thus resulting a decrease in Yφ as well as Yχ [57]. +Now in order to find a consistent parameter space satisfying observed relic density mea- +sured by PLANCK[6], we perform a numerical scan of the relevant parameters for scenario-II +in the following range: +Mφ : {10 − 100 GeV}, +λφH : {10−3 − 10−1} ; +(21) +whereas the other parameters are kept fixed as µφ/Mφ = 0.1, λφ = 0.1 and y1 = 10−12. +The choices of dark sector parameters in eq.(21) ensure that Γ2φ→2SM ≫ Γ3φ→2φ which +is required for the scenario-II. We consider Mφ up to 100 GeV, beyond that Yφ is more +suppressed resulting in a negligible contribution to ∆Neff which will be discussed in due +course of time. +In Fig.8 we plot correct relic density satisfied points in the Mφ vs. λφH plane. The +variation of color gradient represents the variation of MDM considered here. The correct +relic density constraint sets the DM mass in the range MDM: ∼ {10 MeV − 10 GeV} +for our chosen parameters. The gray shaded region on lower left corner of the above figure +represents the region where scenario-II does not work paving way to Scenario-I. With increase +in λφ(> 0.1), Scenario-I will start to dominate over scenario-II even with higher value of +Mφ and the shaded region will move towards right accordingly. For Mφ < mh/2 , h → φφ∗ +21 + +MDM(MeV) +10 +100 +1000 +104 +Br(h→ invisible)>11%) +Scenario-I +λφ H +10−3 +10−2 +10−1 +Mφ(GeV) +20 +40 +60 +80 +100 +FIG. 8: DM relic density satisfied points for scenario-II are shown in the λφH vs. +Mφ plane +with λφ = 0.1, µφ = 0.1Mφ, y1 = 10−12 and the color gradient represents the variation in MDM +satisfying the correct relic density. The gray shaded region corresponds to the parameter space +where scenario-I is dominating. +decay opens up and contributes to the SM Higgs invisible decay width (Γinv. +h ) which is very +precisely measured by CMS [93]. The bound from Γinv. +h +(discussed in Appendix B) excludes +a significant part of the parameter space as shown by the light cyan region in Fig.8. As +understood from the figure, scenario-II works in the higher range of Mφ and the moderate +values λφH leading to lower Yχ as discussed earlier in this subsection. +The 2φ → 2SM +annihilation cross-section near Higgs pole, Mφ ∼ mh/2, causes further suppression in Yχ. +For Mφ > MW, more final states open up resulting in even larger ⟨σv⟩2φ→2SM. Thus to +satisfy the observed DM density one has to reduce λφH in that region as shown in top right +corner (white area) of the figure. Therefore the scenario-II allows higher DM mass to satisfy +the correct relic density upto few GeV. +22 + +B. +Contribution to ∆Neff at CMB +In earlier sections, we have established that our main thrust of this whole exercise is to +calculate contributions to ∆Neff by extra active neutrinos produced in association with FIMP +like DM from the late time decay of a self interacting dark scalar φ. Simultaneously, we +have also emphasized the possibility of correlating the dark matter mass with the measured +value of ∆Neff. Thus, any precise determination of ∆Neff would provide an indirect probe of +the dynamics of dark sector ivolving a strongly self interactiong particle φ as well as FIMP +like DM. Based on our discourse in sec-IV we will now investigate dependence of dark sector +model parameters in ∆Neff which is completely determined by the ratio ρ′ +ν/ρSM +ν . +Mφ=1 GeV +Mφ=0.1 GeV +Δ Neff +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +T(GeV) +10−7 +10−6 +10−5 +10−4 +10−3 +(a) +λφ=0.1 +λφ=1.0 +Δ Neff +10−2 +10−1 +100 +101 +T(MeV) +10−4 +10−3 +10−2 +10−1 +100 +(b) +FIG. 9: Evolution of ∆Neff with temperature(T) for two different values of Mφ keeping λφ = 1.0 +fixed (a) and for two different values of λφ keeping Mφ = 0.1 GeV fixed in scenario-I. Other +parameters are kept fix as y1 = 10−12 and MDM = 400 keV. +In Fig.9 we show the evolution of ∆Neff with temperature T for different set of model +parameters as shown in the figure caption. We first numerically evaluate ρ′ +ν/ρSM +ν +by solving +eq.(19) along with eq.(11) and then plug it into eq.(16) to estimate ∆Neff. From both the +figures Fig.9(a) and 9(b) we notice that at high T, the ∆Neff is almost negligible because +the entropy injection to neutrino bath is very small during the earlier epoch of φ → χν +decay. With the decrease in temperature, φ freezes out from the thermal bath and decays +into χ + ν after BBN, generating a new source of active neutrinos that inject extra energy +density to neutrino bath. This added neutrino density causes continuous growth of ∆Neff +with lowering of temperature. With further decrease in the temperature, at some point +φ decay is completed and any auxiliary neutrino production also stops. +Thus no more +23 + +supplementary energy transfer to neutrino bath takes place and the ratio ρ′ +ν/ρSM +ν +attains its +maximum possible value at that temperature. After that both ρ′ +ν and ρSM +ν +dilutes in the +same fashion with further decrease in temperature, resulting in a fixed ratio ρ′ +ν/ρSM +ν +which +corresponds to a constant value of ∆Neff. Since ρ′ +ν ∝ Yφ (following eq.(17)), higher value of +Yφ leads to higher energy transfer to neutrino bath resulting in larger ∆Neff and vice-versa. +We plot the evolution of ∆Neff in Fig.9(a) for Mφ = 1 GeV (red line) and Mφ = 0.1 GeV +(blue line) keeping λφ = 1.0 fixed. In Fig.9(b) we show the similar plot as in Fig.9(a) but +this time for a fixed Mφ = 0.1 GeV and taking two values of λφ = 1.0 (red line) and 0.1 +(blue line). While generating these two plots, we fix MDM = 400 MeV, and y1 = 10−12. +The behavior of ∆Neff with the model parameters (Mφ, λφ and µφ/Mφ) is same as of Yφ as +discussed earlier for scenario-I (Fig.6(a) and 6(b)) and the same dependence is depicted in +Fig.9(a) and 9(b). +The effect of λφH on ∆Neff in Scenario-II is similar to Scenario-I. Following our previous +argument, for any increase in the value of Higgs portal coupling λφH, φ number density Yφ +decreases and that leads to a diminished contribution of active neutrinos in ∆Neff. However, +Mφ dependence of ∆Neff shows opposite behaviour in Scenario-II than Scenario-I, here, ∆Neff +decreases with an increase in Mφ. The reason for this contrary nature follows the same +argument as we revealed in the context of relic density calculation. For heavier Mφ, due to +enhanced phase space one gets larger ⟨σv⟩2φ→2SM that leads to lower Yφ and finally lower +∆Neff. Hence the energy transferred to neutrino sector is too less to contribute significantly in +∆Neff and the ∆Neff for scenario-II will be far below the sensitivity of the current and future +generation experiments. In this paper we do not display the explicit parameter dependence +in ∆Neff in scenario-II, however similar study could be found in [57]. +Finally, we calculate the ∆Neff for different values of the model parameters in scenario-I +and displayed our findings in Fig.10. We present ∆Neff as a function of Mφ and the color +gradient represents the range of MDM allowed by observed DM relic density [6]. In the figure, +we show different existing exclusion bounds as well as future sensitivities on ∆Neff depicted +by different coloured patches. We notice that a decrease in MDM yields a increase in ∆Neff +also. This is easily understood as lower value of MDM requires higher value of Yχ to satisfy +the observed relic density. As we analyzed earlier, Yχ is governed by Yφ and higher value +of Yχ corresponds to higher value of Yφ. Thus for a higher value of Yφ, more energy gets +transferred to the neutrino sector, leading to the higher value of ∆Neff. We also notice that +24 + +MDM(MeV) +0.1 +1 +10 +CMB - S4 (2σ) +SPT - 3G (1σ) +Planck 2018 (2σ) +Planck 2018 (1σ) +Δ Neff +0.01 +0.10 +1.00 +Mφ(GeV) +0.1 +1 +10 +FIG. 10: Variation of ∆Neff with Mφ for µφ/Mφ = 0.1, λφH = 10−4 and y1 = 10−12 where the +color gradient represents the range of DM mass in scenario-I. The current 1σ and 2σ upper limits +on ∆Neff from PLANCK 2018 and future sensitivities of two upcoming CMB experiments are also +shown for comparison. +higher values of Mφ corresponds to the points yielding higher value of ∆Neff. This is also +understandable as higher value of Mφ leads to higher value of Yφ resulting higher value of +∆Neff for the same reason discussed above. In the same plot, we present the current upper +limits(1σ and 2σ) on ∆Neff from PLANCK 2018 and future sensitivities of two upcoming +CMB experiments. The present 2σ and 1σ limit on ∆Neff from Planck 2018 excludes DM +mass below few hundred keV. The future generation experiments like SPT-3G [74] in 1σ limit +and CMB-S4 2σ limit [75] may probe heavier DM mass upto 1 MeV. The allowed parameter +space from the constraints of ∆Neff is also consistent with bound on free streaming length +of DM coming from Lyman-α forest[53, 94]. +VI. +CONCLUSION +In this work we have proposed a minimal extension of the Type-I seesaw model with a +complex scalar singlet(φ) and a singlet Dirac fermion (χ). To ensure the stability of the +25 + +lightest dark sector particle, an additional Z3 symmetry has been imposed under which +φ and χ transform non-trivially while the rest of SM particles and three RHNs transform +trivially. +Mass spectrum of the dark sector particles are such that the Dirac fermion χ is the lightest +particle and plays the role of DM, while the singlet scalar φ is the next to lightest particle. +The DM with its tiny coupling with SM bath can only be produced from the late time decay +of φ and obtains its abundance. On the other hand φ remains in thermal bath due to its +strong self coupling and after its freeze out it decays to DM and active neutrinos. Depending +on the thermal history of φ, we have divided the analysis into two scenarios. In the first +Scenario (I), φ gains its number density through freeze out mechanism via the number +changing strong self-interactions within the dark sector whereas, in the second Scenario (II) +φ freezes out via the SM Higgs portal coupling to SM particles . The RHNs(N1,2,3) which are +responsible for generating light neutrino masses and mixing angles by type-I seesaw model, +are sufficiently heavy (MN1,2,3 ≫ TRH) such that their number densities do not contribute +to DM relic. However, the presence of RHNs in the particle content allows an effective +interaction between φ, χ and active neutrinos(ν) which leads to extra neutrino production +from the late time decay of φ. To track the abundances of φ and χ we have solved two coupled +Boltzmann equations. We have first checked the effects of different model parameters on +the relic density of DM by solving those Boltzmann equations and identifying the parameter +space giving correct relic density in both scenarios (I & II). Apart from producing the +right amount of DM relic, the late time decay of φ makes significant impact on the total +radiation energy density at the time of CMB formation which is parameterized as ∆Neff. +To compute ∆Neff we have evaluated the extra radiation energy density injected into light +neutrino bath from φ by solving the required Boltzmann equation. In scenario-I DM mass +up to a few hundred keV is excluded from the present 1σ limit on ∆Neff from Planck 2018 +data. The future generation experiments like SPT-3G, CMB-IV will be sensitive enough +to test DM mass up to a few MeV. However, in scenario-II where the abundance of the +mother particle (φ) is suppressed due to sizable interactions with SM bath, we have found +that the entropy injection is insensitive to the bounds on ∆Neff coming from present and +future-generation experiments. Thus in this paper we have explicitly shown an alternative +way of probing FIMP dark matter from the precise measurement of ∆Neff even when the +mother particles do not have sizable interactions with SM bath which is otherwise absent in +26 + +literature. Consequently, we are expecting some very exciting results from next generation +CMB experiments, like SPT-3G and CMB-IV which can shed some light on various dark +sector models, like the one discussed in this paper. +Acknowledgement +SJ and PG thanks D. Nanda for the helpful discussions during this project. The authors +would like to thank Abhijit Kumar Saha, Sougata Ganguly and Deep Ghosh for useful +discussion and comments. SJ is funded by CSIR, Government of India, under the NET JRF +fellowship scheme with Award file No. 09/080(1172)/2020-EMR-I. +Appendix A: Theoretical constraints +Stability +The scalar potential is bounded from below when the quartic couplings of the scalar +potential satisfy these co-positivity conditions[95]: +λH ≥ 0, +λφ ≥ 0, +λφH + 2 +� +λφλH ≥ 0 . +(A1) +The estimation of the lifetime of the desired the stable vacuum which essentially puts an +upper bound on the trilinear dark coupling as [96] +µφ/Mφ < 2 +� +λφ. +(A2) +Perturbative unitarity +The tree-level unitarity of the theory, coming from all possible 2 → 2 scattering ampli- +tudes will form the S matrix and constrain the quartic couplings of the scalar potential[97]. +The eigenvalues of the S matrix are bounded from above as[98]: +|λH| ≤ 4π, +|λφH| ≤ 8π, +|λφ| ≤ 4π, +|2λφ + 3λH ± +� +2λ2 +φH + (2λφ − 3λH)2| ≤ 8π . +(A3) +27 + +The quartic and Yukawa couplings of the interaction Lagrangian should also obey following +inequality equations to maintain perturbativity[99]: +|λH| ≲ 2π +3 , |λφ| ≲ π, |λφH| ≲ 4π, +and +|yφN| < +√ +4π . +(A4) +Appendix B: Constraint from Higgs invisible decay +The dark complex scalar, φ is very weakly coupled with SM Higgs via the Higgs portal +interaction. The late time decay of φ decides both the relic abundance of DM and the +contribution to the ∆Neff which require a light scalar mass of the order of MeV-few GeV +which is well below Mh/2 (will be discussed in the next section). In that case, Higgs can +decay to the dark scalar, φ, and contribute to Higgs’s invisible decay width. The Higgs +invisible decay width is given by +Γh→φφ∗ = (λφHv)2 +16πMh +� +1 − 4M 2 +φ +M 2 +h +, +(B1) +where Mh = 125.06 GeV and v = 246 GeV. The current analysis of the CMS collaboration +[93] at LHC puts a strong constraint on the Higgs invisible decay in the following form +BRinv = +Γinv +h +Γinv +h ++ ΓSM +h +< 11% , +(B2) +where ΓSM +h += 4 MeV. If Mh < 2Mφ then this decay is absent. In this work the Higgs invisible +constraint only applicable for scenario-II where we require relatively large λφH. +Appendix C: 3φ → 2φ +In our setup 3φ → 2φ number changing processes in dark sector occur through φ φ φ → +φ φ∗, φ φ∗ φ∗ → φ φ and their conjugate processes i.e. φ∗ φ∗ φ∗ → φ∗ φ, φ∗ φ∗ φ → φ∗ φ∗ +respectively. Some of these processes are mediated by φ only and the rest are mediated +by both φ and h. However, for light Mφ(≲ O(GeV)), h-mediated diagrams are heavily +suppressed due to heavy propagator suppression and small Higgs portal coupling, λφH. +Therefore for simplicity, one can ignore the Higgs-mediated diagrams. All the φ mediated +28 + +FIG. 11: Feynman diagrams for φ φ φ → φ φ∗ number changing processes. Note that for each +t-channel, there is an u-channel diagram. +FIG. 12: Feynman diagrams for φ φ∗ φ∗ → φ φ number changing processes. Note that for each +t-channel, there is an u-channel diagram. +Feynman diagrams for φ φ φ → φ φ∗ and φ φ∗ φ∗ → φ φ processes are shown in Fig.11 and +Fig.12 respectively. +The amplitude for φ φ φ → φ φ∗ number changing scattering processes is given by +Mφφφ→φφ∗ = M1 + Mt +2 + Mu +2 + Mt +3 + Mu +3 += +� +4µφλφ +� +s − M 2 +φ +� + +4µφλφ +� +t − M 2 +φ +� + +4µφλφ +� +u − M 2 +φ +� + +µ3 +φ +� +s − M 2 +φ +�� +t − M 2 +φ +� + +µ3 +φ +� +s − M 2 +φ +�� +u − M 2 +φ +� +� +. +(C1) +29 + +D +D力And the amplitude for φ φ∗ φ∗ → φ φ number changing scattering processes is given by +Mφ∗φ∗φ→φφ = M1 + Mt +2 + Mu +2 + Mt +3 + Mu +3 + M4 + Mt +5 + Mu +5 + +Mt +6 + Mu +6 += +� +4µφλφ +� +s − M 2 +φ +� + +µ3 +φ +� +t − M 2 +φ +�2 + +µ3 +φ +� +u − M 2 +φ +�2 + +4µφλφ +� +t − M 2 +φ +� + +4µφλφ +� +u − M 2 +φ +� + +4µφλφ +� +s − M 2 +φ +� ++ +4µφλφ +� +t − M 2 +φ +� + +4µφλφ +� +u − M 2 +φ +� + +µ3 +φ +� +t − M 2 +φ +�� +s − M 2 +φ +� + +µ3 +φ +� +u − M 2 +φ +�� +s − M 2 +φ +� +� +. +(C2) +The total thermal averaged cross section for 3φ → 2φ number changing processes can be +expressed using non-relativistic approximation as[100]: +⟨σv2⟩3φ→2φ = ⟨σv2⟩φφφ→φφ∗ + ⟨σv2⟩φφ∗φ∗→φφ +≈ +√ +5 +192πM 3 +φ +� +|Mφφφ→φφ∗|2 + |Mφ∗φ∗φ∗→φφ∗|2� ++ +√ +5 +192πM 3 +φ +� +|Mφφ∗φ∗→φφ|2 + |Mφ∗φφ→φ∗φ∗|2� += +√ +5 +192πM 3 +φ +� +2|Mφφφ→φφ∗|2 + 2|Mφφ∗φ∗→φφ|2� +, +(C3) +where |Mφφφ→φφ∗|2 = |Mφ∗φ∗φ∗→φφ∗|2 and |Mφφ∗φ∗→φφ|2 = |Mφ∗φφ→φ∗φ∗|2. +Appendix D: 2φ → 2 SM and φ SM → φ SM +There is another type of number-changing process between the dark sector, φ, and the +visible sector, SM where two dark scalar φ annihilates into two SM particles via h mediated +diagram. Note that our analysis mostly focuses on the light-dark scalar with mass up to a +few GeV. Therefore φ can only annihilate into light fermion pairs. The Feynman diagrams +of corresponding number-changing processes are shown in Fig.13. +The thermal averaged cross-section for 2φ → 2SM number changing process is given by: +⟨σv⟩2φ→2SM = +� +f +⟨σv⟩φφ∗→ff += +� +f +x +16TM 4 +φK2(x)2 +� ∞ +4M2 +φ +� +σv +� +φφ∗→ffK1 +�√s +T +� +s +� +s − 4M 2 +φ ds +(D1) +30 + +FIG. 13: Feynman Diagrams for φ φ∗ → ff where f stands for SM fermions excluding top quark. +where x = Mφ +T +and +� +σv +� +φφ∗→ff can be written as: +(σv)φφ→ff = +� +1 +4πs√s +Ncλ2 +φHm2 +f +(s − m2 +h)2 + m2 +hΓ2 +h +(s − 4m2 +f) +3 +2 +� +Θ(Mφ − mf). +(D2) +In the above expression Nc = 1 for leptons and Nc = 3 for quarks. +FIG. 14: Feynman Diagrams for φ f → φf where f stands for SM fermion. +The scattering between DM and SM, φ SM → φ SM is also important for our discussion +which is required for analysing the kinetic equilibrium of the DM in early universe. 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Paris-Saclay (2018), 1901.05822. +36 + diff --git a/q9FST4oBgHgl3EQfPDjc/content/tmp_files/load_file.txt b/q9FST4oBgHgl3EQfPDjc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5ec8c0bd2d683c020421cc0ff0795bfc12c56436 --- /dev/null +++ b/q9FST4oBgHgl3EQfPDjc/content/tmp_files/load_file.txt @@ -0,0 +1,1373 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf,len=1372 +page_content='CMB signature of non-thermal Dark Matter produced from self-interacting dark sector Dilip Kumar Ghosh,1, ∗ Purusottam Ghosh,1, † and Sk Jeesun1, ‡ 1School of Physical Sciences, Indian Association for the Cultivation of Science, 2A & 2B Raja S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Mullick Road, Kolkata 700032, India Abstract The basic idea of this work is to achieve the observed relic density of a non-thermal dark mat- ter(DM) and its connection with Cosmic Microwave Background (CMB) via additional relativistic degrees of freedom which are simultaneously generated during the period TBBN to TCMB from a long-lived dark sector particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' To realize this phenomena we minimally extend the type-I seesaw scenario with a Dirac fermion singlet(χ) and a complex scalar singlet (φ) which transform non- trivially under an unbroken symmetry Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' χ being the lightest particle in the dark sector acts as a stable dark matter candidate while the next to lightest state φ operates like a long lived dark scalar particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The initial density of φ can be thermally produced through either self-interacting number changing processes (3φ → 2φ) within dark sector or the standard annihilation to SM particles (2φ → 2 SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The late time (after neutrino decoupling) non-thermal decay of φ can produce dark matter in association with active neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The presence of extra relativistic neutrino degrees of freedom at the time of CMB can have a significant impact on ∆Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus the precise measure- ment of ∆Neff by current PLANCK 2018 collaboration and future experiments like SPT-3G and CMB-IV can indirectly probe this non-thermal dark matter scenario which is otherwise completely secluded due its tiny coupling with the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' ∗Electronic address: tpdkg@iacs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='in †Electronic address: spspg2655@iacs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='in ‡Electronic address: skjeesun48@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='com 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='13754v1 [hep-ph] 31 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' INTRODUCTION The standard model (SM) of particle physics has been extraordinarily victorious in ex- plaining properties of elementary particles of the universe and their interactions through strong, electromagnetic and weak forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The SM seems complete after the discovery of Higgs-like particle with mass Mh = 125 GeV at the Large Hadron Collider (LHC)[1, 2], which is responsible for mass generation mechanism through electroweak symmetry break- ing in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Inspite of the great triumph of the SM, several theoretical and experimental issues still persist, that demands physics beyond the framework of the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Based on numerous astrophysical and cosmological observations at a wide range of length scales, it is now well established fact that about 80% of total mass of the universe con- sists of Dark matter (DM)[3–6] with relic density (ΩDMh2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='120 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='001) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Another astonishing experimental evidence is the observation of tiny but non-zero neutrino masses (mν ≲ O(10−10) GeV) and neutrino flavour oscillations [7–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' To address these issues, vari- ous theoretical as well as phenomenological ideas have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The issue of neutrino masses and their mixing angles can be resolved by the Seesaw mechanisms [13–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' However, any direct experimental verification of these ideas are yet to be confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' While in the dark matter sector, weakly interacting massive particles (WIMP) [23–28] is the most popular and widely studied thermal DM candidate whose interaction strength with SM particles is of the order of electroweak interactions and via freeze-out mechanism it fits nicely the observed relic density of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Nevertheless, null measurements from various dark matter detec- tion experiments [29–41] severely restricts the WIMP freeze-out mechanism and forcing us to think if the standard WIMP paradigm is just waning or it is already deceased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' To bypass this deadlock, an alternative framework, coined as freeze-in mechanism has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In this framework DM is a feebly interacting massive particle (FIMP) whose interactions with SM plasma is too small ≲ O(10−10) to keep them in thermal bath [42–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Rather FIMPs are produced non-thermally either from decay or annihilation of bath particles in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The FIMP freezes in once the temperature of the universe becomes lower than the FIMP mass and produces DM relic abundance in the correct ball-park as observed today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Moreover, FIMPs having such a petite coupling with SM particles can easily ac- commodate various non-observational signature of DM in different detection experiments like Panda[29], XENON[30], LUX[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' However, some attempts have been made to test the 2 FIMP scenario indirectly using observational data from big bang nucleosynthesis (BBN) or cosmic microwave background (CMB) [48–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Furthermore, non-thermal production of DM from the decay of heavier dark sector particles have also been studied in literature [55–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Apart from FIMP, strongly interacting massive particle(SIMP) is another alternate paradigm to explain the DM abundance [59–61] as well as the structure formation of the universe[62–64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' SIMPs are produced thermally in the early universe by number changing processes within itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' SIMP scenario requires strong self interaction and very small anni- hilation rate to SM particles contrary to WIMPs to successfully satisfy the correct DM relic density [65–67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' On the other hand Cosmic Microwave Background (CMB) is an ideal probe of the physics in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The very precise measurement of anisotropies in the temperature of photons which dissociate from visible sector in the recombination phase of the thermal evolution of our universe, leads to the determination of the energy density in that particular era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' From this one can estimate the number of light species in the universe and in the massless limit this is provided by the relativistic degrees of freedom g∗ [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' On the other hand after neutrino decoupling, one recasts the number of light degrees of freedom associated with neutrino bath as Neff and in the SM it is roughly number of active neutrinos (Nν = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus any physics scenarios beyond the SM (BSM) with new light degrees of freedom with masses O (eV) or less can subscribe to Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We have very precise information of Neff from recent Planck 2018 [6], which suggests Neff at the time of CMB formation to be NCMB eff = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='33 at 95% confidence level (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='L), whereas in the SM, NSM eff = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The quantity Neff is parameterized as Neff ≡ (ρrad − ργ)/ρν, where, ργ, ρν, and ρrad denote the photon energy density, active neutrino energy density and total radiation energy density of the universe respectively [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The deviation from 3, the number of active neutrinos can be attributed to various non-trivial effects like non-instantaneous neutrino decoupling, finite temperature QED corrections to the electromagnetic plasma, flavour oscillations of neutrinos [70–73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Multiple upcoming experiments like SPT-3G[74], CMB-IV[75] are going to be extremely sensitive to the presence of any new radiation /light degrees of freedom and will put stringent bound on ∆Neff = Neff − NSM eff ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='06 at 95% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Various BSM scenarios that entail additional entropy injection to the neutrino sector can face a tough challange from the measurement of ∆Neff by both the present and future generation CMB experiments [76–81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' This precise measurement of ∆Neff has also non-trivial implications 3 on various new physics models that produce dark matter in associated with the injection of additional light degrees of freedom [82–85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In this work, we are interested in non thermal production of dark matter from heavier dark sector, where the dark sector may or may not have sizeable interaction with the SM bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' To realize this picture we extend the SM by one complex SM gauge singlet scalar (φ), one gauge singlet Dirac fermion (χ) and 3 right handed neutrinos (RHN)(N1,2,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The three RHNs are responsible for neutrino mass generation through well known Type-I seesaw mechanism[14, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' φ and χ are dark sector particles and an additional discrete Z3 symmetry has been imposed under which they transform non trivially while the rest of the particles transform trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In our analysis lightest dark sector particle χ can play the role of DM whereas the heavy dark sector particle (φ) is a long lived owing to its very small coupling (≲ 10−12), which will eventually allows φ → χν decay at temperature below neutrino decoupling temperature (∼ 1 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Non thermal decay of φ is the only source of DM(χ) production whereas φ freezes out thermally and gains non-zero number density via either of these two mechanisms : (i) the number changing self interactions (3φ → 2φ), (ii) annihilation to SM particles (2φ → 2SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In this work we emphasise on the first scenario where φ has strong self interactions but very weak interaction with the SM bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The implication of this particular scenario has been so far overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Through our detailed numerical analysis we will highlight the importance of this mechanism in both DM phenomenology and its footprint on CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For the shake of completeness of the analysis, we will also consider the second process as well to showcase region of parameter space where these two scenarios are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' It should be noted that in both cases φ particle maintain kinetic equilibrium with SM bath via the elastic scattering processes and share common temperature with SM bath contrary to studies that deal with secluded or decoupled dark sector scenarios [87, 88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' If the decay of φ is happening after neutrino decoupling then it will increase neutrino bath entropy and contribute to ∆Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' If the decay is completed before CMB we can trace the signature of DM from ∆Neff at the time of CMB and find some interesting correlation of freeze in DM and ∆Neff in our proposed set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The rest of the paper is structured as follows: In section II we introduce the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The possible dynamics of DM production have been discussed in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In section IV we discuss the light neutrino production from late time decay of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The outcome of 4 DM relic density together with the contribution to ∆Neff at CMB for both scenarios-I and II have been discussed in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Finally, we summarize our results in section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We show relevant theoretical constraints and limit from the SM Higgs invisible decay width in Appendix A and Appendix B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Feynman diagrams and corresponding thermal averaged cross-section for 3φ → 2φ and 2φ → 2SM processes are explicitly demonstrated in Appendix C and Appendix D respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' THE MODEL In order to explain DM production from dark sector and its cosmological imprints in CMB, we extend the SM by a complex scalar φ, one Dirac fermion χ and three neutral Majorana fermions, N1,2,3 which are singlet under the SM gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' An additional Z3 symmetry provides the stability of the lightest dark sector particle, under which the field φ and χ transform non-trivially i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' {φ, χ} → {ei 2π 3 φ, ei 2π 3 χ} while all the SM fields including N1,2,3 transform trivially i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' {N1,2,3, SM} → {N1,2,3, SM} 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The lightest dark state, χ acts as a stable DM candidate which is produced from the late time decay of the other dark sector particle, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The right handed neutrinos (RHN) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' N1,2,3 which do not transform under Z3, will be responsible for light neutrino mass via Type-I seesaw mechanism [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' All the BSM fields and their corresponding charge assignments under the extended SM Electroweak (EW) gauge group are tabulated in table-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' BSM Fields SU(2)L U(1)Y Z3 Dark scalar (DS) φ 1 0 ω(≡ ei 2π 3 ) DM χ 1 0 ω(≡ ei 2π 3 ) RHN N1,2,3 1 0 1 TABLE I: Charge assignment of BSM fields under the extended SM EW gauge group, GEW SM ⊗ Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 1 In general any ZN symmetry can serve similar kind of scenario with different self interacting number changing processes, m φ → 2 φ (m ≥ 3), as well as the standard annihilation to SM particles, 2φ → 2SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For example, Z2 will provide 4φ → 2φ interactions which are more phase space suppressed for Mφ ∼ O(MeV) compare to 3φ → 2φ interactions realised in Z3 symmetry [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 5 The Lagrangian of this model takes the following form : L = LK+Y SM − V (H) � �� � SM +LN + LDS + LDS−H + LDS−ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (1) Here, V (H) represents the SM Higgs potential which is given by V (H) = −µ2 H|H|2 + λH|H|4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (2) The BSM part encapsulate interactions of heavy RHN sector (LN), dark sector(LDS) as well as their connection with the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The interaction of heavy RHN sector is described by, LN = � i i ¯Niγµ∂µNi − � i,j 1 2MNij ¯ N c i Nj − � ℓ,j Yℓj ¯Lℓ ˜HNj + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (3) where i, j = 1, 2, 3 and ℓ = e, µ, τ are lepton flavour indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Lℓ = (νℓ ℓ)T are left handed the SM lepton doublet and H is the SM scalar doublet with ˜H = iσ2H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The second term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (3) is the Majorana mass term associated with N1,2,3 and the last term is the Dirac Yukawa interactions with N1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' After electroweak symmetry breaking (EWSB), the SM scalar doublet, H can be expressed in unitary gauge as H = � 0 h+v √ 2 �T where v = 246 GeV is vaccum expectation value (VEV) of SM Higgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Active neutrino masses can be generated via Type-I seesaw mechanism followed from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (3) as � mν � 3×3 ≈ � Y v/ √ 2 �� MN �−1� Y Tv/ √ 2 � and the mixing angle between active neutrino and RHN is then θmix ∼ � Y v/ √ 2 �� MN �−1 , where MN ≈ � MN � 3×3 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The dark sector of this model consists of a complex scalar (φ) and a Dirac fermion (χ) with similar transformation property under Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The lightest state behave as a stable DM particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The Lagrangian of the dark sector is described as follows: LBSM ⊃ LDS + LDS−H + LDS−ν = � |∂µφ|2 − µ2|φ|2 + i¯χγµ∂µχ − MDM ¯χχ − λφ|φ|4 − µφ 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (φ3 + φ∗3) − yφχχcχφ � + � − λφH|H|2|φ|2� + � − � i yφNi ¯χφNi + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' � , (4) where, i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In the above equation µ is the bare mass term of φ and MDM is the mass of dark fermion χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For simplicity, in this work we consider all parameters to be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In the dark scalar sector, we assume µ > 0 and λφ > 0 so that ⟨φ⟩ = 0 which implies unbroken Z3 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' After EWSB the physical mass of φ can be expressed as, M 2 φ = µ2 + λφH v2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (5) 6 The most important interaction as far as our analysis is concerned, is given by the Yukawa interaction involving the dark scalar (φ), the DM (χ) and SM neutrinos (ν): Lint DS−ν = y1χνφ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (6) This Lagrangian can be realized from the last term in braces in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (4) via small mixing angles(θmix) with RHNs(N1,2,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The effective Yukawa coupling, y1 can be understood as � i yφNiθi mix, where i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We choose the dark sector of our model parameters in such a manner that we always get χ as the lightest dark sector particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' This mass pattern and the underlying discrete symmetry ensure us that the Dirac fermion (χ) with mass MDM is the DM particle and φ with mass Mφ is the next to ligtest particle (NLP) in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The DM interacts with the SM bath only through φ via the Yukawa interaction shown in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus for a given mass hierarchy between φ and χ, the life-time of φ is determined by the strength of the Yukawa coupling y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For our analysis, we assume NLP (φ) to be a long-lived (τφ > τBBN) particle and for this to happen one requires a very tiny Yukawa coupling y1 ≲ 10−12 (for Mφ ∼ O(GeV)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The NLP φ can be thermally produced via the sizable Higgs-portal interaction or through number changing self interaction processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The production of the DM in the thermal bath through scattering process is highly suppressed because of it feeble coupling (y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' However, it can be produced non-thermally from the decay of long-lived φ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 1: Diagram of DM production with active neutrinos from NLP φ decay width of φ to DM and a light neutrino is given by, Γφ→χν = y2 1 Mφ 16 π � 1 − M 2 DM M 2 φ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (7) Besides this, there are two more production channels of the DM χ: (a) N1,2,3 → χφ and (b) φ → ¯χcχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The main aim of this work is to connect non-thermal DM and ∆Neff producing 7 Xfrom self-interacting dark sector (NLP) which is achievable via the decay φ → χν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' But the presence of those new channels (a &b) will dilute the effect of the late time decay of φ in ∆Neff and may even completely imperil our non-thermal dark matter scenario by thermalizing the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' To avoid DM production from RHNs we set MN1,2,3 ≫ TRH so that their number densities get Boltzmann suppressed(e−MN/T) [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Therefore for our discussion, we choose the following hierarchy MN1,2,3 ≫ TRH > Mφ > MDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (8) In order to get active neutrino mass of the order ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 eV, we require MN1,2,3 ∼ O(1010) GeV and θmix ∼ O(10−10) [89] and to satisfy the criteria of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='8 we set TRH = 103 GeV which is consistent with the bound obtained from BBN [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Following this argument and masses of relevant particles of this model, in the rest of our analysis we can safely ignore the production of DM from RHN decay in the computation of Yφ and Yχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Moreover to suppress the process (b) we consider yφχ ≪ y1, and this is necessary to exalt φ → χν decay so that φ can have the maximal contribution to ∆Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Interestingly, active neutrinos(ν) produced from the decay of NLP φ along with DM(χ) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 can have very intriguing consequences in the observation of CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We assume the value of Yukawa coupling (y1) such that φ → χν decay mostly happens be- tween neutrino decoupling temperature (T < 2 MeV) and CMB formation (T ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' This promptly opens up the possibility of probing the impact of extra neutrino production from CMB radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' And this can be achieved if y1 varies in the range (10−12 − 10−15) and for such a tiny coupling φ becomes a long-lived particle (τφ > τBBN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus the afore- mentioned supplementary active neutrino (ν) injection in our proposed scenario increases neutrino sector entropy and which in turn contribute significantly to additional neutrino degrees of freedom or ∆Neff which is very precisely measured at the time of CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus any experimental observation on ∆Neff can have very intriguing impact on the dynamics of dark scalar φ which in turn can influence the dark matter (χ) abundance via φ → χν decay process, thus affecting two disjoint (FIMP dark matter & Neff) sectors simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' To explore this phenomenology, we perform a detailed numerical scan over model parameters to show that the precise measurement of ∆Neff at CMB can indeed restrict certain region of parameter space of non-thermal DM production which is otherwise remains elusive to visible sector due to extremely tiny strength of interactions involved in such non-thermal DM production process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 8 While doing our numerical analysis, we use the following model parameters: {MDM, Mφ, λφH, λφ, µφ, y1}, (9) Here, the Higgs portal coupling λφH which decides the interaction between φ and SM, plays a significant role in deciding φ’s number density through 2φ → 2 SM annihilation and also in (φ SM → φ SM) elastic scattering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' On the other hand, the scalar sector parameters λφ and µφ decide the self interactions of φ which is relevant for the number changing processes like 3φ → 2φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' And finally the effective Yukawa coupling, y1 dictataes both DM abundance and additional contribution to Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' DYNAMICS OF DARK SECTOR In this section, we discuss the dynamics of the dark sector that leads to the early time production of the heavy NLP dark scalar (φ) followed by the late time non-thermal produc- tion of DM (χ) from the decay of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The number density of DM will be generated at some later epoch (after the neutrino decoupling temperature) of the Universe via the following two steps : Step I: thermal production of heavy dark scalar φ at the early time of Universe (τ < τBBN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Step II: non-thermal production of DM, χ from the late time decay of φ (τ > τBBN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 104 102 1 10-2 10-4 10-6 10-8 10-10 10-12 10-14 10-10 10-6 10-4 10-2 T(GeV) Yi Yϕ eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 3ϕ→2ϕ / 2ϕ→2SM ϕ → χ(DM) + ν Yϕ YDM (Ωh2≃0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='12) BBN CMB 10-4 10-6 10-8 10-10 10-12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='20 T(GeV) ΔNeff BBN CMB Planck 2018(1σ)CMB FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 2: A cartoon diagram of DM production(left) and the impact in ∆Neff at the time of CMB(right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 9 A cartoon of our proposed setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In the left panel, we show the variation of co-moving density as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The purple and red solid lines correspond to the thermal production of φ (Step I) and the non thermal production of DM (χ) (Step II) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We also show two important temperatures, namely, the BBN and CMB that play crucial role in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Active neutrinos produced in the aforementioned decay of φ make substantial contributions to Neff, which can attract severe constraints from various observational limits on ∆Neff, as shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The gray rectangular band is excluded by the Planck 2018 data at 1σ [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Having this broad picture in mind we now provide details of the thermal production of NLP followed by non-thermal production of DM in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Step-I: Thermal production of φ We consider a scenario in the early universe, when the interaction rate (Γint φ ) of the NLP (φ) dominates over the expansion rate (H) of the Universe, (Γint φ >> H) so that φ remains in thermal and chemical equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' As the temperature of the universe cools down, the interaction rate of φ falls below the expansion rate of universe (Γint φ < H), thus the system departs from thermal equilibrium and the number density of φ freezes out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The number density of φ is mainly provided by the following two types of number changing processes: (i) 3φ ↔ 2φ via φ self interactions (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='3(a)) and (ii) 2φ ↔ 2 SM via the SM Higgs portal interactions (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' As a result of these two number changing processes, the NLP (φ) keeps its chemical equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' On the other hand, the kinetic equilibrium is maintained between φ and the SM bath via elastic scatterings, generically expressed as φ SM ↔ φ SM which help φ to keep same temperature with SM bath till freeze out takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The complete dynamics of thermal production of φ can be described by the following Boltz- mann equation(BEQ): dYφ dx = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='116 g2 s √gρ M 4 φ x5 Mpl � σv2� 3φ→2φ (Y 3 φ − Y 2 φ Y eq φ ) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='264 gs √gρ Mφ x2 Mpl ⟨σv⟩2φ→2SM (Y 2 φ − Y eq φ 2) − � 45 4π3 ⟨Γφ→χν⟩ x M 2 φ Mpl √gρ Yφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (10) 10 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 3: A cartoon of number changing process of φ: (a) three φ annihilate to two φ (3φ → 2φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' and (b) two φ annihilate to two SM particles (2φ → 2SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Let us first describe various notations used in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Yφ(= nφ s ) is the co-moving number density of φ where s is the entropy density and x is the dimensionless parameter defined as x = Mφ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Y eq φ is the equilibrium co-moving number density of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' gs(x) and gρ(x) are the ef- fective relativistic degrees of freedom associated with entropy density and the energy density respectively and finally Mpl is the Planck mass(Mpl = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='22 × 1019GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The thermal aver- aged cross-section of 2φ → 2 SM process is denoted by ⟨σv⟩2φ→2SM and for self-interacting number changing process (3φ → 2φ), it is defined as ⟨σv2⟩3φ→2φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The first two terms in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (10) lead to non zero density of φ via thermal freeze-out mechanism and it occurs at x = xtot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' F , where tot in the superscript implies that both number changing processes i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 3φ → 2φ and 2φ → 2SM are involved in φ freeze-out process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The last term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (10) pro- vides the late time (after BBN) decay of φ into DM (χ) and SM neutrinos, resulting the dilution of number density of φ into χ and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (10) it is clear that two number changing processes of NLP (φ) as discussed above are present to keep φ in the thermal bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' However, depending upon the mass and couplings of NLP, it can be shown very easily that one of those two number changing processes is infact sufficient for the freeze-out and the final yield of NLP (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' To justify our argument quantitatively we define the interaction rate of 3φ → 2φ process as: Γ3φ→2φ = n2 φ⟨σv2⟩3φ→2φ and of 2φ → 2SM as: Γ2φ→2SM = nφ⟨σv⟩2φ→2SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In addition to these number changing processes, φ SM → φ SM number preserving scattering process is also present to keep φ in kinetic equilibrium with SM bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The interaction rate of this process is defined as: Γ[φ SM→φ SM] = nSM⟨σv⟩[φ SM→φ SM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Depending on the relative interaction strength between 11 0SM SMtwo number changing processes of φ, we are interested in the following two production modes of φ: Scenario I : Γ[φ SM→φ SM] > Γ3φ→2φ ≫ Γ2φ→2SM , Scenario II : Γ[φ SM→φ SM] > Γ2φ→2SM ≫ Γ3φ→2φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In the above hierarchy of scattering processes, Γ[φ SM→φ SM] plays a decisive part in main- taining kinetic equilibrium of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' During the freeze-out of φ through processes like: nφ → 2φ, (for n > 2) the rest mass energy of initial state particles can significantly enhance the ki- netic energy of final state particles, which in turn can heat up the dark sector [66], leading to an imbalance between the dark sector temperature (Tφ) and SM bath temperature (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus, in general, to take into account this temperature imbalance one should consider a new parameter (Tφ) in the evolution equation of NLP number density (Yφ) [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' However, in our study we can avoid this paradigm by considering kinetic equilibrium between φ and SM bath, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' by taking Tφ = T at least upto the temperature at which φ freezes out from the thermal bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' And to achieve this, Γ[φ SM→φ SM] must be larger than interaction rate of the other processes as well as the expansion rate H of the universe (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='e Γ[φ SM→φ SM]|xF ≳ H(xF)) [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Most importantly this condition must be satisfied in both Scenario I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The rel- evant Feynman diagrams and thermal averaged cross-sections for 3φ → 2φ, 2φ → 2SM and φ SM → φ SM processes are shown in Appendices C and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Scenario I: In this scenario we consider the interaction rate of 3φ → 2φ number changing process (Γ3φ→2φ) is significantly higher than 2φ → 2SM process (Γ2φ→2 SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus 3φ → 2φ process successfully keeps φ in thermal bath for longer duration in comparison to the process 2φ → 2 SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Hence freeze-out of φ is mainly governed by the 3φ → 2φ process and it occurs at x = x3φ→2φ F ≈ xtot F > x2φ→2SM F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Here x3φ→2φ F (x2φ→2SM F ) signifies the inverse freeze out temperature of φ when only 3φ → 2φ (2φ → 2SM) is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In our model, the interaction rate of 3φ → 2φ (2φ → 2SM) process depends on the couplings λφ, µφ/Mφ (λφH ) and mass Mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' To demonstrate the dynamics (where Γ3φ→2φ ≫ Γ2φ→2 SM), we show the variation of the co-moving number den- sity Yφ as a function of x(= Mφ/T) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='4(a) for a sample Benchmark point: {Mφ, µφ/Mφ, λφH, λφ} = {20 GeV, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1, 10−2, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The black solid line corresponds 12 3ϕ→2ϕ 2ϕ→2SM 3ϕ→2ϕ+2ϕ→2SM 5 10 50 100 10-12 10-10 10-8 10-6 10-4 10-2 x Y mϕ=20GeV,λϕ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='0,λϕH=10-2 (a) λφ H=10 4 λφ H=10-2 λφ 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 1 Mφ (GeV) 10 20 30 40 50 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 4: (a)Thermal freeze-out of φ governed by 3φ → 2φ and (b)Parameter space for scenario-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For other details see the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' to the equilibrium co-moving density of φ (Y eq φ ) and the blue dashed line corresponds to the co-moving number density of φ considering contributions from both the num- ber changing processes: 3φ → 2φ and 2φ → 2 SM in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The brown solid line (red dotted line) depicts the variation of number density of φ when only 3φ → 2φ (2φ → 2 SM) process is present in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The relative contribution of these two processes in the evolution of Yφ is clearly seen in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' If we consider only the sub-dominant 2φ → 2 SM process, φ freezes-out earlier (red dotted line) due to small Γ2φ→2 SM, whereas, the dominant 3φ → 2φ process maintains φ in thermal bath for longer duration (brown solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus the freeze-out abundance of φ (blue dashed line) is governed mainly by the dominant 3φ → 2φ process due to larger Γ3φ→2φ for our choice of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Therefore, we can safely ignore the second term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (10) and the modified BEQ takes the following form: dYφ dx = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='116 g2 s √gρ M 4 φ x5 Mpl � σv2� 3φ→2φ (Y 3 φ − Y 2 φ Y eq φ ) − � 45 4π3 ⟨Γφ→χν⟩ x M 2 φ Mpl √gρ Yφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (11) Based on the above argument we can identify the parameter space for scenario-I satis- fying the criteria: x3φ→2φ F > x2φ→2SM F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='4(b) we display the parameter space for this scenario in Mφ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' λφ plane with µφ/Mφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 for two different values λφH = {10−2, 10−4} depicted by the blue and red shaded region respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The 13 criteria for scenario-I holds only for the region left to individual lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' With an in- crease in Mφ ,Γ3φ→2φ becomes more mass suppressed compared to Γ2φ→2SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Hence for fixed values of λφ, and λφH with increasing Mφ, Γ3φ→2φ falls below Γ2φ→2SM and scenario-I doesn’t hold anymore for the parameter space right to the colored lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' With an increase in λφH, Γ2φ→2SM increases and eventually Γ3φ→2φ falls below Γ2φ→2SM even with lower Mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For that reason we see the shaded region moves toward lower Mφ (towards left) with an increase in λφH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The regions right to the colored lines demand a different treatment which will be discussed shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Before we conclude this part of our analysis, it is worth noting that the present dark sector dynamics also allows 4φ → 2φ number changing process involving the same λφ coupling that is responsible for 3φ → 2φ process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Inspite of the same interaction strength (λφ), 4φ → 2φ process is more phase space suppressed compared to that of 3φ → 2φ, hence, we neglect it in our numerical calculation of Yφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Scenario II: In this picture we consider Γ2φ→2SM ≫ Γ3φ→2φ, which is contrary to the previous scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In this case freeze-out of φ is dictated by 2φ → 2SM annihilation process that keeps φ in thermal bath for a longer period compared to 3φ → 2φ process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Hence the freeze-out of φ occurs at x = xtot F ≈ x2φ→2SM F > x3φ→2φ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 3ϕ→2ϕ 2ϕ→2SM 3ϕ→2ϕ+2ϕ→2SM 5 10 50 100 10-12 10-10 10-8 10-6 10-4 10-2 x Y mϕ=20GeV,λϕ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='01,λϕH=10-2 (a) λφ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 λφ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='01 λφ H 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 1 Mφ (GeV) 10 20 30 40 50 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 5: (a)Thermal freeze-out of φ governed by 2φ → 2SM and (b)Parameter space for scenario-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For other details see the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='5(a) we report the evolution of Yφ as a function of x = Mφ T for λφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='01 keeping other parameters same as in Scenario-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' From this figure it is evident that Yφ is entirely 14 decided by 2φ → 2SM number changing processes contrary to the previous scenario where 3φ → 2φ process was controlling the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (10) can be simplified by neglecting the sub-dominant 3φ → 2φ process: dYφ dx = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='264 gs √gρ Mφ x2 Mpl � σv2� 2φ→2SM (Y 2 φ − (Y eq φ )2) − � 45 4π3 ⟨Γφ→χν⟩ x M 2 φ Mpl √gρ Yφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (12) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='5(b) we display parameter space for this scenario in Mφ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' λφH plane with µφ/Mφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 for two different values λφ = (10−1 & 10−2) depicted by the blue and red shaded region respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For the same reason discussed in the context of scenario-I, in this case also Γ3φ→2φ decreases with decrease in λφ and finally falls below Γ2φ→2SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' And this phenomena is true even for lower Mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For this reason here also we see that the shaded region shifts towards lower Mφ (left) with decrease in λφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In summary the main observation of this whole subsection are the following: Scenario I : Γ3φ→2φ ≫ Γ2φ→2SM =⇒ xtot F ≈ x3φ→2φ F > x2φ→2SM F , Scenario II : Γ3φ→2φ ≪ Γ2φ→2SM =⇒ xtot F ≈ x2φ→2SM F > x3φ→2φ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (13) It is worth mentioning that scenario-II is more common and has already been studied in different literature [57, 84], where mother particles are considered to have sizable annihilation cross-section with SM bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In this work our main focus is on scenario-I, although for the sake of completeness of the analysis we also discuss scenario-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Step-II: Non thermal DM production Following our previous discussion we now focus on the non-thermal production of DM (χ) from the dilution of φ density described by the last term in the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='S of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We solve the following Boltzmann equation to get the evolution of DM(χ) abundance, dYχ dx = � 45 4π3 ⟨Γ⟩φ→χν x M 2 sc Mpl √gρ Yφ, (14) where, Yχ is the co-moving number density of DM χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In general the solution of Yφ comes from the BEQ in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (11) for scenario-I and in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (12) for scenario-II respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In the calculation of Γ(φ → χν) we consider the Yukawa coupling y1 in the range (∼ 10−12 −10−15) 15 so that the decay of φ → χ+ν happens in post BBN and pre CMB era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' At this stage, we find it worth discussing one subtle issue regarding the thermal averaged decay width ⟨Γ⟩φ→χν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' As we have pointed out before, that at the time when φ freezes-out, it maintains the same temperature as the SM bath via the elastic scattering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' However, this may not be true at the time of decay(< TBBN) if Γ[φ SM→φ SM] < H at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' This results the dark sector to acquire a different temperature T ′ (̸= T) than the thermal bath and this must be evaluated in order to get ⟨Γ⟩φ→χν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In this work, as we are studying the dark sector dynamics at low temperature (T ′ ≪ Mφ), and in this limit the thermally averaged decay width can simply be approximated as ⟨Γ⟩φ→χν (T ′) ≈ Γφ→χν [42, 91], thus reducing the complication of tracking temperature dependence of the evolution of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' After solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (14) we get the complete picture of DM production as shown by red solid line in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' LIGHT NEUTRINO PRODUCTION BEFORE CMB Now we discuss the production of supplementary light neutrinos from the late time decay of φ and the relevant mechanism of verifying those light degrees of freedom at CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' As revealed earlier, neutrinos that are produced after neutrino decoupling (T ≲ 2 MeV) would inject entropy in the neutrino bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' At the time of CMB, the number of relativistic neutrino degrees of freedom is expressed as, N CMB eff = 8 7 �11 4 �4/3 ρSM ν ργ ����� T=TCMB , (15) where, ρSM ν = 3 × 2 × 7 8 × π2 30(T SM ν )4 and ργ = 2 × π2 30T 4 are energy densities of neutrino and photon respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Due to the extra neutrino injection from the non-thermal decay of φ, the energy density of the neutrino bath increases to ρ′ ν (ρ′ ν > ρSM ν ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In this case, the relativistic neutrino degrees of freedom(N ′ eff) also differs from the prediction of SM at the time of CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We parameterise this deviation at the time of CMB in the following manner, ∆Neff = � ρ′ ν ρSM ν − 1 � NSM eff ����� T=TCMB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (16) We now solve the following Boltzmann equation to estimate the evolution of ρ′ ν with tem- perature, dρ′ ν dx = −4 β ρ′ ν x + 1 xH(x) ⟨EΓ⟩φ→χν Yφ s , (17) 16 where the term β indicates the variation of gs(T) with T and is defined as β(T) = 1 + 1 3 T gs(T) d gs(T) dT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (18) where, x is the dimensionless variable as mentioned earlier in context of equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Yφ is the co-moving number density of φ which is computed by solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (11) or eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (12) depending on the scenario we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' gs(x) is the number of effective degrees of freedom related to the entropy density and s is the co-moving entropy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The term ⟨EΓ⟩φ→χν in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (17) is the most crucial ingredient in this analysis which represents the thermal averaged energy density transferred to neutrino sector and is defined as ⟨EΓ⟩φ→χν = |M|2 φ→χν 32π � M 2 φ − M 2 DM � M 2 φ � 1 − M 2 DM M 2 φ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The first term in the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='S of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (17) is responsible for the dilution of ρ′ ν due to expansion of the universe while the second term decides the evolution of augmented contribution to ρ′ ν from φ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The evolution of ρSM ν after the decoupling of neutrinos is governed by only the expansion effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus in the absence of any new source ρSM ν can be computed by setting the second term of the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='S of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (17) equal to zero and considering only the dilution of energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' RELIC DENSITY AND ∆Neff So far we have built up the basic framework of the underlying dynamics of dark sector particles (φ and χ) that provided freeze-in DM as well as yielded extra active light neutrino that with its possible footprints in ∆Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In this section, we perform an exhaustive nu- merical analysis of Scenario-I and Scenario-II to quantitatively estimate phenomenological consequences of the late-time decay of φ in the light of current and future measurements of ∆Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For this we first scrutiny the dependence of DM relic density and the ∆Neff on various model parameters as elaborated in sec-III and sec-IV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Relic density To calculate the DM relic density, we numerically solve eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (14) along with either eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (11) (for scenario-I) or eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (12) (for scenario-II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The solution of the coupled BEQs for each sce- nario yields Yφ and Yχ as a function of x(= Mφ/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Using the co-moving density of DM, Yχ 17 at x → ∞, one finds out DM relic density as [23]: Ωχh2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='755 × 108 × �MDM GeV � × Y today χ , (19) where Y today χ = Yχ(x → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The precise determination of Y today χ is highly model dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In the following two sub-sections we pin down Y today χ for scenario-I and II and corresponding relic densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Scenario-I: As shown before, densities of φ and χ for the scenario-I are mainly driven by 3φ → 2φ number changing process in the dark scalar sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Based on this number changing process, we calculate the co-moving abundances of φ and χ and show their evolution with x(= Mφ/T) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The solid, dashed and dotted lines signify Yφ, Y eq φ and Yχ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' It can be seen from these figures that the late-time decay of φ (solid lines) produces the abundance of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' As the φ → χ + ν decay proceeds, the number density of φ slowly changes into χ number density and eventually at the end of the decay, the density of φ completely dilutes to χ number density (Yχ ≡ Yφ at τ ≫ τφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' One can easily understand this from the fact that φ → χν decay is the only possible decay mode of the NLP (φ) [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus, in the generation of Yχ from Yφ, the magnitude of the Yukawa coupling (y1 > 0) has hardly any role to play, except for setting the lifetime of φ and this provides Yφ � x3φ→2φ F � ≃ Y today χ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Therefore the relic density of DM given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (19) turns out to be Ωχh2 ∝ MDM×Yφ � x3φ→2φ F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Consequently in order to get fixed Ωχh2, any increase in MDM demands a decrease in Yφ � x3φ→2φ F � and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We have pointed out before that φ → χ ν decay to happen between BBN and CMB the value of y1 should lie in the range : {10−12 − 10−15}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' As a sample representative value we set y1 = 10−12 throughout our numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' To show the evolution of Yφ and Yχ with temperature, we fix µφ/Mφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 and MDM = 400 keV for both plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We consider λφH = 10−4 to realize Scenario-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='6(a) we present evolution of densities Yφ (solid line) and Yχ (dotted line) as a function of dimension less parameter x(= Mφ/T) for two different Mφ and a fixed self- interaction coupling λφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The dynamics of dark sector particles for Mφ = 1 GeV and Mφ = 10 GeV are depicted by red and blue colors respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' With the increase of Mφ, ⟨σv2⟩3φ→2φ encounters phase space and propagator suppression which is also understood from the expression given in appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' As Yφ goes like Yφ ∝ 1/ ⟨σv2⟩3φ→2φ (using analytical solution [61]), with the smaller ⟨σv2⟩3φ→2φ, the thermal freeze-out of φ happens at earlier time with higher abundance Yφ and eventually this Yφ is transfered to the Yχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' As a result Yχ 18 yΦ =10-12 MΦ =10 GeV MΦ=1 GeV Y 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 x 10−2 10−1 100 101 102 103 104 105 (a) yΦ =10-12 MΦ=1 GeV, MN1= 420 KeV YΦ eq λΦ =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='0 λΦ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 Y 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 x 10−2 10−1 100 101 102 103 104 105 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 6: Evolution of co-moving abundances of φ (solid line) and DM(χ) (dotted line) with x(≡ 1/T) (T in GeV) for scenario-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In (a)for a fixed λφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='0 with two different values of Mφ and in (b) for a fixed Mφ = 1 GeV with two different values of λφ are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The Higgs portal coupling is considered here to small, λφH = 10−4 in order to realise the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The other parameters like y1 = 10−12, MDM = 400 keV and µφ/Mφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 are kept same for both plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' is also higher for higher Mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' This feature is portrayed in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='6(a) where higher(lower) value of Mφ leads to the higher(lower) abundance Yχ represented by the red(blue) dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' To study the role of dark scalar self-coupling,λφ on DM abundance, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='6(b) we show the variation in Yχ for two different values of λφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 (red line) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='0 (blue line) keeping Mφ = 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' It is obvious, that as the value of λφ increases, the thermal averaged cross- section ⟨σv2⟩3φ→2φ also increases which eventually reduces the abundance Yφ and finally this reduced Yφ generates lower Yχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' This is elucidated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='6(b), where a higher(lower) value of λφ gives a lower(higher) Yχ as it is shown by the blue(red) dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The other parameter µφ with mass dimension is also responsible for 3φ → 2φ processes as ⟨σv2⟩3φ→2φ ∝ (µφ/Mφ)2 (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' With an increase in the ratio µφ/Mφ, the cross- section will enhance leading to a decrease in Yφ as well as Yχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For simplicity, we consider the ratio µφ/Mφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 throughout our analysis and is consistent with the theoretical upper bound on µφ coming from stable vacuum as discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' After describing the dependence of relic abundance on different model parameters, we now present the allowed region of dark sector parameter space from DM observed density, (ΩDMh2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='120±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='001) given by PLANCK [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We perform a numerical scan on the model 19 MDM(MeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 1 10 Scenario-II λφ 10−2 10−1 100 Mφ(GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 1 10 100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 7: DM relic density satisfied points in Mφ − λφ plane for scenario-I with µφ/Mφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1, λφH = 10−4 and y1 = 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The color gradient indicates the range of MDM satisfying the correct relic density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The shaded region corresponds to the parameter space where scenario-II is dominating over scenario-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' White regions are just computational artifact associated with the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' parameters in the following range Mφ : {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 − 100 GeV}, λφ : {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='001 − 1} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (20) to calculate Ωχh2 using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='(19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We keep other parameters fixed as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' and to allow the on shell decay φ → χ + ν we set Mφ > MDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Our scan result is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='7, where we show points satisfying relic density con- straints in λφ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Mφ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The grey shaded region corresponds to the parameter space where scenario-II dominates which demands a different analysis and will be discussed shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The color gradient in the above figure represents DM mass range varying from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 MeV to 10 MeV set by the observed relic density constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' One can see from this figure that the higher value of λφ prefers to higher value of MDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' As explained in the context of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='6, with the increase of λφ, Yχ decreases and hence higher value of MDM is required in order to satisfy the correct relic density as depicted in above Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For simplicity we restrict our scan within the specified range of λφ mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' However one can make the scan even 20 for higher value of λφ ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='0 within the perturbativity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For those values of λφ even heavier DM mass,(10 MeV ≲ MDM < Mφ) will be allowed by the relic density constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Scenario-II: We shall now move to the second scenario where the density of φ is mainly driven by 2φ → 2 SM number changing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Following our earlier discussion we know that the density of φ converts into the density of DM � Yφ � x2φ→2SM F � ≃ Y today χ � via the late time decay of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Therefore the relic density of DM given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (19) becomes Ωχh2 ∝ MDM × Yφ � x2φ→2SM F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Similar to previous scenario Yφ(x2φ→2SM F ), decreases with increase in MDM in order to get fixed density and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' One can also analytically express the yield of φ at freeze-out as: Yφ ∝ 1/⟨σv⟩2φ→2SM [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For heavier Mφ, more annihilation processes of φ to SM pairs kinematically open up and enhance the cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus ⟨σv⟩2φ→2SM can be expressed as � X=SM⟨σv⟩φφ→XX Θ � Mφ − MX � where Θ is the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For a fixed Mφ, Yφ as well as Yχ decreases as one increases λφH since ⟨σv⟩2φ→2SM ∝ λ2 φH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' However, here the dependence of Yχ on Mφ is contrary to that of scenario-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In this case with the increase in Mφ, the annihilation cross-section, ⟨σv⟩2φ→SM also increases for the aforementioned reasons and thus resulting a decrease in Yφ as well as Yχ [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Now in order to find a consistent parameter space satisfying observed relic density mea- sured by PLANCK[6], we perform a numerical scan of the relevant parameters for scenario-II in the following range: Mφ : {10 − 100 GeV}, λφH : {10−3 − 10−1} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (21) whereas the other parameters are kept fixed as µφ/Mφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1, λφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 and y1 = 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The choices of dark sector parameters in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (21) ensure that Γ2φ→2SM ≫ Γ3φ→2φ which is required for the scenario-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We consider Mφ up to 100 GeV, beyond that Yφ is more suppressed resulting in a negligible contribution to ∆Neff which will be discussed in due course of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='8 we plot correct relic density satisfied points in the Mφ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' λφH plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The variation of color gradient represents the variation of MDM considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The correct relic density constraint sets the DM mass in the range MDM: ∼ {10 MeV − 10 GeV} for our chosen parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The gray shaded region on lower left corner of the above figure represents the region where scenario-II does not work paving way to Scenario-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' With increase in λφ(> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1), Scenario-I will start to dominate over scenario-II even with higher value of Mφ and the shaded region will move towards right accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For Mφ < mh/2 , h → φφ∗ 21 MDM(MeV) 10 100 1000 104 Br(h→ invisible)>11%) Scenario-I λφ H 10−3 10−2 10−1 Mφ(GeV) 20 40 60 80 100 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 8: DM relic density satisfied points for scenario-II are shown in the λφH vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Mφ plane with λφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1, µφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1Mφ, y1 = 10−12 and the color gradient represents the variation in MDM satisfying the correct relic density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The gray shaded region corresponds to the parameter space where scenario-I is dominating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' decay opens up and contributes to the SM Higgs invisible decay width (Γinv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' h ) which is very precisely measured by CMS [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The bound from Γinv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' h (discussed in Appendix B) excludes a significant part of the parameter space as shown by the light cyan region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' As understood from the figure, scenario-II works in the higher range of Mφ and the moderate values λφH leading to lower Yχ as discussed earlier in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The 2φ → 2SM annihilation cross-section near Higgs pole, Mφ ∼ mh/2, causes further suppression in Yχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For Mφ > MW, more final states open up resulting in even larger ⟨σv⟩2φ→2SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus to satisfy the observed DM density one has to reduce λφH in that region as shown in top right corner (white area) of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Therefore the scenario-II allows higher DM mass to satisfy the correct relic density upto few GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 22 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Contribution to ∆Neff at CMB In earlier sections, we have established that our main thrust of this whole exercise is to calculate contributions to ∆Neff by extra active neutrinos produced in association with FIMP like DM from the late time decay of a self interacting dark scalar φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Simultaneously, we have also emphasized the possibility of correlating the dark matter mass with the measured value of ∆Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus, any precise determination of ∆Neff would provide an indirect probe of the dynamics of dark sector ivolving a strongly self interactiong particle φ as well as FIMP like DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Based on our discourse in sec-IV we will now investigate dependence of dark sector model parameters in ∆Neff which is completely determined by the ratio ρ′ ν/ρSM ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Mφ=1 GeV Mφ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 GeV Δ Neff 10−5 10−4 10−3 10−2 10−1 100 101 T(GeV) 10−7 10−6 10−5 10−4 10−3 (a) λφ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 λφ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='0 Δ Neff 10−2 10−1 100 101 T(MeV) 10−4 10−3 10−2 10−1 100 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 9: Evolution of ∆Neff with temperature(T) for two different values of Mφ keeping λφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='0 fixed (a) and for two different values of λφ keeping Mφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 GeV fixed in scenario-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Other parameters are kept fix as y1 = 10−12 and MDM = 400 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='9 we show the evolution of ∆Neff with temperature T for different set of model parameters as shown in the figure caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We first numerically evaluate ρ′ ν/ρSM ν by solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (19) along with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (11) and then plug it into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (16) to estimate ∆Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' From both the figures Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='9(a) and 9(b) we notice that at high T, the ∆Neff is almost negligible because the entropy injection to neutrino bath is very small during the earlier epoch of φ → χν decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' With the decrease in temperature, φ freezes out from the thermal bath and decays into χ + ν after BBN, generating a new source of active neutrinos that inject extra energy density to neutrino bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' This added neutrino density causes continuous growth of ∆Neff with lowering of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' With further decrease in the temperature, at some point φ decay is completed and any auxiliary neutrino production also stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus no more 23 supplementary energy transfer to neutrino bath takes place and the ratio ρ′ ν/ρSM ν attains its maximum possible value at that temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' After that both ρ′ ν and ρSM ν dilutes in the same fashion with further decrease in temperature, resulting in a fixed ratio ρ′ ν/ρSM ν which corresponds to a constant value of ∆Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Since ρ′ ν ∝ Yφ (following eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (17)), higher value of Yφ leads to higher energy transfer to neutrino bath resulting in larger ∆Neff and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We plot the evolution of ∆Neff in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='9(a) for Mφ = 1 GeV (red line) and Mφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 GeV (blue line) keeping λφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='0 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='9(b) we show the similar plot as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='9(a) but this time for a fixed Mφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 GeV and taking two values of λφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='0 (red line) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' While generating these two plots, we fix MDM = 400 MeV, and y1 = 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The behavior of ∆Neff with the model parameters (Mφ, λφ and µφ/Mφ) is same as of Yφ as discussed earlier for scenario-I (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='6(a) and 6(b)) and the same dependence is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='9(a) and 9(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The effect of λφH on ∆Neff in Scenario-II is similar to Scenario-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Following our previous argument, for any increase in the value of Higgs portal coupling λφH, φ number density Yφ decreases and that leads to a diminished contribution of active neutrinos in ∆Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' However, Mφ dependence of ∆Neff shows opposite behaviour in Scenario-II than Scenario-I, here, ∆Neff decreases with an increase in Mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The reason for this contrary nature follows the same argument as we revealed in the context of relic density calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' For heavier Mφ, due to enhanced phase space one gets larger ⟨σv⟩2φ→2SM that leads to lower Yφ and finally lower ∆Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Hence the energy transferred to neutrino sector is too less to contribute significantly in ∆Neff and the ∆Neff for scenario-II will be far below the sensitivity of the current and future generation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In this paper we do not display the explicit parameter dependence in ∆Neff in scenario-II, however similar study could be found in [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Finally, we calculate the ∆Neff for different values of the model parameters in scenario-I and displayed our findings in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We present ∆Neff as a function of Mφ and the color gradient represents the range of MDM allowed by observed DM relic density [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In the figure, we show different existing exclusion bounds as well as future sensitivities on ∆Neff depicted by different coloured patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We notice that a decrease in MDM yields a increase in ∆Neff also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' This is easily understood as lower value of MDM requires higher value of Yχ to satisfy the observed relic density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' As we analyzed earlier, Yχ is governed by Yφ and higher value of Yχ corresponds to higher value of Yφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus for a higher value of Yφ, more energy gets transferred to the neutrino sector, leading to the higher value of ∆Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We also notice that 24 MDM(MeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 1 10 CMB - S4 (2σ) SPT - 3G (1σ) Planck 2018 (2σ) Planck 2018 (1σ) Δ Neff 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='00 Mφ(GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1 1 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 10: Variation of ∆Neff with Mφ for µφ/Mφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='1, λφH = 10−4 and y1 = 10−12 where the color gradient represents the range of DM mass in scenario-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The current 1σ and 2σ upper limits on ∆Neff from PLANCK 2018 and future sensitivities of two upcoming CMB experiments are also shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' higher values of Mφ corresponds to the points yielding higher value of ∆Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' This is also understandable as higher value of Mφ leads to higher value of Yφ resulting higher value of ∆Neff for the same reason discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In the same plot, we present the current upper limits(1σ and 2σ) on ∆Neff from PLANCK 2018 and future sensitivities of two upcoming CMB experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The present 2σ and 1σ limit on ∆Neff from Planck 2018 excludes DM mass below few hundred keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The future generation experiments like SPT-3G [74] in 1σ limit and CMB-S4 2σ limit [75] may probe heavier DM mass upto 1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The allowed parameter space from the constraints of ∆Neff is also consistent with bound on free streaming length of DM coming from Lyman-α forest[53, 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' CONCLUSION In this work we have proposed a minimal extension of the Type-I seesaw model with a complex scalar singlet(φ) and a singlet Dirac fermion (χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' To ensure the stability of the 25 lightest dark sector particle, an additional Z3 symmetry has been imposed under which φ and χ transform non-trivially while the rest of SM particles and three RHNs transform trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Mass spectrum of the dark sector particles are such that the Dirac fermion χ is the lightest particle and plays the role of DM, while the singlet scalar φ is the next to lightest particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The DM with its tiny coupling with SM bath can only be produced from the late time decay of φ and obtains its abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' On the other hand φ remains in thermal bath due to its strong self coupling and after its freeze out it decays to DM and active neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Depending on the thermal history of φ, we have divided the analysis into two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In the first Scenario (I), φ gains its number density through freeze out mechanism via the number changing strong self-interactions within the dark sector whereas, in the second Scenario (II) φ freezes out via the SM Higgs portal coupling to SM particles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The RHNs(N1,2,3) which are responsible for generating light neutrino masses and mixing angles by type-I seesaw model, are sufficiently heavy (MN1,2,3 ≫ TRH) such that their number densities do not contribute to DM relic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' However, the presence of RHNs in the particle content allows an effective interaction between φ, χ and active neutrinos(ν) which leads to extra neutrino production from the late time decay of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' To track the abundances of φ and χ we have solved two coupled Boltzmann equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' We have first checked the effects of different model parameters on the relic density of DM by solving those Boltzmann equations and identifying the parameter space giving correct relic density in both scenarios (I & II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Apart from producing the right amount of DM relic, the late time decay of φ makes significant impact on the total radiation energy density at the time of CMB formation which is parameterized as ∆Neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' To compute ∆Neff we have evaluated the extra radiation energy density injected into light neutrino bath from φ by solving the required Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In scenario-I DM mass up to a few hundred keV is excluded from the present 1σ limit on ∆Neff from Planck 2018 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The future generation experiments like SPT-3G, CMB-IV will be sensitive enough to test DM mass up to a few MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' However, in scenario-II where the abundance of the mother particle (φ) is suppressed due to sizable interactions with SM bath, we have found that the entropy injection is insensitive to the bounds on ∆Neff coming from present and future-generation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Thus in this paper we have explicitly shown an alternative way of probing FIMP dark matter from the precise measurement of ∆Neff even when the mother particles do not have sizable interactions with SM bath which is otherwise absent in 26 literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Consequently, we are expecting some very exciting results from next generation CMB experiments, like SPT-3G and CMB-IV which can shed some light on various dark sector models, like the one discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Acknowledgement SJ and PG thanks D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Nanda for the helpful discussions during this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The authors would like to thank Abhijit Kumar Saha, Sougata Ganguly and Deep Ghosh for useful discussion and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' SJ is funded by CSIR, Government of India, under the NET JRF fellowship scheme with Award file No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 09/080(1172)/2020-EMR-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Appendix A: Theoretical constraints Stability The scalar potential is bounded from below when the quartic couplings of the scalar potential satisfy these co-positivity conditions[95]: λH ≥ 0, λφ ≥ 0, λφH + 2 � λφλH ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (A1) The estimation of the lifetime of the desired the stable vacuum which essentially puts an upper bound on the trilinear dark coupling as [96] µφ/Mφ < 2 � λφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (A2) Perturbative unitarity The tree-level unitarity of the theory, coming from all possible 2 → 2 scattering ampli- tudes will form the S matrix and constrain the quartic couplings of the scalar potential[97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The eigenvalues of the S matrix are bounded from above as[98]: |λH| ≤ 4π, |λφH| ≤ 8π, |λφ| ≤ 4π, |2λφ + 3λH ± � 2λ2 φH + (2λφ − 3λH)2| ≤ 8π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (A3) 27 The quartic and Yukawa couplings of the interaction Lagrangian should also obey following inequality equations to maintain perturbativity[99]: |λH| ≲ 2π 3 , |λφ| ≲ π, |λφH| ≲ 4π, and |yφN| < √ 4π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (A4) Appendix B: Constraint from Higgs invisible decay The dark complex scalar, φ is very weakly coupled with SM Higgs via the Higgs portal interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The late time decay of φ decides both the relic abundance of DM and the contribution to the ∆Neff which require a light scalar mass of the order of MeV-few GeV which is well below Mh/2 (will be discussed in the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In that case, Higgs can decay to the dark scalar, φ, and contribute to Higgs’s invisible decay width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The Higgs invisible decay width is given by Γh→φφ∗ = (λφHv)2 16πMh � 1 − 4M 2 φ M 2 h , (B1) where Mh = 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='06 GeV and v = 246 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The current analysis of the CMS collaboration [93] at LHC puts a strong constraint on the Higgs invisible decay in the following form BRinv = Γinv h Γinv h + ΓSM h < 11% , (B2) where ΓSM h = 4 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' If Mh < 2Mφ then this decay is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' In this work the Higgs invisible constraint only applicable for scenario-II where we require relatively large λφH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Appendix C: 3φ → 2φ In our setup 3φ → 2φ number changing processes in dark sector occur through φ φ φ → φ φ∗, φ φ∗ φ∗ → φ φ and their conjugate processes i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' φ∗ φ∗ φ∗ → φ∗ φ, φ∗ φ∗ φ → φ∗ φ∗ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Some of these processes are mediated by φ only and the rest are mediated by both φ and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' However, for light Mφ(≲ O(GeV)), h-mediated diagrams are heavily suppressed due to heavy propagator suppression and small Higgs portal coupling, λφH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Therefore for simplicity, one can ignore the Higgs-mediated diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' All the φ mediated 28 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 11: Feynman diagrams for φ φ φ → φ φ∗ number changing processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Note that for each t-channel, there is an u-channel diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 12: Feynman diagrams for φ φ∗ φ∗ → φ φ number changing processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Note that for each t-channel, there is an u-channel diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Feynman diagrams for φ φ φ → φ φ∗ and φ φ∗ φ∗ → φ φ processes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='11 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='12 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The amplitude for φ φ φ → φ φ∗ number changing scattering processes is given by Mφφφ→φφ∗ = M1 + Mt 2 + Mu 2 + Mt 3 + Mu 3 = � 4µφλφ � s − M 2 φ � + 4µφλφ � t − M 2 φ � + 4µφλφ � u − M 2 φ � + µ3 φ � s − M 2 φ �� t − M 2 φ � + µ3 φ � s − M 2 φ �� u − M 2 φ � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='(C1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='D力And the amplitude for φ φ∗ φ∗ → φ φ number changing scattering processes is given by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='Mφ∗φ∗φ→φφ = M1 + Mt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='2 + Mu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='2 + Mt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='3 + Mu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='3 + M4 + Mt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='5 + Mu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='5 + +Mt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='6 + Mu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='4µφλφ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='s − M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='µ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='t − M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='�2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='µ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='u − M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='�2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='4µφλφ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='t − M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='4µφλφ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='u − M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='4µφλφ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='s − M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='4µφλφ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='t − M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='4µφλφ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='u − M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='µ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='t − M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='s − M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='µ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='u − M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='s − M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (C2) The total thermal averaged cross section for 3φ → 2φ number changing processes can be expressed using non-relativistic approximation as[100]: ⟨σv2⟩3φ→2φ = ⟨σv2⟩φφφ→φφ∗ + ⟨σv2⟩φφ∗φ∗→φφ ≈ √ 5 192πM 3 φ � |Mφφφ→φφ∗|2 + |Mφ∗φ∗φ∗→φφ∗|2� + √ 5 192πM 3 φ � |Mφφ∗φ∗→φφ|2 + |Mφ∗φφ→φ∗φ∗|2� = √ 5 192πM 3 φ � 2|Mφφφ→φφ∗|2 + 2|Mφφ∗φ∗→φφ|2� , (C3) where |Mφφφ→φφ∗|2 = |Mφ∗φ∗φ∗→φφ∗|2 and |Mφφ∗φ∗→φφ|2 = |Mφ∗φφ→φ∗φ∗|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Appendix D: 2φ → 2 SM and φ SM → φ SM There is another type of number-changing process between the dark sector, φ, and the visible sector, SM where two dark scalar φ annihilates into two SM particles via h mediated diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Note that our analysis mostly focuses on the light-dark scalar with mass up to a few GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Therefore φ can only annihilate into light fermion pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The Feynman diagrams of corresponding number-changing processes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The thermal averaged cross-section for 2φ → 2SM number changing process is given by: ⟨σv⟩2φ→2SM = � f ⟨σv⟩φφ∗→ff = � f x 16TM 4 φK2(x)2 � ∞ 4M2 φ � σv � φφ∗→ffK1 �√s T � s � s − 4M 2 φ ds (D1) 30 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 13: Feynman Diagrams for φ φ∗ → ff where f stands for SM fermions excluding top quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' where x = Mφ T and � σv � φφ∗→ff can be written as: (σv)φφ→ff = � 1 4πs√s Ncλ2 φHm2 f (s − m2 h)2 + m2 hΓ2 h (s − 4m2 f) 3 2 � Θ(Mφ − mf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (D2) In the above expression Nc = 1 for leptons and Nc = 3 for quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 14: Feynman Diagrams for φ f → φf where f stands for SM fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The scattering between DM and SM, φ SM → φ SM is also important for our discussion which is required for analysing the kinetic equilibrium of the DM in early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The Feynman diagram for the scattering between DM and SM fermions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' The thermal averaged scattering cross-section between DM and SM is followed by: ⟨σv⟩φ SM→φ SM = � f � σv⟩φf→φf + ⟨σv⟩φ∗f→φ∗f � = 2 � f ⟨σv⟩φf→φf = � f x 16TM 2 φm2 fK2(Mφ/T)K2(mf/T) × � ∞ � Mφ+mf �2 � σv � φf→φfK1 �√s T � s � s − � Mφ + mf �2 ds ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (D3) 31 h fh where x = Mφ T and the scattering cross-section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' � σv � φf→φf is given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' � σv � φf→φf = 1 4πs√s 1 2√s �� s − � Mφ + mf �2�� s − � Mφ − mf �2� × � − 2(t − 4M 2 φ) � λφHv t − m2 h �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (D4) [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Chatrchyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (CMS), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' B 716, 30 (2012), 1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='7235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' (ATLAS), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' B 716, 1 (2012), 1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='7214.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' thesis, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' Paris-Saclay (2018), 1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content='05822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} +page_content=' 36' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FST4oBgHgl3EQfPDjc/content/2301.13754v1.pdf'} diff --git a/qtAzT4oBgHgl3EQf5_4s/content/tmp_files/2301.01867v1.pdf.txt b/qtAzT4oBgHgl3EQf5_4s/content/tmp_files/2301.01867v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f5e32cd7e7f22d5c9dd628e90531ffa338b67507 --- /dev/null +++ b/qtAzT4oBgHgl3EQf5_4s/content/tmp_files/2301.01867v1.pdf.txt @@ -0,0 +1,979 @@ +1 +Unsupervised High Impedance Fault Detection +Using Autoencoder and Principal Component +Analysis +Yingxiang Liu, Mohammad Razeghi-Jahromi and James Stoupis +Abstract—Detection of high impedance faults (HIF) has +been one of the biggest challenges in the power distribution +network. The low current magnitude and diverse charac- +teristics of HIFs make them difficult to be detected by +over-current relays. Recently, data-driven methods based +on machine learning models are gaining popularity in HIF +detection due to their capability to learn complex patterns +from data. Most machine learning-based detection methods +adopt supervised learning techniques to distinguish HIFs +from normal load conditions by performing classifications, +which rely on a large amount of data collected during HIF. +However, measurements of HIF are difficult to acquire in +the real world. As a result, the reliability and generalization +of the classification methods are limited when the load +profiles and faults are not present in the training data. +Consequently, this paper proposes an unsupervised HIF +detection framework using the autoencoder and principal +component analysis-based monitoring techniques. The pro- +posed fault detection method detects the HIF by monitoring +the changes in correlation structure within the current +waveforms that are different from the normal loads. The +performance of the proposed HIF detection method is +tested using real data collected from a 4.16 kV distribution +system and compared with results from a commercially +available solution for HIF detection. The numerical re- +sults demonstrate that the proposed method outperforms +the commercially available HIF detection technique while +maintaining high security by not falsely detecting during +load conditions. +Index Terms—High impedance fault detection, Unsuper- +vised Learning, Neural Network +I. INTRODUCTION +H +IGH impedance fault (HIF) is a group of power +system disturbances that typically occurs when a +live conductor contacts a surface with high impedance. +The HIF current magnitude typically ranges from 0 to 75 +Y. Liu is with the Ming Hsieh Department of Electrical and Com- +puter Engineering , University of Southern California, Los Angeles, +CA, 90089 USA (e-mail: yingxian@usc.edu) +M.Razeghi-Jahromi is with ABB Corporate Research United +States +(USCRC), +Raleigh, +NC +27606 +USA +(e-mail: +moham- +mad.razeghijahromi@us.abb.com) +J.Stoupis is with ABB Corporate Research United States (USCRC), +Raleigh, NC 27606 USA (e-mail: james.stoupis@us.abb.com) +A, and the characteristics of HIFs are affected by various +factors such as surface type and load conditions [1]. The +low current magnitudes and diverse characteristics make +the HIFs difficult to be detected using conventional over- +current relays [2]. It is estimated that between 5% and +10% of the distribution faults are HIF [3], and about 25% +of the HIFs are not detected using the over-current relays +[4]. Since over-current relays usually cannot detect HIFs, +the arcs and flashover caused by HIFs can cause fires +and jeopardize human safety [5]. Therefore, effectively +detecting HIFs remains a non-negligible challenge. +Recent advances in the industrial internet of things +and smart grid allow increasing computation resources +and data analysis capabilities within the power grid [6], +[7]. As a result, machine learning approaches have been +gaining popularity for HIF detection. Ghaderi et al. [8] +trained a support vector machine (SVM) classifier with +features of current waveform energy and normalized +joint time-frequency moments. Baqui et al. [9] combined +artificial neural network (ANN) with discrete wavelet +transforms (DWT) for HIF detection in medium-voltage +networks. Features were extracted from current mea- +surements using DWTs and then fed into the ANN for +classification. Wang et al. [10] first applied a modified +Gabor WT to the input signal to extract two-dimensional +scalograms and then applied a two-dimensional convolu- +tional neural network (CNN) for classification. In [11], a +Long Short Term Memory (LSTM) classifier was trained +with features obtained from DWT analysis to detect the +HIFs in the solar photovoltaic integrated power system. +The machine learning-based studies above use su- +pervised learning methods to detect HIF by training +classification models that map the input to a set of +labels corresponding to different HIF types. However, +there are some limitations to using classification for HIF +detection. The first one is the generalization problem of +the models. The supervised learning-based HIF detection +methods detect the occurrence of HIF by performing +classification using models trained with labeled data +collected under various normal load conditions and dur- +arXiv:2301.01867v1 [cs.LG] 5 Jan 2023 + +ing different HIFs. However, when the supervised HIF +detection methods are deployed in the grid, the classifiers +may produce undependable results if the load profile +or HIFs are not present in the training set. Another +limitation is scalability. Since the HIF detection method +needs to be deployed to different parts of the grids with +various load profiles, the model needs to be trained with +data collected from different utilities from different parts +of the grid to ensure the data-driven model works for +all of them. Therefore, the supervised learning methods +are different to scale in real-world applications. To deal +with the limitations mentioned above, the unsupervised +learning methods can be used for fault detection since +they do not require labels and can easily adapt to +different load conditions. The fault detection methods +based on unsupervised learning methods have been +successfully applied to various engineering applications +such as chemical and semiconductor manufacturing [12]. +However, their applications for HIF detection are still +limited. In recent years, Rai et al. [13] applied a convolu- +tional autoencoder trained with simulated HIF scenarios. +Then cross-correlation between the reconstructed signal +and the original signal was used to discriminate HIFs +from loads. Although the proposed method showed good +fault detection performance on the simulated dataset, +it relies on training using faulty HIF data, which is +difficult to acquire in real-world applications. Sarwar +et al. [14] introduced principal component analysis- +based statistical process monitoring techniques to detect +HIF. The proposed methods can successfully detect the +occurrence of HIF. However, instead of analyzing the +measurements collected from one location in the grid, it +applies PCA to 29 variables simulated from the IEEE +13-node test feeder. As a result, it requires resource- +intensive communication and data storage between mul- +tiple measurement devices. +Consequently, this paper proposed an unsupervised +HIF detection framework based on the autoencoder (AE) +and principal component analysis, which are trained us- +ing historical measurements collected from one location +in the grid. First, the univariate current measurement +is augmented into a data matrix consisting of multiple +variables. Then the autoencoder extracts nonlinear fea- +tures from the data matrix to capture the correlations +among different variables. Next, a PCA model is built +based on the autoencoder’s reconstruction errors. Finally, +the PCA-based statistical monitoring technique is used +to characterize the residuals from the AE model of +the normal load data and establish thresholds based on +various statistics. The autoencoder and the PCA can then +be deployed online to monitor the new current mea- +surement. If HIF occurs, the correlation structure of the +augmented data matrix will deviate from the correlation +learned by the AE from the normal loads, thus leading +to abnormal reconstruction errors from the autoencoder +and reflected in indices of the PCA-based monitoring +model. The main contributions of this study are: (1) +combine autoencoder and PCA model to characterize the +correlations structure of univariate current measurement; +(2) introduce statistical process monitoring techniques +for detecting HIF using data collected from a single +location in the grid; and (3) The proposed unsupervised +method only relies on the measurements of the normal +loads. In addition, since the number of parameters in +the AE model is small, the proposed method can be +trained rapidly and thus can be easily adapted and +deployed to computing devices located across the grid. +The remainder of this paper is organized as follows. +Section II introduces autoencoder and PCA-based pro- +cess monitoring technique, followed by the details of the +proposed HIF detection method in Section III. Section +IV presents a real dataset collected from a 4.16 kV +distribution system to evaluate the effectiveness of the +proposed method. Finally, the conclusions are presented +in Section V. +II. PRELIMINARIES +A. Autoencoder +Autoencoder is an unsupervised neural network that +learns to compress and reconstruct the input data ef- +fectively. It has been widely used for fault detection +in various applications such as electric motors [15], +wind turbines [16], and chemical processes [17]. An +autoencoder consists of two parts: an encoder followed +by a decoder which can be represented using different +neural network structures such as multi-layer perceptron +(MLP), convolutional neural network (CNN), and recur- +rent neural network (RNN). In this study, we used the +MLP as the encoder and decoder due to its simplicity. +For an autoencoder composed of a single hidden layer, +the encoder maps the input vector x ∈ RM in the hidden +representation h ∈ RP as follows. +h = f(W1x + b1) +(1) +where f is an non-linear activation function, W1 ∈ +RP ×M is a weight matrix, and b1 ∈ RP is a bias vector. +The decoder then tries to reconstruct the input x by using +˜x = f(W2h + b2) +(2) +where W2 ∈ RM×P is the decoder weight matrix, and +b2 ∈ RM is the bias vector, and ˜x is the reconstructed +input vector. To avoid the autoencoder learning to copy +the input to the output and to capture the correlation +2 + +among different input variables, the dimension of the +hidden layer h is chosen to be smaller than the dimension +of the input. Training of the autoencoder is performed by +minimizing the mean squared error (MSE) loss function: +L(θ) = ||x − ˜x||2 +(3) +where θ represents all the network parameters. +B. PCA for Fault Detection +Principal Component Analysis (PCA) is widely used +as a dimensional reduction tool in different domains such +as computer science and electrical engineering [18]– +[20]. It produces a low-dimensional representation of +multivariate data by finding a direction or subspace of +the largest variance in the original measurement space. +Let X ∈ RN×M denotes a data matrix with each row +representing a sample x ∈ RM. After applying PCA to +the data matrix X, it can be decomposed as, +X = TP⊤ + ˜T˜P⊤ +(4) +where P consists of the first l loading vectors that +contain most variance of the data and ˜P is the last +M − l loading vectors. The subspace spanned by P is +known as the principal component subspace (PCS) and +that spanned by ˜P is called the residual subspace (RS). +Consequently, the measurement space can be divided +into the PCS and the RS, where the PCS contains normal +or major variations, and the RS contains small variations +or noises. +PCA has been widely used for statistical process mon- +itoring [12], [21], [22] and fault detection of multivariate +data collected from chemical processes. It is used to +model the normal static variation from data related to +normal operation. To perform fault detection, the general +idea is first to build models using data collected during +normal operations. Then control limits are established to +define normal operation regions. Finally, the models and +the control limits are applied to new data for online fault +detection. With a PCA model, different fault detection +indices such as Hotelling’s T 2 index, the SPE (or Q +index) index and the combined index ϕ can be defined +to monitor various aspects of the data. It is important +to note that these indices and the corresponding limits +assume that the data samples are independent in time. +1) Hotelling’s T 2 index +Hotelling’s T 2 index measures variations in the +PCS, +T 2 = x⊤PΛ−1P⊤x +(5) +where Λ is the convariance matrix of the latent +scores matrix T. It can be proven that T 2 statistic +follows a F distribution, +N(N − l) +l(N 2 − 1)T 2 ∼ Fl,N−l +(6) +where Fl,N−l is an F distribution with l and +N − l degrees of freedom [23]. As a result, for +a given confidence level α, the control limit can +be calculated based on the Fl,N−l distribution. The +index is considered normal if +T 2 ≤ T 2 +α ≡ l(N 2 − 1) +N(N − l)Fl,N−l;α +(7) +If the number of data points N is large, the +T 2 index can be well approximated with a χ2 +distribution with l degrees of freedom [12] and +T 2 +α = χ2 +l;α +(8) +The T 2 index measures the distance to the origin in +the principal component subspace, which contains +normal process variations with large variance. The +variation of the projection of a sample vector x on +the PCS is considered normal if its T 2 index is +less than the control limit T 2 +α. +2) SPE (Squared Prediction Error) index +The SPE index measures the projection of a sample +vector x ∈ RM onto the residual space. It is +defined as the squared norm of the residual vector +˜x. +SPE(x) = ||˜x||2 = x⊤ ˜P˜P⊤x +(9) +The control limit of the SPE index can be derived +using the result in [24], +δ2 +α = gχ2 +h;α +(10) +where +g = +�M +i=l+1 λ2 +i +�M +i=l+1 λi +, h = (�M +i=l+1 λi)2 +�M +i=l+1 λ2 +i +(11) +α is confidence level. l is the number of PC in +the principal component subspace, and λi is the +ith eigenvalue of the sample convariance matrix +1 +N−1X⊤X. +Since the SPE index focuses on the residual sub- +space, it measures the variability that breaks the +static process relations. If the SPE index is above +the control limit δα, it indicates a fault occurs that +breaks the normal correlation structure. +3) Combined index +If both the T 2 index and SPE index are equally +important, a global index can be used to combine +the two indices, such as the combined index ϕ +[25], [26]. This results in monitoring one index +3 + +instead of two. The combined index is defined as +follows, +ϕ = T 2(x) + g−1SPE(x) ∼ χ2 +l+h +(12) +where g and h come from the calculation of the +SPE control limit. With α as the confidence level, +the control limit of the combined index is χ2 +l+h;α. +As a result, a fault is detected if the value of ϕ is +greater than the control limit. +III. PROPOSED HIF DETECTION PROCEDURE +The occurrence of HIF introduces minor random dis- +tortions in current waveforms. As a result, the correlation +between the current measurements between different +cycles will show inconsistency from the correlation +structure of the measurements collected during normal +load conditions. Therefore, the proposed fault detection +procedure detects the HIF by monitoring the changes +in correlation structure within the current waveforms. +The workflow of the proposed HIF detection is shown +in Figure 1. +Fig. 1. Workflow of the proposed HIF detection method. +A. Data Preprocessing +The proposed HIF detection method first converts +the single-phase current waveform to a data matrix +by sampling at the same locations within each cycle +across the historical measurement of loads. Let ts be +the number of samples per cycle and the length of the +original signal to be N × ts, the original signal can be +represented as S = [s(1), s(2), ..., s(N×ts−1), s(N×ts)]. +With M to be the number of variables and ∆ = ts/M +be the gap when sampling from the original signal S, +the matrix X can be written as +X = +� +� +s(1) +s(1+∆) +... +s(ts) +s(1+ts) +s(1+∆+ts) +... +s(2ts) +... +... +... +... +� +� +(13) +The resulting matrix X has M columns and N rows. +Since the autoencoder will be trained to reconstruct each +row of the data matrix, the sampling is used to reduce +the network’s input dimension and thus decrease the total +number of parameters in the neural network model to +prevent overfitting and improve training speed. +B. Offline Training +In the offline training step, an autoencoder and a +PCA model are built to characterize the correlation +structures of the current waveforms of normal loads. +An autoencoder model is trained to extract the normal +correlation and nonlinear features from the augmented +data matrix by minimizing the MSE loss in Equation 3. +After the autoencoder is trained to reconstruct the data +matrix formed using normal load current waveforms, +it can remove common features from the data matrix, +leaving small residuals for all the variables in the data +matrix. As a result, the autocorrelations within the input +data matrix are eliminated, and the residuals only contain +static variations, which can be modeled using the PCA +and lend themselves to detect faults. PCA-based process +monitoring techniques are applied to the reconstruction +errors or the residuals of the fault-free data matrix +produced by the trained autoencoder. Let ˜X be the output +of the trained autoencoder. The reconstruction error of +the data matrix can be written as, +E = X − ˜X +(14) +After normalizing each column of E to have zero mean +and unit variance, a PCA model can be built from the +normalized reconstruction error. Then the number of +latent variables l can be selected based on cumulative +percent variance (CPV) +CPV (l) = +�l +i=1 λi +�M +i=1 λi +(15) +With the selected l and confidence level α, the control +limits for SPE, T 2, and ϕ indices can be established +using Equations 10, 8, and 12. +C. Online HIF Detection +The trained autoencoder and PCA model are ap- +plied to three phases separately for detecting the high +impedance fault in new measurements. For each phase, +4 + +Historic Load Data +New Data +Data Preprocessing +Data Preprocessing +Train AE Model +Apply AE Model +Reconstruction Error +Reconstruction Error +Build PCA Model +Apply PCA Model +Establish Monitor Indices +Compare Monitor Indices +and Limits +with Established Limits +Offline Training +Counter and Detection +Result +Online Detectionafter acquiring the new current measurement of a cycle, +a vector x with M variables is constructed by sampling +from the cycle. Then the new vector is passed as an +input to the trained autoencoder model to get a vector +of reconstruction errors e ∈ RM. Since the autoen- +coder is trained using data from normal loads, abnormal +reconstruction errors of the vector can be observed if +the occurrence of HIF distorts the correlation structure +within a cycle. After scaling reconstruction errors e +with the mean and variances calculated when building +the PCA model in the offline training step, SPE, T 2, +and ϕ index for the reconstruction error vector can be +calculated using 9, 5, and 12. This study uses the ϕ +index for HIF detection since it can effectively combine +the SPE and T 2 indices. If the combined index of e is +above the control limit calculated in the offline training +phase, it indicates that there are abnormal distortions +that break the normal correlation structure in the cycle +corresponding to the vector x. To account for the noise +and transient disturbances in the measurements, we use +a counter to record the number of cycles with indices +above the control limit. The counter is incremented +when the combined index corresponding to one cycle +exceeds the control limit and decreases if the index drops +below the control limit. A trip signal is issued when the +counter exceeds a predetermined threshold, which means +the trip signal will be generated if the combined index +consistently stays above the control limit. +IV. EVALUATION +A. Dataset +The dataset used in this study was collected during +the testing and evaluation of ABB’s feeder protection +system REF 550 [27], [28]. The measurements of three- +phase voltages and currents were collected in a 4.16 +kV distribution system near a hospital. High impedance +faults in phase A were stages at about 12 miles from +the hospital by dropping the conductor on four different +surfaces: grass, water puddle, soil, and asphalt. In ad- +dition, the faults were created multiple times for each +surface under different load conditions. In each case, a +fault was introduced at around 100 seconds and lasted +for 60 seconds before the conductor was lifted off the +test surface. In addition to the fault cases, measurements +of normal load were recorded. The number of samples +per cycle ts for all the measurements is 320. +Figure 2 shows the root mean square (RMS) current +waveforms of a section of normal loads. It can be +observed that the variations in the load are dynamic +and complex, with the occasional presents of spikes. In +addition, the three phases are unbalanced with distinct +patterns. Figure 3 shows the comparison between the +current waveform of the normal load and the waveform +during HIF. Unlike the simulated cases used in previous +publications [10], [13], the load waveform is distorted +and dynamic. As a result, distinguishing the HIF from +the normal load is more challenging since the magni- +tudes of distortion in the two cases are similar. +Fig. 2. RMS currents of normal load. +Fig. 3. Current waveform of the normal load and the waveform during +the HIF. +B. Results and Analysis +The first step in implementing the proposed fault +detection procedure is to augment the current waveforms +to a data matrix. There are four load cases in the dataset. +Three load cases containing around 580 seconds of mea- +surements are used to train and validate the autoencoder +model, and the last load case is left for testing. First, +for each phase in each load case, the univariate current +5 + +Phase A +350 +Amplitude (A) +340 +330 +25 +50 +75 +100 +125 +150 +175 +0 +Time (s) +Phase B +Amplitude (A) +245 +240 +235 +75 +25 +50 +100 +125 +150 +0 +175 +Time (s) +Phase C +Amplitude (A) +315 +310 +305 +50 +25 +75 +100 +125 +150 +175 +0 +Time (s)Current before HIF +Amplitude (A) +250 +250 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +Time (s) +Current during HIF +500 +Amplitude (A) +250 +0 +250 +0.01 +0.02 +0.00 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +Time (s)measurement is converted to a data matrix. Since the +number of samples per cycle ts is 320, the sampling +gap ∆ is selected to be 10, resulting in a data matrix +consisting of 32 variables. As a result, each row in the +matrix corresponds to the measurement sampled from +one cycle. Next, all the data matrices formed from all +three phases in three load cases are concatenated. After +scaling each column of the concatenated data matrix to +have values between 0 and 1, 80% of the data is used +for training, and 20% is used for validation. +The autoencoder model used in this study has five +layers. The dimension of the input and output layers are +32, and the dimension of the three hidden layers are 15, +10, and 15, respectively. The rectified linear unit (ReLU) +is used as the activation function for the input and hidden +layers. The model is trained using Adam optimizer in +PyTorch with a learning rate of 0.001 is used to minimize +the MSE loss. The autoencoder model is trained for 100 +Epochs with a batch size of 32. +After the autoencoder model is trained using the +normal load data, the reconstruction errors of the training +and validation data are used to build a PCA model. The +number of leading PCs l is selected so that the first l +PCs captured 95% of the variances, and the confidence +level α is chosen to be 99%. +The autoencoder and PCA models are applied to the +load and HIF cases staged on different surface types. +The proposed HIF detection method is first applied to the +load case that is not used during training to show that the +proposed method does not generate false alarms for new +load profiles. Figure 4 shows the combined indices and +trip signals generated from a counter with a threshold of +60 for all three phases. It can be observed that most of the +indices stay below the control limit, with a few outliers +caused by spikes present in the current waveform. As a +result, no trip signal is generated for all three phases, +which is expected for the normal load. +Three HIF cases were staged on the grass surface +when ABB tested the REF 550 for HIF Detection, and +the REF 550 failed to detect one of them. On the +contrary, our proposed HIF detection can successfully +detect all the HIF cases when the conductor of phase A +contacts the grass. Figure 5 shows the detection result +of the proposed method for the case that REF 550 failed +to detect. It can be observed that the combined index of +phase A rises above the control limit after introducing +HIF at around 100 seconds. The index stays above the +control limit until the conductor is lifted off the grass at +around 160 seconds. As a result, a trip signal is generated +for phase A after the index stays above the control limit +longer than 60 cycles. Phase C is also affected by the +HIF. However, since the magnitude of its monitoring +Fig. 4. Detection result of normal load. +Fig. 5. Detection result of HIF on grass. +index is much smaller than phase A, HIF is determined +to have occurred in phase A. +In addition to the tests conducted on the grass surface, +four HIF cases were staged by dropping the conductor of +phase A on the soil surface. When these four cases were +tested, the REF 550 detected three of them, and one was +not detected. To compare our proposed HIF detection +method, we apply the trained autoencoder and PCA +models to these four cases, and the results show that all +the HIFs can be detected. Figure 6 shows the detection +result of the proposed method for the HIF case that REF +6 + +Load +4 +3 +Phase A +2 +0 +Trip +0 +4 +3 +Phase B +2 +0 +Trip +0 +4 +3 +0 +Trip +0 +25 +75 +100 +125 +150 +175 +0 +50 +Time (s)Grass +7.5 +Phase A +5.0 +2.5 +mmhha +0.0 +Trip +0 +4 +3 +B +Phase I +2 +0 +Trip +0 +4 +3 +Phase ( +2 +0 +Trip +0 +25 +75 +50 +100 +125 +150 +175 +0 +Time (s)Fig. 6. Detection result of HIF on soil. +550 failed to detect. It can be seen that before the fault +is introduced at around 100 seconds, the indices for all +three phases stay below the control limit, indicating that +the current waveforms are normal and there is no fault. +However, after the conductor of phase A contacts the +soil, the monitoring index of phase A immediately rises +and stays above the control limit. As a result, a trip +signal is generated for phase A. Similar to the HIF cases +staged on grass, phase C also shows minor abnormal +distortions since the corresponding monitoring index +oscillates around the control limit. However, the trip +signal is not generated for phase C since the number of +abnormal cycles does not reach the predefined threshold +of 60. +The dataset also contains measurements of high +impedance faults on asphalt and puddle filled with drink- +able water. During the testing, the REF 550 could not +detect any HIF on asphalt and water puddle. Like the +detection results from REF 550, our proposed method +cannot detect any of these cases due to the near-infinite +impedance conditions of the downed conductor test +and the long distance between the fault location and +where the measurements were taken. Figure 7 shows the +detection result of one of the HIF cases on asphalt in +which no trip signal is generated since all the indices +stay below the control limit. Even though the proposed +method cannot detect the faults that occurred on near- +infinite impedance surface types, no false alarms are +generated during various load conditions before and after +the HIFs in all the cases. +The comparison between the HIF detection results +Fig. 7. Detection result of HIF on asphalt. +from the REF 550 and our proposed method can be +summarized using the following metrics: accuracy (Acc), +security (Sec), dependability (Dep), safety (Saf), and +sensibility (Sen) [1]. +Acc = +TP + TN +TP + TN + FP + FN × 100% +(16) +Sec = +TN +TN + FP × 100% +(17) +Dep = +TP +TP + FN × 100% +(18) +Saf = +TN +TN + FN × 100% +(19) +Sen = +TP +TP + FP × 100% +(20) +where true positives (TP) and true negatives (TN) are +the numbers of the correctly detected fault and normal +load cases, and false negatives (FN) and false positives +(FP) are the numbers of the wrongly detected fault and +load cases. We calculate the above metrics based on the +detection results for all the cases in the entire dataset. +The results are shown in Table IV-B. +Since the REF 550 and our proposed method can +correctly identify the load conditions, they achieve 100% +dependability and security, indicating they are robust to +faulty tripping. Furthermore, our proposed HIF detection +method can correctly detect more HIF cases. As a result, +our proposed HIF detection method shows improvement +in the other metrics compared to the REF 550. +7 + +Soil +7.5 +Phase A +5.0 +2.5 +0.0 +Trip +0 +A +3 +Phase B +Trip +4 +3 +Phase C +2 +0 +Trip +50 +100 +150 +200 +0 +Time (s)Asphalt +3 +Phase A +2 +0 +Trip +0 +4 +3 +Phase B +2 +0 +Trip +4 +3 +0 +0 +50 +100 +150 +200 +250 +0 +Time (s)TABLE I +COMPARISON OF REF 550 AND PROPOSED HIF DETECTION +METHOD. +Acc +Sec +Dep +Saf +Sen +REF 550 +68.9% +100% +35.7% +62.5% +100% +AE + PCA +75.9% +100% +50% +68.2% +100% +V. CONCLUSION +This paper proposes an unsupervised HIF detection +method based on the autoencoder and principal com- +ponent analysis, which does not require measurements +during HIFs. The proposed method first converts the +univariate current measurement collected from one lo- +cation in the grid into a data matrix. The data matrix +is then used to train an autoencoder for extracting +nonlinear features from the data matrix and capturing +the correlations among variables in the data matrix. +Finally, the PCA-based statistical monitoring technique +is used to characterize the residuals of the normal load +data from the AE model and establish thresholds based +on various statistics. The proposed method detects high +impedance faults by monitoring the deviation in the +correlation structure of the augmented data matrix from +the correlation learned by the AE from the normal +loads. The proposed HIF detection method is applied to +real data collected from a 4.16 kV distribution system +which contains various normal load cases and HIF cases +staged on four types of surfaces: grass, water puddle, +soil, and asphalt. The detection results are compared +with the results from the commercially available HIF +detection solution REF 550, demonstrating that our pro- +posed method outperforms REF 550 by detecting more +HIF cases while not making false alarms during load +conditions. +REFERENCES +[1] A. Ghaderi, H. L. Ginn, and H. A. Mohammadpour, “High +impedance fault detection: A review,” Electric Power Systems +Research, vol. 143, p. 376–388, 2017. +[2] C. G. Wester, “High impedance fault detection on distribution +systems,” in 1998 rural electric power conference presented at +42nd annual conference. +IEEE, 1998, pp. c5–1. +[3] M. Adamiak, C. Wester, M. Thakur, and C. Jensen, “High +impedance fault detection on distribution feeders,” GE Industrial +solutions, 2006. +[4] B. D. Russell and C. L. Benner, “Arcing fault detection for +distribution feeders: security assessment in long term field trials,” +IEEE Transactions on power delivery, vol. 10, no. 2, pp. 676– +683, 1995. +[5] B. K. Chaitanya, A. 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MacGregor, “Multivariate statistical methods for monitoring +large datasets from chemical processes,” in AIChE Meeting, San +Francisco, CA, 1989. +[23] N. D. Tracy, J. C. Young, and R. L. Mason, “Multivariate +control charts for individual observations,” Journal of Quality +Technology, vol. 24, no. 2, pp. 88–95, 1992. +[24] G. Box, “Some theorems on quadratic forms applied in the +study of analysis of variance problems, I. effect of inequality +8 + +of variance in the one-way classification,” Ann. Math. Statistics, +vol. 25, pp. 290–302, 1954. +[25] H. H. Yue and S. J. Qin, “Reconstruction-based fault identifi- +cation using a combined index,” Industrial & Engineering +Chemistry Research, vol. 40, no. 20, p. 4403–4414, 2001. +[26] Y. Dong and S. J. Qin, “New dynamic predictive monitoring +schemes based on dynamic latent variable models,” Industrial & +Engineering Chemistry Research, vol. 59, no. 6, pp. 2353–2365, +2020. +[27] D. B. Ratan Das, “System for detection of high impedance fault,” +19th International Conference on Electricity Distribution, 2007. +[28] “Ref 550 advanced feeder protection and control - abb,” https: +//library.e.abb.com/public/64e517269f719a5bc12573af006d2dd2/ +REF_550_DB41-902%20Rev.E.pdf. +9 + diff --git a/qtAzT4oBgHgl3EQf5_4s/content/tmp_files/load_file.txt b/qtAzT4oBgHgl3EQf5_4s/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1a06f7d3cf1c70b03683c00158060d9218da6b9 --- /dev/null +++ b/qtAzT4oBgHgl3EQf5_4s/content/tmp_files/load_file.txt @@ -0,0 +1,516 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf,len=515 +page_content='1 Unsupervised High Impedance Fault Detection Using Autoencoder and Principal Component Analysis Yingxiang Liu, Mohammad Razeghi-Jahromi and James Stoupis Abstract—Detection of high impedance faults (HIF) has been one of the biggest challenges in the power distribution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The low current magnitude and diverse charac- teristics of HIFs make them difficult to be detected by over-current relays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Recently, data-driven methods based on machine learning models are gaining popularity in HIF detection due to their capability to learn complex patterns from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Most machine learning-based detection methods adopt supervised learning techniques to distinguish HIFs from normal load conditions by performing classifications, which rely on a large amount of data collected during HIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' However, measurements of HIF are difficult to acquire in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' As a result, the reliability and generalization of the classification methods are limited when the load profiles and faults are not present in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Consequently, this paper proposes an unsupervised HIF detection framework using the autoencoder and principal component analysis-based monitoring techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The pro- posed fault detection method detects the HIF by monitoring the changes in correlation structure within the current waveforms that are different from the normal loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The performance of the proposed HIF detection method is tested using real data collected from a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='16 kV distribution system and compared with results from a commercially available solution for HIF detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The numerical re- sults demonstrate that the proposed method outperforms the commercially available HIF detection technique while maintaining high security by not falsely detecting during load conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Index Terms—High impedance fault detection, Unsuper- vised Learning, Neural Network I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' INTRODUCTION H IGH impedance fault (HIF) is a group of power system disturbances that typically occurs when a live conductor contacts a surface with high impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The HIF current magnitude typically ranges from 0 to 75 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Liu is with the Ming Hsieh Department of Electrical and Com- puter Engineering , University of Southern California, Los Angeles, CA, 90089 USA (e-mail: yingxian@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='edu) M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='Razeghi-Jahromi is with ABB Corporate Research United States (USCRC), Raleigh, NC 27606 USA (e-mail: moham- mad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='razeghijahromi@us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='com) J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='Stoupis is with ABB Corporate Research United States (USCRC), Raleigh, NC 27606 USA (e-mail: james.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='stoupis@us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='com) A, and the characteristics of HIFs are affected by various factors such as surface type and load conditions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The low current magnitudes and diverse characteristics make the HIFs difficult to be detected using conventional over- current relays [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' It is estimated that between 5% and 10% of the distribution faults are HIF [3], and about 25% of the HIFs are not detected using the over-current relays [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Since over-current relays usually cannot detect HIFs, the arcs and flashover caused by HIFs can cause fires and jeopardize human safety [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Therefore, effectively detecting HIFs remains a non-negligible challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Recent advances in the industrial internet of things and smart grid allow increasing computation resources and data analysis capabilities within the power grid [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' As a result, machine learning approaches have been gaining popularity for HIF detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Ghaderi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' [8] trained a support vector machine (SVM) classifier with features of current waveform energy and normalized joint time-frequency moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Baqui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' [9] combined artificial neural network (ANN) with discrete wavelet transforms (DWT) for HIF detection in medium-voltage networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Features were extracted from current mea- surements using DWTs and then fed into the ANN for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' [10] first applied a modified Gabor WT to the input signal to extract two-dimensional scalograms and then applied a two-dimensional convolu- tional neural network (CNN) for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' In [11], a Long Short Term Memory (LSTM) classifier was trained with features obtained from DWT analysis to detect the HIFs in the solar photovoltaic integrated power system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The machine learning-based studies above use su- pervised learning methods to detect HIF by training classification models that map the input to a set of labels corresponding to different HIF types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' However, there are some limitations to using classification for HIF detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The first one is the generalization problem of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The supervised learning-based HIF detection methods detect the occurrence of HIF by performing classification using models trained with labeled data collected under various normal load conditions and dur- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='01867v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='LG] 5 Jan 2023 ing different HIFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' However, when the supervised HIF detection methods are deployed in the grid, the classifiers may produce undependable results if the load profile or HIFs are not present in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Another limitation is scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Since the HIF detection method needs to be deployed to different parts of the grids with various load profiles, the model needs to be trained with data collected from different utilities from different parts of the grid to ensure the data-driven model works for all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Therefore, the supervised learning methods are different to scale in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' To deal with the limitations mentioned above, the unsupervised learning methods can be used for fault detection since they do not require labels and can easily adapt to different load conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The fault detection methods based on unsupervised learning methods have been successfully applied to various engineering applications such as chemical and semiconductor manufacturing [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' However, their applications for HIF detection are still limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' In recent years, Rai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' [13] applied a convolu- tional autoencoder trained with simulated HIF scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Then cross-correlation between the reconstructed signal and the original signal was used to discriminate HIFs from loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Although the proposed method showed good fault detection performance on the simulated dataset, it relies on training using faulty HIF data, which is difficult to acquire in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Sarwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' [14] introduced principal component analysis- based statistical process monitoring techniques to detect HIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The proposed methods can successfully detect the occurrence of HIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' However, instead of analyzing the measurements collected from one location in the grid, it applies PCA to 29 variables simulated from the IEEE 13-node test feeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' As a result, it requires resource- intensive communication and data storage between mul- tiple measurement devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Consequently, this paper proposed an unsupervised HIF detection framework based on the autoencoder (AE) and principal component analysis, which are trained us- ing historical measurements collected from one location in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' First, the univariate current measurement is augmented into a data matrix consisting of multiple variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Then the autoencoder extracts nonlinear fea- tures from the data matrix to capture the correlations among different variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Next, a PCA model is built based on the autoencoder’s reconstruction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Finally, the PCA-based statistical monitoring technique is used to characterize the residuals from the AE model of the normal load data and establish thresholds based on various statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The autoencoder and the PCA can then be deployed online to monitor the new current mea- surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' If HIF occurs, the correlation structure of the augmented data matrix will deviate from the correlation learned by the AE from the normal loads, thus leading to abnormal reconstruction errors from the autoencoder and reflected in indices of the PCA-based monitoring model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The main contributions of this study are: (1) combine autoencoder and PCA model to characterize the correlations structure of univariate current measurement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' (2) introduce statistical process monitoring techniques for detecting HIF using data collected from a single location in the grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' and (3) The proposed unsupervised method only relies on the measurements of the normal loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' In addition, since the number of parameters in the AE model is small, the proposed method can be trained rapidly and thus can be easily adapted and deployed to computing devices located across the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Section II introduces autoencoder and PCA-based pro- cess monitoring technique, followed by the details of the proposed HIF detection method in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Section IV presents a real dataset collected from a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='16 kV distribution system to evaluate the effectiveness of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Finally, the conclusions are presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' PRELIMINARIES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Autoencoder Autoencoder is an unsupervised neural network that learns to compress and reconstruct the input data ef- fectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' It has been widely used for fault detection in various applications such as electric motors [15], wind turbines [16], and chemical processes [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' An autoencoder consists of two parts: an encoder followed by a decoder which can be represented using different neural network structures such as multi-layer perceptron (MLP), convolutional neural network (CNN), and recur- rent neural network (RNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' In this study, we used the MLP as the encoder and decoder due to its simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' For an autoencoder composed of a single hidden layer, the encoder maps the input vector x ∈ RM in the hidden representation h ∈ RP as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' h = f(W1x + b1) (1) where f is an non-linear activation function, W1 ∈ RP ×M is a weight matrix, and b1 ∈ RP is a bias vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The decoder then tries to reconstruct the input x by using ˜x = f(W2h + b2) (2) where W2 ∈ RM×P is the decoder weight matrix, and b2 ∈ RM is the bias vector, and ˜x is the reconstructed input vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' To avoid the autoencoder learning to copy the input to the output and to capture the correlation 2 among different input variables, the dimension of the hidden layer h is chosen to be smaller than the dimension of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Training of the autoencoder is performed by minimizing the mean squared error (MSE) loss function: L(θ) = ||x − ˜x||2 (3) where θ represents all the network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' PCA for Fault Detection Principal Component Analysis (PCA) is widely used as a dimensional reduction tool in different domains such as computer science and electrical engineering [18]– [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' It produces a low-dimensional representation of multivariate data by finding a direction or subspace of the largest variance in the original measurement space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Let X ∈ RN×M denotes a data matrix with each row representing a sample x ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' After applying PCA to the data matrix X, it can be decomposed as, X = TP⊤ + ˜T˜P⊤ (4) where P consists of the first l loading vectors that contain most variance of the data and ˜P is the last M − l loading vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The subspace spanned by P is known as the principal component subspace (PCS) and that spanned by ˜P is called the residual subspace (RS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Consequently, the measurement space can be divided into the PCS and the RS, where the PCS contains normal or major variations, and the RS contains small variations or noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' PCA has been widely used for statistical process mon- itoring [12], [21], [22] and fault detection of multivariate data collected from chemical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' It is used to model the normal static variation from data related to normal operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' To perform fault detection, the general idea is first to build models using data collected during normal operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Then control limits are established to define normal operation regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Finally, the models and the control limits are applied to new data for online fault detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' With a PCA model, different fault detection indices such as Hotelling’s T 2 index, the SPE (or Q index) index and the combined index ϕ can be defined to monitor various aspects of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' It is important to note that these indices and the corresponding limits assume that the data samples are independent in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' 1) Hotelling’s T 2 index Hotelling’s T 2 index measures variations in the PCS, T 2 = x⊤PΛ−1P⊤x (5) where Λ is the convariance matrix of the latent scores matrix T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' It can be proven that T 2 statistic follows a F distribution, N(N − l) l(N 2 − 1)T 2 ∼ Fl,N−l (6) where Fl,N−l is an F distribution with l and N − l degrees of freedom [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' As a result, for a given confidence level α, the control limit can be calculated based on the Fl,N−l distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The index is considered normal if T 2 ≤ T 2 α ≡ l(N 2 − 1) N(N − l)Fl,N−l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='α (7) If the number of data points N is large, the T 2 index can be well approximated with a χ2 distribution with l degrees of freedom [12] and T 2 α = χ2 l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='α (8) The T 2 index measures the distance to the origin in the principal component subspace, which contains normal process variations with large variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The variation of the projection of a sample vector x on the PCS is considered normal if its T 2 index is less than the control limit T 2 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' 2) SPE (Squared Prediction Error) index The SPE index measures the projection of a sample vector x ∈ RM onto the residual space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' It is defined as the squared norm of the residual vector ˜x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' SPE(x) = ||˜x||2 = x⊤ ˜P˜P⊤x (9) The control limit of the SPE index can be derived using the result in [24], δ2 α = gχ2 h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='α (10) where g = �M i=l+1 λ2 i �M i=l+1 λi , h = (�M i=l+1 λi)2 �M i=l+1 λ2 i (11) α is confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' l is the number of PC in the principal component subspace, and λi is the ith eigenvalue of the sample convariance matrix 1 N−1X⊤X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Since the SPE index focuses on the residual sub- space, it measures the variability that breaks the static process relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' If the SPE index is above the control limit δα, it indicates a fault occurs that breaks the normal correlation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' 3) Combined index If both the T 2 index and SPE index are equally important, a global index can be used to combine the two indices, such as the combined index ϕ [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' This results in monitoring one index 3 instead of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The combined index is defined as follows, ϕ = T 2(x) + g−1SPE(x) ∼ χ2 l+h (12) where g and h come from the calculation of the SPE control limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' With α as the confidence level, the control limit of the combined index is χ2 l+h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' As a result, a fault is detected if the value of ϕ is greater than the control limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' PROPOSED HIF DETECTION PROCEDURE The occurrence of HIF introduces minor random dis- tortions in current waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' As a result, the correlation between the current measurements between different cycles will show inconsistency from the correlation structure of the measurements collected during normal load conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Therefore, the proposed fault detection procedure detects the HIF by monitoring the changes in correlation structure within the current waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The workflow of the proposed HIF detection is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Workflow of the proposed HIF detection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Data Preprocessing The proposed HIF detection method first converts the single-phase current waveform to a data matrix by sampling at the same locations within each cycle across the historical measurement of loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Let ts be the number of samples per cycle and the length of the original signal to be N × ts, the original signal can be represented as S = [s(1), s(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=', s(N×ts−1), s(N×ts)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' With M to be the number of variables and ∆ = ts/M be the gap when sampling from the original signal S, the matrix X can be written as X = � � s(1) s(1+∆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' s(ts) s(1+ts) s(1+∆+ts) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' s(2ts) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' � � (13) The resulting matrix X has M columns and N rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Since the autoencoder will be trained to reconstruct each row of the data matrix, the sampling is used to reduce the network’s input dimension and thus decrease the total number of parameters in the neural network model to prevent overfitting and improve training speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Offline Training In the offline training step, an autoencoder and a PCA model are built to characterize the correlation structures of the current waveforms of normal loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' An autoencoder model is trained to extract the normal correlation and nonlinear features from the augmented data matrix by minimizing the MSE loss in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' After the autoencoder is trained to reconstruct the data matrix formed using normal load current waveforms, it can remove common features from the data matrix, leaving small residuals for all the variables in the data matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' As a result, the autocorrelations within the input data matrix are eliminated, and the residuals only contain static variations, which can be modeled using the PCA and lend themselves to detect faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' PCA-based process monitoring techniques are applied to the reconstruction errors or the residuals of the fault-free data matrix produced by the trained autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Let ˜X be the output of the trained autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The reconstruction error of the data matrix can be written as, E = X − ˜X (14) After normalizing each column of E to have zero mean and unit variance, a PCA model can be built from the normalized reconstruction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Then the number of latent variables l can be selected based on cumulative percent variance (CPV) CPV (l) = �l i=1 λi �M i=1 λi (15) With the selected l and confidence level α, the control limits for SPE, T 2, and ϕ indices can be established using Equations 10, 8, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Online HIF Detection The trained autoencoder and PCA model are ap- plied to three phases separately for detecting the high impedance fault in new measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' For each phase, 4 Historic Load Data New Data Data Preprocessing Data Preprocessing Train AE Model Apply AE Model Reconstruction Error Reconstruction Error Build PCA Model Apply PCA Model Establish Monitor Indices Compare Monitor Indices and Limits with Established Limits Offline Training Counter and Detection Result Online Detectionafter acquiring the new current measurement of a cycle, a vector x with M variables is constructed by sampling from the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Then the new vector is passed as an input to the trained autoencoder model to get a vector of reconstruction errors e ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Since the autoen- coder is trained using data from normal loads, abnormal reconstruction errors of the vector can be observed if the occurrence of HIF distorts the correlation structure within a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' After scaling reconstruction errors e with the mean and variances calculated when building the PCA model in the offline training step, SPE, T 2, and ϕ index for the reconstruction error vector can be calculated using 9, 5, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' This study uses the ϕ index for HIF detection since it can effectively combine the SPE and T 2 indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' If the combined index of e is above the control limit calculated in the offline training phase, it indicates that there are abnormal distortions that break the normal correlation structure in the cycle corresponding to the vector x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' To account for the noise and transient disturbances in the measurements, we use a counter to record the number of cycles with indices above the control limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The counter is incremented when the combined index corresponding to one cycle exceeds the control limit and decreases if the index drops below the control limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' A trip signal is issued when the counter exceeds a predetermined threshold, which means the trip signal will be generated if the combined index consistently stays above the control limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' EVALUATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Dataset The dataset used in this study was collected during the testing and evaluation of ABB’s feeder protection system REF 550 [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The measurements of three- phase voltages and currents were collected in a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='16 kV distribution system near a hospital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' High impedance faults in phase A were stages at about 12 miles from the hospital by dropping the conductor on four different surfaces: grass, water puddle, soil, and asphalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' In ad- dition, the faults were created multiple times for each surface under different load conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' In each case, a fault was introduced at around 100 seconds and lasted for 60 seconds before the conductor was lifted off the test surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' In addition to the fault cases, measurements of normal load were recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The number of samples per cycle ts for all the measurements is 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Figure 2 shows the root mean square (RMS) current waveforms of a section of normal loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' It can be observed that the variations in the load are dynamic and complex, with the occasional presents of spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' In addition, the three phases are unbalanced with distinct patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Figure 3 shows the comparison between the current waveform of the normal load and the waveform during HIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Unlike the simulated cases used in previous publications [10], [13], the load waveform is distorted and dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' As a result, distinguishing the HIF from the normal load is more challenging since the magni- tudes of distortion in the two cases are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' RMS currents of normal load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Current waveform of the normal load and the waveform during the HIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Results and Analysis The first step in implementing the proposed fault detection procedure is to augment the current waveforms to a data matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' There are four load cases in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Three load cases containing around 580 seconds of mea- surements are used to train and validate the autoencoder model, and the last load case is left for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' First, for each phase in each load case, the univariate current 5 Phase A 350 Amplitude (A) 340 330 25 50 75 100 125 150 175 0 Time (s) Phase B Amplitude (A) 245 240 235 75 25 50 100 125 150 0 175 Time (s) Phase C Amplitude (A) 315 310 305 50 25 75 100 125 150 175 0 Time (s)Current before HIF Amplitude (A) 250 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='08 Time (s) Current during HIF 500 Amplitude (A) 250 0 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='08 Time (s)measurement is converted to a data matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Since the number of samples per cycle ts is 320, the sampling gap ∆ is selected to be 10, resulting in a data matrix consisting of 32 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' As a result, each row in the matrix corresponds to the measurement sampled from one cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Next, all the data matrices formed from all three phases in three load cases are concatenated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' After scaling each column of the concatenated data matrix to have values between 0 and 1, 80% of the data is used for training, and 20% is used for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The autoencoder model used in this study has five layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The dimension of the input and output layers are 32, and the dimension of the three hidden layers are 15, 10, and 15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The rectified linear unit (ReLU) is used as the activation function for the input and hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The model is trained using Adam optimizer in PyTorch with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='001 is used to minimize the MSE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The autoencoder model is trained for 100 Epochs with a batch size of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' After the autoencoder model is trained using the normal load data, the reconstruction errors of the training and validation data are used to build a PCA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The number of leading PCs l is selected so that the first l PCs captured 95% of the variances, and the confidence level α is chosen to be 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The autoencoder and PCA models are applied to the load and HIF cases staged on different surface types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The proposed HIF detection method is first applied to the load case that is not used during training to show that the proposed method does not generate false alarms for new load profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Figure 4 shows the combined indices and trip signals generated from a counter with a threshold of 60 for all three phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' It can be observed that most of the indices stay below the control limit, with a few outliers caused by spikes present in the current waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' As a result, no trip signal is generated for all three phases, which is expected for the normal load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Three HIF cases were staged on the grass surface when ABB tested the REF 550 for HIF Detection, and the REF 550 failed to detect one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' On the contrary, our proposed HIF detection can successfully detect all the HIF cases when the conductor of phase A contacts the grass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Figure 5 shows the detection result of the proposed method for the case that REF 550 failed to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' It can be observed that the combined index of phase A rises above the control limit after introducing HIF at around 100 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The index stays above the control limit until the conductor is lifted off the grass at around 160 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' As a result, a trip signal is generated for phase A after the index stays above the control limit longer than 60 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Phase C is also affected by the HIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' However, since the magnitude of its monitoring Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Detection result of normal load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Detection result of HIF on grass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' index is much smaller than phase A, HIF is determined to have occurred in phase A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' In addition to the tests conducted on the grass surface, four HIF cases were staged by dropping the conductor of phase A on the soil surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' When these four cases were tested, the REF 550 detected three of them, and one was not detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' To compare our proposed HIF detection method, we apply the trained autoencoder and PCA models to these four cases, and the results show that all the HIFs can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Figure 6 shows the detection result of the proposed method for the HIF case that REF 6 Load 4 3 Phase A 2 0 Trip 0 4 3 Phase B 2 0 Trip 0 4 3 0 Trip 0 25 75 100 125 150 175 0 50 Time (s)Grass 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='5 Phase A 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='5 mmhha 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='0 Trip 0 4 3 B Phase I 2 0 Trip 0 4 3 Phase ( 2 0 Trip 0 25 75 50 100 125 150 175 0 Time (s)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Detection result of HIF on soil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' 550 failed to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' It can be seen that before the fault is introduced at around 100 seconds, the indices for all three phases stay below the control limit, indicating that the current waveforms are normal and there is no fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' However, after the conductor of phase A contacts the soil, the monitoring index of phase A immediately rises and stays above the control limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' As a result, a trip signal is generated for phase A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Similar to the HIF cases staged on grass, phase C also shows minor abnormal distortions since the corresponding monitoring index oscillates around the control limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' However, the trip signal is not generated for phase C since the number of abnormal cycles does not reach the predefined threshold of 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The dataset also contains measurements of high impedance faults on asphalt and puddle filled with drink- able water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' During the testing, the REF 550 could not detect any HIF on asphalt and water puddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Like the detection results from REF 550, our proposed method cannot detect any of these cases due to the near-infinite impedance conditions of the downed conductor test and the long distance between the fault location and where the measurements were taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Figure 7 shows the detection result of one of the HIF cases on asphalt in which no trip signal is generated since all the indices stay below the control limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Even though the proposed method cannot detect the faults that occurred on near- infinite impedance surface types, no false alarms are generated during various load conditions before and after the HIFs in all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The comparison between the HIF detection results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Detection result of HIF on asphalt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' from the REF 550 and our proposed method can be summarized using the following metrics: accuracy (Acc), security (Sec), dependability (Dep), safety (Saf), and sensibility (Sen) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Acc = TP + TN TP + TN + FP + FN × 100% (16) Sec = TN TN + FP × 100% (17) Dep = TP TP + FN × 100% (18) Saf = TN TN + FN × 100% (19) Sen = TP TP + FP × 100% (20) where true positives (TP) and true negatives (TN) are the numbers of the correctly detected fault and normal load cases, and false negatives (FN) and false positives (FP) are the numbers of the wrongly detected fault and load cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' We calculate the above metrics based on the detection results for all the cases in the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The results are shown in Table IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Since the REF 550 and our proposed method can correctly identify the load conditions, they achieve 100% dependability and security, indicating they are robust to faulty tripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Furthermore, our proposed HIF detection method can correctly detect more HIF cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' As a result, our proposed HIF detection method shows improvement in the other metrics compared to the REF 550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' 7 Soil 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='5 Phase A 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='0 Trip 0 A 3 Phase B Trip 4 3 Phase C 2 0 Trip 50 100 150 200 0 Time (s)Asphalt 3 Phase A 2 0 Trip 0 4 3 Phase B 2 0 Trip 4 3 0 0 50 100 150 200 250 0 Time (s)TABLE I COMPARISON OF REF 550 AND PROPOSED HIF DETECTION METHOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Acc Sec Dep Saf Sen REF 550 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='9% 100% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='7% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='5% 100% AE + PCA 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='9% 100% 50% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='2% 100% V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' CONCLUSION This paper proposes an unsupervised HIF detection method based on the autoencoder and principal com- ponent analysis, which does not require measurements during HIFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The proposed method first converts the univariate current measurement collected from one lo- cation in the grid into a data matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The data matrix is then used to train an autoencoder for extracting nonlinear features from the data matrix and capturing the correlations among variables in the data matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' Finally, the PCA-based statistical monitoring technique is used to characterize the residuals of the normal load data from the AE model and establish thresholds based on various statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The proposed method detects high impedance faults by monitoring the deviation in the correlation structure of the augmented data matrix from the correlation learned by the AE from the normal loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content=' The proposed HIF detection method is applied to real data collected from a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAzT4oBgHgl3EQf5_4s/content/2301.01867v1.pdf'} +page_content='16 kV distribution system which contains various normal load cases and HIF cases staged on four types of surfaces: grass, water puddle, soil, and asphalt.' metadata={'source': 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Ride-hailing services have skyrocketed in popularity due to +the convenience they offer, but recent research has shown that their pric- +ing strategies can have a disparate impact on some riders, such as those +living in disadvantaged neighborhoods with a greater share of residents of +color or residents below the poverty line. Since these communities tend to +be more dependent on ride-hailing services due to lack of adequate public +transportation, it is imperative to address this inequity. To this end, this +paper presents the first thorough study on fair pricing for ride-hailing +services by devising applicable fairness measures and corresponding fair +pricing mechanisms. By providing discounts that may be subsidized by +the government, our approach results in an increased number and more +affordable rides for the disadvantaged community. Experiments on real- +world Chicago taxi data confirm our theoretical findings which provide +a basis for the government to establish fair ride-hailing policies. +Keywords: Fairness · Ride-hailing · Pricing · Government policy. +1 +Introduction +Since Uber first launched in 2009, ride-hailing companies have evolved into a +global service that has become an indelible aspect of our lives [28,27]. While +such services have had positive effects, providing consumers with more choices +and comfort, some aspects of these services might have negative social effects +[17,8,5,32]. For example, research has shown that the pricing algorithms of ride- +hailing services can lead to disparate impact: neighborhoods with a higher pro- +portion of people of color, higher levels of poverty, and younger residents are +significantly associated with higher fare prices [17]. This disparate impact is es- +pecially troubling because it further harms communities who tend to be more +dependent on these services in the first place due to poor public transporta- +tion connectivity and who have historically been systemically marginalized in a +myriad of ways. +∗ Preconference version: Not final. +∗∗ Correspondence should be directed to: nsaxena@usc.edu. +arXiv:2301.03489v1 [cs.CY] 9 Jan 2023 + +2 +N. Saxena et al. +City governments all over the country already recognize a disparity in trans- +portation options available to its disadvantaged residents, and ride-hailing apps +worsen this inequity. The City of Chicago has even made reducing inequities +in mobility for all its residents one of the five essential elements of its devel- +opment plan On To 2050 [19] which focuses on inclusive growth. Many local +governments are enacting policies to help address this injustice [10,3,25,14]. Un- +fortunately, none of these policies help alleviate the additional disparate impact +caused by the pricing mechanisms of ride-hailing companies. While they are a +step in the right direction, these policies are not enough on their own since dis- +advantaged neighborhoods do not have enough efficient public transportation +coverage [7,11,22,33], which leads to greater dependence on ride-hailing [4]. In- +deed, researchers analyzing Uber trip data for six large cities in the U.S.A. and +Europe found that 20 to 40% of ride-hailing trips had no viable public trans- +portation alternative available [4]. This work aims to provide a technique for the +government to help address the worsened inequity. Discounts proposed by our +pricing mechanisms can be covered by the government as subsidies to reduce +disparate impact for disadvantaged neighborhoods, in the same vein as other +subsidy initiatives for low-income residents for necessary services [10,3,25,14]. +Although research into pricing mechanisms for ride-hailing services is a bur- +geoning area, most work looks at improving the efficiency of matching drivers +and riders and pricing rides at times of low supply [24], or optimizing pricing to +regulate demand on the platform [31]. There is little work examining fair pricing +to reduce disparate impact. Relevantly, fares for Uber rides in USA from ma- +jor airports to hotels were found to be significantly correlated with the prices of +hotel rooms in [5], although they do not attempt to address this price discrimina- +tion in ride-hailing. Fair benefits for drivers on the ride-hailing platform has also +been investigated, but fair prices for the riders are not considered [23]. In [13], +the hypothetical scenario where ride-hailing services use personal data about +customers to determine customized prices for each customer is studied. While +this can help address a form of discrimination, it does not explicitly address the +disparate impact caused by current pricing techniques. +Despite its importance and necessity, unveiling and mitigating the disparate +impact of ride-hailing apps’ pricing mechanisms, with the goal of providing +a foundation for the government for drafting fair ride-hailing policy presents +unique challenges. First, there is a difference in affordability. Different neighbor- +hoods may have different average levels of income, and thus what may be afford- +able for a resident of one neighborhood may not be affordable for the resident of +another. A fair, equitable solution should price rides according to affordability +so that no one gets priced out. Second, there are data-related challenges since +all the data ideally needed to model the problem holistically is unavailable. For +example, metrics such as price elasticity, which in our case is the measure of +change in the number of trips with a change in the price of trips, is unknown, +as is the surge level of trips, and thus these quantities must be estimated, and +require making assumptions at times. + +Unveiling and Mitigating Bias in Ride-Hailing Pricing +3 +To address the aforementioned challenges, this paper introduces novel defini- +tions of fair pricing and corresponding fair-pricing strategies to address discrim- +ination in ride-hailing. More specifically, the main contributions of this paper +are: i) We use the concepts of price elasticity and consumer surplus from the +economics literature to explore the affordability of trips for different groups and +the relationship between pricing and the number of rides that take place to +formally define bias in ride-hailing. ii) We propose a fair-pricing strategy that +effectively reduces disparate impact and is versatile in accommodating differ- +ent requirements for multiple scenarios. iii) We use real-world ride-hailing data +from Chicago and census data to conduct several experiments comparing our +pricing mechanisms to other models commonly used for pricing, validating the +real-world utility of our method. +2 +Bias in Ride-Hailing +Biased pricing in ride-hailing results in higher prices for people who are likely +to already be struggling financially. To address this disparate impact due to +the ride-hailing services’ black-box pricing algorithms, we consider bias in ride- +hailing to be a significant difference in the average rides taken by the disadvan- +taged versus those of non-disadvantaged groups and corresponding affordability. +To formally define disadvantaged and non-disadvantaged groups, we use the +classification by the Chicago Metropolitan Agency for Planning (CMAP) which +identifies census tracts in the Chicago region, called Economically Disconnected +Areas (EDAs) that are disconnected from economic growth and prosperity, and +may be experiencing disinvestment [18]. In other words, EDAs are neighborhoods +with high concentrations of low-income households and minorities or households +with limited English proficiency speakers. Approximately one-third of the city +lives in an EDA [18]. In the remainder of this paper, we interchangeably use +the terms EDA regions and disadvantaged neighborhoods as well as non-EDA +regions and non-disadvantaged neighborhoods for readability. We further define +EDA-trips to denote trips (or rides) that either begin or end in an EDA region. +Non-EDA-trips denotes trips (or rides) that do not. +3 +Unveiling Ride-Hailing Bias +3.1 +Relative Rideability +Aligned with the p%-rule [2] used by the U.S. Equal Employment Opportunity +Commission (EEOC) to evaluate disparate impact, we introduce the Relative +Rideability (R2) score to quantify the previously discussed bias in ride-hailing +shown as the significant difference in the average rides across different groups. +Specifically, the p%-rule states that if the selection rate for a certain group is +less than p% of the selection rate for the group with the highest selection rate, +then there is a substantially different rate of selection and may be considered +disparate impact. Analogously, R2 can be mathematically defined as below: + +4 +N. Saxena et al. +R2 = +min{d1, . . . , di} +max{¬d1, . . . , ¬dj} +(1) +where di is the average number of trips by residents of disadvantaged group i, +while ¬dj is the average number of trips by residents of non-disadvantaged group +j. With ¬dj commonly greater than di in inequitable ride-hailing services, the +higher the R2 the fairer the model. +3.2 +Affordability +Next we look at another way of quantifying bias in ride-hailing: difference in ride +affordability. A measure of affordability can be captured by consumer surplus +[30], a concept from economics, that captures the difference between a consumer’s +willingness to pay for a certain product or service, and the price of that product +or service. If the price of the product or service is less than the amount the +customer is willing to pay for it, then the consumer surplus is positive. Otherwise, +it is negative, which can happen if the good or service is necessary (e.g., food, +or life-saving medication). In other words, the higher the consumer surplus, the +more easily the consumer can afford that product or service. +However, the information needed to quantify consumer surplus in ride-hailing, +such as the number of people who looked at the quoted price but did not make a +request and what that price was, is unavailable in publicly released ride-hailing +data. We circumvent this challenge by computing price elasticity (Ep) instead, +which measures the change in demand of a product or service with respect to a +change in its price, to obtain an estimation of consumer surplus: +Ep = δQ +δP +(2) +where δQ is the percentage change in the quantity of the product or service +demanded, and δP is the percentage change in its price. As a note, price elasticity +will also be used in our following fair pricing mechanisms to estimate how the +number of trips might change with a change in price. +Price elasticity is considered at a point where there is a change in the price of +a good or service. However, ride-hailing trips are not assigned a flat per-mile rate, +thus simply studying trips at different prices is not ideal since the difference in +price of two trips might be caused by multiple factors. We therefore study similar +trips that were shown different prices due to the way ride-hailing platforms com- +pute surge level, a multiplier that is used to multiply and increase the estimated +price of a ride at the time of high demand or low supply. In addition, ride-hailing +platforms typically compute a continuous surge level for rides, but show a dis- +crete value to consumers for simplicity and ease of understanding. For example, +a trip for which a surge level was computed as 1.449 will result in a discretized +surge of 1.4x, whereas a surge level of 1.451x will result in a discretized surge +of 1.5x. We make use of regression discontinuity design around these discretized + +Unveiling and Mitigating Bias in Ride-Hailing Pricing +5 +surge points to estimate price elasticities when surge levels go from 1.2x to 1.3x, +1.3x to 1.4x, 1.4x to 1.5x, and so forth. We run a linear probability model, a +special case of ordinary least squares regression, that is commonly used in eco- +nomics in which the outcome variable is binary, and the dependent variables may +be binary or continuous. We fit the linear probability model regression below for +each surge discontinuity, with Ride indicating whether a particular request leads +to a trip, and include all trips on either side of the surge discontinuity: +Ride = β0+(α∗i1∗i2)+(β1∗i1)+(β2∗(1−i1)∗i2)+(β3(1−i2)∗x1)+(β4∗i2∗x1)+ϵ +(3) +where α indicates the drop in rides around a discontinuity, i1 is a decision variable +indicating whether the surge of that particular trip lies within 0.01 of a surge +price discontinuity, i2 is a decision variable that denotes whether the surge for +this particular trip is to the right of the price discontinuity (i.e. its discretized +surge level is higher than the discretized surge point, thus a trip with a continuous +surge level of 1.451 discretized to 1.5x surge will have a value of 1, whereas 1.449 +discretized to 1.4x surge will have a value of 0), and x1 is the actual (non- +discretized) surge value, and the βs are the coefficients, and ϵ the error. To +compute this at each jump level, we make use of trips that are on either side +of the discontinuity. Since we compute this for each surge discontinuity, α helps +capture the change in number of trips due to change in price because of the +surge level. Our calculation of consumer surplus and price elasticity is taken +from the approach detailed in an economics paper by [6] to compute consumer +surplus for Uber across four major markets in the United States. To sum up, +price elasticities are first estimated at different surge levels using the regression +equation we applied in Equation 3, and then we utilize those price elasticities +for computing consumer surplus. +This regression discontinuity design we compute at each price discontinuity +(Equation 3) helps estimate α, which indicates the change in the number of trips +that occur at that discontinuity due to the change in price. We can then make +use of this α to compute price elasticities for price discontinuities as below: +Ep = δQ +δP += +α +Np +δP +(4) +where Np is the proportion of trips that occur at a particular price p. +At this point we run into another data-related challenge: surge levels are +unavailable in ride-hailing data. To get around this, we make use of RANdom +SAmple Consensus (RANSAC) regression to determine when surge pricing was in +effect and what the surge levels were. When data contains outliers, RANSAC can +be used for the robust estimation of model parameters from a subset of ‘inliers’ +(i.e. the data points that are not outliers) from the dataset. The intuition behind +using RANSAC is as follows. According to Uber, surge pricing goes into effect +when there are an unusually large number of people requesting trips at the same +time [26]. In other words, surge pricing occurs when a non-standard or much +higher than normal number of customers try to book trips simultaneously. If all + +6 +N. Saxena et al. +trips that take place at 1.0x (no surge) are standard trips, or inliers, then trips +that occur at surge pricing would be considered the outliers. Once we have the +surge level in effect for each ride (1.0x or higher), we can use rides on either side +of a surge level in the regression continuity design to estimate price elasticities. +Algorithm 1 Affordability +Input: Ride-hailing data (d) for a group +Output: Total consumer surplus for the group +1: Compute surge level of rides: Surge ← RANSAC(d) +2: for surge level s ← 1.1x to N do +3: +Compute regression discontinuity design around s (Equation 3) +4: +Compute Ep ← +α +Np +δP at s (Equation 4) +5: end for +6: for trips at surge s ← 1.0x to N do +7: +consumer surplus ∆c ← �N +i=s+0.1x Epi ∗ numTripsi ∗ (( i−s +s ) ∗ 100) ∗ avgFares +8: end for +To reiterate, consumer surplus is the difference between the price a consumer +is willing to pay for a product or service, and the price they are actually charged. +Algorithm 1 details its sketch. Specifically, we look at ride-hailing data for a +group (EDA or non-EDA residents), and first estimate trips’ surge levels (line +1). Next, we compute a regression discontinuity design around each successive +surge level, and then estimate price elasticity at that surge level (lines 2-5). +When considering trips that take place at 1.0x surge (i.e., no surge), we take the +price elasticity at the next surge level, 1.1x, and calculate the number of trips +that would have happened if the customers shown 1.0x surge had been shown +a 1.1x surge instead. We then multiply this number of potential trips with the +difference in fare (which is 10% in this case) and the average price actually paid +at 1.0x surge. This is the surplus for up to surge 1.1x for those riders that were +charged 1.0x surge. We then replicate this for each pair of successive surge levels, +and sum it all up to arrive at the total consumer surplus estimate (lines 6-8). +Finally, Algorithm 1 outputs consumer surplus for trips for the group. +4 +Fair Pricing Mechanisms +With the tailored fairness metrics for ride-hailing, we now detail fair pricing +mechanisms to help reduce disparate impact on disadvantaged communities. We +focus our efforts on pricing EDA trips to help address the disparate impact on +EDA residents which can help the City of Chicago in addressing concerns about +mobility inequity for its residents [19]. In addition, to allow for flexibility when +accounting for different situations, we define two variants of our pricing mecha- +nism: variable discounting and fixed discounting. With variable discounting, the +discount given to EDA-trips may vary depending on many factors, such as time + +Unveiling and Mitigating Bias in Ride-Hailing Pricing +7 +of day, demand, etc., while all EDA-trips receive the same discount (e.g., 15%) +with the alternative fixed discounting. Such a dual approach reflects the needs of +the platform and/or the government’s policy-making. Specifically, the platform +might require the fixed discounting route because that may be easier and more +straightforward to implement; on the other hand, the government may think it +is justified to give different trips different discounts depending on the conditions +at the time, and the increased complexity that may come with implementing +such a system is not a major concern. +4.1 +Variable discounting: FairRide +We first propose a new pricing mechanism called FairRide. The intuition behind +FairRide is that a pricing mechanism should take different riders’ differing ability +to afford rides into account in order to reduce disparate impact on the most +disadvantaged in society. To that effect, we propose a mechanism which prices +rides for EDA-regions separately. We focus on EDA-trips, and run a multiple +regression model to determine pricing only for rides that begin or end in an +EDA (EDA-trips). Considering only trips by EDA residents leads to rides to +be priced in accordance with the riders’ ability to afford them. In Section 4 +we compare FairRide with machine learning models commonly employed for +pricing in the literature [9,21,15,16,29,1], and find that looking only at EDA- +trips leads to most models resulting in more trips for EDA residents than the +current pricing mechanism, and lead to a higher relative rideability (R2), and +FairRide outperforms them all. A naive baseline of simply applying a $5 discount +on all rides is also implemented. +4.2 +Fixed discounting: FixedFairRide +The second fair pricing mechanism, called FixedFairRide, solves an optimization +problem to maximize EDA-trips while maintaining or increasing revenue in a +manner that offers a consistent, fixed, discount (δ) to all EDA-trips. In other +words, we determine a fixed amount to discount EDA-trips by to ensure a fixed +discounting policy. Our overarching goal is to determine the optimal value, δ, for +discounting EDA-trips such that relative rideability increases and the number +of EDA trips is maximized. This can be mathematically formulated as: +η = max +δ +� +t +�� +l +N(δ ∗ p) +� +(5) +where η represents the total number of rides, t denotes a particular time period, +l represents EDA location pairs, δ denotes the discount for EDA-trips, p is the +price of the EDA trip, and N is the number of EDA trips at that price. +Now we introduce two constraints to this optimization according to two pos- +sible scenarios. In the first setting, the government covers the discount for EDA +residents as a subsidy. Thus, the platform and the driver receive the same amount + +8 +N. Saxena et al. +of revenue they receive currently from a ride, but the EDA resident would get +a discount from the original price that will be topped up by the government +subsidy. Here we do not need to consider revenue, since the price charged by +the platform does not change. Nor do we need to consider the feasibility of +the trip for the driver, since the driver still receives the same exact fare as the +pre-discount price. However, the government may wish to set a ceiling for the +discount they offer (say, e.g., 30%), which can be a constraint for this setting: +0 < δ < n +(6) +where n is the maximum discount the government is willing to subsidize. +In the second setting, we assume the platform offers to cover the discount +themselves to help address the disparate impact. However, the platform would +likely prefer the total revenue to not drop. Thus, the alternative constraint is: +Revenue ≥ r, +(7) +0 < δ < 1 +(8) +where r denotes the total revenue by current pricing, while δ must be greater +than 0 indicating the discount for EDA-trips is always positive. In addition, we +observe that we can increase revenue beyond r simply by adding a tremendous +amount of new, heavily discounted trips. However, below a certain price, a trip +may not be worth it for the driver since they may not earn enough from it to +cover their costs and make a profit. To consider the feasibility of the trip for the +driver, we enforce another floor for revenue so that the driver is not in danger +of not earning anything from the trip and the floored constraint thus becomes: +Revenue ≥ max(r, η ∗ pmin) +(9) +where r is the total revenue with current pricing, η the total number of rides. +pmin is the minimum price for a trip which can be a function of factors like travel +time, distance, demand, etc, deemed important by the platform for a trip. +The discontinuities in the price due to surge levels make this a non-convex +problem, and not straightforward to solve. We therefore employ grid search [12] +while varying values for δ. +5 +Experimental Evaluation +In this section, we use the real-world data for the City of Chicago to run ex- +periments. We use this dataset since this is, as far as we are aware, the only +publicly available dataset that contains price (or fare) information for trips, +which is necessary to address bias in pricing in ride-hailing [20]. The data from +all ride-hailing platforms that operate within the city of Chicago are aggregated, +standardized, and anonymized before release. Which ride-hailing platform ser- +viced a trip is not identified. For each trip, the dataset provides the trip start (or +‘pick-up’) time, trip end (or ‘drop-off’) time, each rounded off to the nearest 15 + +Unveiling and Mitigating Bias in Ride-Hailing Pricing +9 +minutes; trip pick-up and drop-off locations at the level of Chicago census tract +or community area; duration of the trip in seconds; and the fare of trip, rounded +to the nearest $2.50. Locations for pick-ups or drop-offs outside the city limits of +Chicago are unavailable. Drivers and riders are not identified. We look at trips +from January to October 2021, and focus our attention on trips that were not +authorized as ‘shared.’ Since the Covid-19 pandemic began in early 2020, most +ride-hailing services disabled the option for shared rides. +Finally, we perform a spatial-join between the pick-up and drop-off coordi- +nates of this dataset with a spatial dataset released by the City of Chicago that +identifies EDA regions in Chicago, to help identify trips within our data that +begin or end in EDA- or non-EDA-regions. +5.1 +The Profound Bias +We use the metrics we proposed in Section 4 to unveil the intrinsic discrimination +of the ride-hailing companies’ current pricing mechanisms. We find that R2 = +0.33. Such a low R2 indicates the pervasiveness of real bias, and the extent of +disparate impact it could cause on disadvantaged communities. We also observe +below a stark difference in consumer surplus and affordability. +Next we look at affordability via consumer surplus. We first use RANSAC +regression to estimate when surge pricing was in effect, and the surge levels for +rides. The general trend can be observed in Figure 1a: generally, as the surge +multiplier increases, the number of trips that occur decrease. While we do not +have ground truth for surge levels to measure how accurate the predictions from +our model are, the trend we observe is in line with the surge level trends in [6], +who had access to ground truth as the work was in conjunction with Uber. We +can also make the following observations from Figure 1a. First, a greater share +of EDA-trips take place at lower surge levels as compared to non-EDA-trips. +Second, the number of EDA-trips falls below 1,000 at surge level 4.4x, when the +average trip price is $62.06. But for non-EDA-trips, it takes until surge level 5.9x +for the number of trips to fall below 1,000 trips, at an average price of $97.02. +Now that we know the surge levels for trips, we use the equations from +Section 3.2 to calculate price elasticities and consumer surplus. Price elasticity +is negative for both EDA-trips and non-EDA-trips, except in cases when surge +levels are very high (> 8.0x) and with very few rides (typically <100). This +indicates that as prices increase, demand typically decreases, and vice-versa. We +find that the consumer surplus of non-EDA-trips is much higher than that of +EDA-trips. Examining only surge levels with a reasonable number of rides (at the +highest surge levels we observe less than a 100 rides at each level, so we do not +include these in the following figures), the total consumer surplus for non-EDA- +trips is $17,972,629.829, while the surplus for EDA-trips is $2,326,555.876 over +the 10 month period we look at (Figure 1b). On dividing by the total number +of rides serviced in each category, we get an average consumer surplus of $67.11 +for non-EDA-trips, and an average consumer surplus of $36.76 for EDA-trips. +In 2016, [6] estimated that the consumer surplus for UberX across the entire +United States in the year 2015 was $6.8 billion, thus one day’s consumer surplus +across all cities in the US by their estimates would be $18 million. Further, these +calculations are for 7 years ago, and they would likely be higher now. Compared + +10 +N. Saxena et al. +(a) Share of trips at surge levels for the +first ten months of 2021. +(b) Consumer Surplus for trips. +Fig. 1: Surge levels and affordability (∆c). +to these figures, and considering that Chicago is a major city in the United +States, our estimates are likely lower than the true values. Thus riders beginning +or ending trips in non-EDA areas are able to afford ride-hailing services far more +easily than riders that begin or end trips in EDA neighborhoods. +5.2 +Pricing Mechanisms +FairRide We compare FairRide to machine learning models used for pricing +[9,21,15,16,29,1]. We also compare against a naive baseline, which is a discount +of $5 applied to all EDA rides. +We find that training on data organized by region leads to lower prices for +approximately 35.7% of EDA-trips, and a 35.59% increase in rides for the period +we examine in 2021 for rides beginning or ending in an EDA. In other words, as +rides become more affordable, there are an additional 1,803,514 rides that begin +or end in an EDA. As a result, with FairRide Relative Rideability (R2) increases +by 35.6% to 0.457, and affordability (∆c) increases by 22.5% to $2,850,552.77. +FixedFairRide We then test FixedFairRide with different values of pmin (the +minimum price for a trip) and observe that as pmin increases, the value of δ (the +trip discount) decreases (Table 2). +Thus, as the minimum price per trip increases, the lower the discount will be. +Each of these values of pmin results in more EDA-trips than the current pricing +mechanism, all also lead to more trips than FairRide up till pmin = 15, which +results in 6,824,375 trips. There is a significant increase in Relative Rideability +(R2) as well as affordability (∆c or consumer surplus) for EDA residents. In the +dataset we use for our experiments the lowest fare value is $2.50, and the highest +proportion of trips (approximately 14%) happen around $10. Thus it is likely a +minimum trip price much higher than $10 will not be practical in the real world. + +EDA trips + Non-EDA trips +20 +15 +f Trips +Percentage +5 +0 +Surge level20,000,000 +15,000,000 +(in USD) +10,000,000 +Value ( +5,000,000 +0 +EDA trips +Non-EDA tripsUnveiling and Mitigating Bias in Ride-Hailing Pricing +11 +Table 1: Number of trips, R2, and affordability from different models. +Model +η (% change) +R2 (% change) +∆c (% change) +Original +5,066,849 +0.337 +2,326,555.87 +FairRide +6,870,363 +0.457 (+35.60) 2,850,552.77 (+22.52) +Mohd et al. (2020) [16] +6,867,535 +0.456 (+35.31) +2,722,461.33 (+17.01) +Miao (2017) [15] +6,868,152 +0.456 (+35.31) +2,818,648.50 (+21.15) +Wolk (2020) [29] +6,864,923 +0.456 (+35.31) +2,534,351.94 (+8.93) +Baseline (-$5) +6,618,494 +0.440 (+30.56) +2,739,784.93 (+17.76) +Gu et al. (2020) [9] +6,843,205 +0.455 (+35.01) +2,612,239.28 (+12.27) +Rathan et al. (2019) [21] +6,385,843 +0.424 (+25.81) +2,667,079.84 (+14.63) +Alkhatib et al. (2013) [1] +6,608,789 +0.439 (+30.26) +2,476,128.68 (+6.42) +FixedFairRide(pmin = 12) +8,142,519 +0.541 (+60.53) 3,201,963.73 (+37.62) +Table 2: Discount δ different minimum trip price pmin. +pmin +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +δ +0.77 0.72 0.67 0.62 0.57 0.52 0.47 0.42 0.36 0.30 0.24 +6 +Conclusion +Although a number of studies in the recent past have explored pricing for ride- +hailing services, including looking at fair compensation for drivers, and insurance +policy as a safeguard against ride-hailing using personal information for pricing, +none have looked at fairer pricing to reduce disparate impact on disadvantaged +residents. We proposed applicable fairness metrics to unveil the intrinsic bias in +ride-hailing along with a flexible pricing mechanism to price rides more fairly +and make trips more affordable for disadvantaged residents who are under-served +by public transportation. 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In: Proceedings of +the AAAI Conference on Artificial Intelligence. vol. 36, pp. 12235–12243 (2022) + diff --git a/s9E1T4oBgHgl3EQf3gUP/content/tmp_files/load_file.txt b/s9E1T4oBgHgl3EQf3gUP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c797926090933513067a90314bafc85a3cad8ce4 --- /dev/null +++ b/s9E1T4oBgHgl3EQf3gUP/content/tmp_files/load_file.txt @@ -0,0 +1,529 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf,len=528 +page_content='Unveiling and Mitigating Bias in Ride-Hailing Pricing for Equitable Policy Making∗ Nripsuta Ani Saxena ∗∗,1, Wenbin Zhang2, and Cyrus Shahabi1 1 University of Southern California, Los Angeles, CA 90089, USA 2 Michigan Technological University, Houghton, MI 49931, USA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Ride-hailing services have skyrocketed in popularity due to the convenience they offer, but recent research has shown that their pric- ing strategies can have a disparate impact on some riders, such as those living in disadvantaged neighborhoods with a greater share of residents of color or residents below the poverty line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Since these communities tend to be more dependent on ride-hailing services due to lack of adequate public transportation, it is imperative to address this inequity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' To this end, this paper presents the first thorough study on fair pricing for ride-hailing services by devising applicable fairness measures and corresponding fair pricing mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' By providing discounts that may be subsidized by the government, our approach results in an increased number and more affordable rides for the disadvantaged community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Experiments on real- world Chicago taxi data confirm our theoretical findings which provide a basis for the government to establish fair ride-hailing policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Keywords: Fairness · Ride-hailing · Pricing · Government policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' 1 Introduction Since Uber first launched in 2009, ride-hailing companies have evolved into a global service that has become an indelible aspect of our lives [28,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' While such services have had positive effects, providing consumers with more choices and comfort, some aspects of these services might have negative social effects [17,8,5,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' For example, research has shown that the pricing algorithms of ride- hailing services can lead to disparate impact: neighborhoods with a higher pro- portion of people of color, higher levels of poverty, and younger residents are significantly associated with higher fare prices [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' This disparate impact is es- pecially troubling because it further harms communities who tend to be more dependent on these services in the first place due to poor public transporta- tion connectivity and who have historically been systemically marginalized in a myriad of ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' ∗ Preconference version: Not final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' ∗∗ Correspondence should be directed to: nsaxena@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='03489v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='CY] 9 Jan 2023 2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Saxena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' City governments all over the country already recognize a disparity in trans- portation options available to its disadvantaged residents, and ride-hailing apps worsen this inequity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' The City of Chicago has even made reducing inequities in mobility for all its residents one of the five essential elements of its devel- opment plan On To 2050 [19] which focuses on inclusive growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Many local governments are enacting policies to help address this injustice [10,3,25,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Un- fortunately, none of these policies help alleviate the additional disparate impact caused by the pricing mechanisms of ride-hailing companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' While they are a step in the right direction, these policies are not enough on their own since dis- advantaged neighborhoods do not have enough efficient public transportation coverage [7,11,22,33], which leads to greater dependence on ride-hailing [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In- deed, researchers analyzing Uber trip data for six large cities in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' and Europe found that 20 to 40% of ride-hailing trips had no viable public trans- portation alternative available [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' This work aims to provide a technique for the government to help address the worsened inequity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Discounts proposed by our pricing mechanisms can be covered by the government as subsidies to reduce disparate impact for disadvantaged neighborhoods, in the same vein as other subsidy initiatives for low-income residents for necessary services [10,3,25,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Although research into pricing mechanisms for ride-hailing services is a bur- geoning area, most work looks at improving the efficiency of matching drivers and riders and pricing rides at times of low supply [24], or optimizing pricing to regulate demand on the platform [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' There is little work examining fair pricing to reduce disparate impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Relevantly, fares for Uber rides in USA from ma- jor airports to hotels were found to be significantly correlated with the prices of hotel rooms in [5], although they do not attempt to address this price discrimina- tion in ride-hailing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Fair benefits for drivers on the ride-hailing platform has also been investigated, but fair prices for the riders are not considered [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In [13], the hypothetical scenario where ride-hailing services use personal data about customers to determine customized prices for each customer is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' While this can help address a form of discrimination, it does not explicitly address the disparate impact caused by current pricing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Despite its importance and necessity, unveiling and mitigating the disparate impact of ride-hailing apps’ pricing mechanisms, with the goal of providing a foundation for the government for drafting fair ride-hailing policy presents unique challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' First, there is a difference in affordability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Different neighbor- hoods may have different average levels of income, and thus what may be afford- able for a resident of one neighborhood may not be affordable for the resident of another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' A fair, equitable solution should price rides according to affordability so that no one gets priced out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Second, there are data-related challenges since all the data ideally needed to model the problem holistically is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' For example, metrics such as price elasticity, which in our case is the measure of change in the number of trips with a change in the price of trips, is unknown, as is the surge level of trips, and thus these quantities must be estimated, and require making assumptions at times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Unveiling and Mitigating Bias in Ride-Hailing Pricing 3 To address the aforementioned challenges, this paper introduces novel defini- tions of fair pricing and corresponding fair-pricing strategies to address discrim- ination in ride-hailing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' More specifically, the main contributions of this paper are: i) We use the concepts of price elasticity and consumer surplus from the economics literature to explore the affordability of trips for different groups and the relationship between pricing and the number of rides that take place to formally define bias in ride-hailing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' ii) We propose a fair-pricing strategy that effectively reduces disparate impact and is versatile in accommodating differ- ent requirements for multiple scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' iii) We use real-world ride-hailing data from Chicago and census data to conduct several experiments comparing our pricing mechanisms to other models commonly used for pricing, validating the real-world utility of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' 2 Bias in Ride-Hailing Biased pricing in ride-hailing results in higher prices for people who are likely to already be struggling financially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' To address this disparate impact due to the ride-hailing services’ black-box pricing algorithms, we consider bias in ride- hailing to be a significant difference in the average rides taken by the disadvan- taged versus those of non-disadvantaged groups and corresponding affordability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' To formally define disadvantaged and non-disadvantaged groups, we use the classification by the Chicago Metropolitan Agency for Planning (CMAP) which identifies census tracts in the Chicago region, called Economically Disconnected Areas (EDAs) that are disconnected from economic growth and prosperity, and may be experiencing disinvestment [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In other words, EDAs are neighborhoods with high concentrations of low-income households and minorities or households with limited English proficiency speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Approximately one-third of the city lives in an EDA [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In the remainder of this paper, we interchangeably use the terms EDA regions and disadvantaged neighborhoods as well as non-EDA regions and non-disadvantaged neighborhoods for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We further define EDA-trips to denote trips (or rides) that either begin or end in an EDA region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Non-EDA-trips denotes trips (or rides) that do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' 3 Unveiling Ride-Hailing Bias 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='1 Relative Rideability Aligned with the p%-rule [2] used by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Equal Employment Opportunity Commission (EEOC) to evaluate disparate impact, we introduce the Relative Rideability (R2) score to quantify the previously discussed bias in ride-hailing shown as the significant difference in the average rides across different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Specifically, the p%-rule states that if the selection rate for a certain group is less than p% of the selection rate for the group with the highest selection rate, then there is a substantially different rate of selection and may be considered disparate impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Analogously, R2 can be mathematically defined as below: 4 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Saxena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' R2 = min{d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' , di} max{¬d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' , ¬dj} (1) where di is the average number of trips by residents of disadvantaged group i, while ¬dj is the average number of trips by residents of non-disadvantaged group j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' With ¬dj commonly greater than di in inequitable ride-hailing services, the higher the R2 the fairer the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='2 Affordability Next we look at another way of quantifying bias in ride-hailing: difference in ride affordability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' A measure of affordability can be captured by consumer surplus [30], a concept from economics, that captures the difference between a consumer’s willingness to pay for a certain product or service, and the price of that product or service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' If the price of the product or service is less than the amount the customer is willing to pay for it, then the consumer surplus is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Otherwise, it is negative, which can happen if the good or service is necessary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=', food, or life-saving medication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In other words, the higher the consumer surplus, the more easily the consumer can afford that product or service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' However, the information needed to quantify consumer surplus in ride-hailing, such as the number of people who looked at the quoted price but did not make a request and what that price was, is unavailable in publicly released ride-hailing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We circumvent this challenge by computing price elasticity (Ep) instead, which measures the change in demand of a product or service with respect to a change in its price, to obtain an estimation of consumer surplus: Ep = δQ δP (2) where δQ is the percentage change in the quantity of the product or service demanded, and δP is the percentage change in its price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' As a note, price elasticity will also be used in our following fair pricing mechanisms to estimate how the number of trips might change with a change in price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Price elasticity is considered at a point where there is a change in the price of a good or service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' However, ride-hailing trips are not assigned a flat per-mile rate, thus simply studying trips at different prices is not ideal since the difference in price of two trips might be caused by multiple factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We therefore study similar trips that were shown different prices due to the way ride-hailing platforms com- pute surge level, a multiplier that is used to multiply and increase the estimated price of a ride at the time of high demand or low supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In addition, ride-hailing platforms typically compute a continuous surge level for rides, but show a dis- crete value to consumers for simplicity and ease of understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' For example, a trip for which a surge level was computed as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='449 will result in a discretized surge of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='4x, whereas a surge level of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='451x will result in a discretized surge of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='5x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We make use of regression discontinuity design around these discretized Unveiling and Mitigating Bias in Ride-Hailing Pricing 5 surge points to estimate price elasticities when surge levels go from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='2x to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='3x, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='3x to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='4x, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='4x to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='5x, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We run a linear probability model, a special case of ordinary least squares regression, that is commonly used in eco- nomics in which the outcome variable is binary, and the dependent variables may be binary or continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We fit the linear probability model regression below for each surge discontinuity, with Ride indicating whether a particular request leads to a trip, and include all trips on either side of the surge discontinuity: Ride = β0+(α∗i1∗i2)+(β1∗i1)+(β2∗(1−i1)∗i2)+(β3(1−i2)∗x1)+(β4∗i2∗x1)+ϵ (3) where α indicates the drop in rides around a discontinuity, i1 is a decision variable indicating whether the surge of that particular trip lies within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='01 of a surge price discontinuity, i2 is a decision variable that denotes whether the surge for this particular trip is to the right of the price discontinuity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' its discretized surge level is higher than the discretized surge point, thus a trip with a continuous surge level of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='451 discretized to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='5x surge will have a value of 1, whereas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='449 discretized to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='4x surge will have a value of 0), and x1 is the actual (non- discretized) surge value, and the βs are the coefficients, and ϵ the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' To compute this at each jump level, we make use of trips that are on either side of the discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Since we compute this for each surge discontinuity, α helps capture the change in number of trips due to change in price because of the surge level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Our calculation of consumer surplus and price elasticity is taken from the approach detailed in an economics paper by [6] to compute consumer surplus for Uber across four major markets in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' To sum up, price elasticities are first estimated at different surge levels using the regression equation we applied in Equation 3, and then we utilize those price elasticities for computing consumer surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' This regression discontinuity design we compute at each price discontinuity (Equation 3) helps estimate α, which indicates the change in the number of trips that occur at that discontinuity due to the change in price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We can then make use of this α to compute price elasticities for price discontinuities as below: Ep = δQ δP = α Np δP (4) where Np is the proportion of trips that occur at a particular price p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' At this point we run into another data-related challenge: surge levels are unavailable in ride-hailing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' To get around this, we make use of RANdom SAmple Consensus (RANSAC) regression to determine when surge pricing was in effect and what the surge levels were.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' When data contains outliers, RANSAC can be used for the robust estimation of model parameters from a subset of ‘inliers’ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' the data points that are not outliers) from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' The intuition behind using RANSAC is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' According to Uber, surge pricing goes into effect when there are an unusually large number of people requesting trips at the same time [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In other words, surge pricing occurs when a non-standard or much higher than normal number of customers try to book trips simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' If all 6 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Saxena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' trips that take place at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='0x (no surge) are standard trips, or inliers, then trips that occur at surge pricing would be considered the outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Once we have the surge level in effect for each ride (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='0x or higher), we can use rides on either side of a surge level in the regression continuity design to estimate price elasticities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Algorithm 1 Affordability Input: Ride-hailing data (d) for a group Output: Total consumer surplus for the group 1: Compute surge level of rides: Surge ← RANSAC(d) 2: for surge level s ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='1x to N do 3: Compute regression discontinuity design around s (Equation 3) 4: Compute Ep ← α Np δP at s (Equation 4) 5: end for 6: for trips at surge s ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='0x to N do 7: consumer surplus ∆c ← �N i=s+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='1x Epi ∗ numTripsi ∗ (( i−s s ) ∗ 100) ∗ avgFares 8: end for To reiterate, consumer surplus is the difference between the price a consumer is willing to pay for a product or service, and the price they are actually charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Algorithm 1 details its sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Specifically, we look at ride-hailing data for a group (EDA or non-EDA residents), and first estimate trips’ surge levels (line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Next, we compute a regression discontinuity design around each successive surge level, and then estimate price elasticity at that surge level (lines 2-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' When considering trips that take place at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='0x surge (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=', no surge), we take the price elasticity at the next surge level, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='1x, and calculate the number of trips that would have happened if the customers shown 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='0x surge had been shown a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='1x surge instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We then multiply this number of potential trips with the difference in fare (which is 10% in this case) and the average price actually paid at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='0x surge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' This is the surplus for up to surge 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='1x for those riders that were charged 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='0x surge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We then replicate this for each pair of successive surge levels, and sum it all up to arrive at the total consumer surplus estimate (lines 6-8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Finally, Algorithm 1 outputs consumer surplus for trips for the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' 4 Fair Pricing Mechanisms With the tailored fairness metrics for ride-hailing, we now detail fair pricing mechanisms to help reduce disparate impact on disadvantaged communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We focus our efforts on pricing EDA trips to help address the disparate impact on EDA residents which can help the City of Chicago in addressing concerns about mobility inequity for its residents [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In addition, to allow for flexibility when accounting for different situations, we define two variants of our pricing mecha- nism: variable discounting and fixed discounting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' With variable discounting, the discount given to EDA-trips may vary depending on many factors, such as time Unveiling and Mitigating Bias in Ride-Hailing Pricing 7 of day, demand, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=', while all EDA-trips receive the same discount (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=', 15%) with the alternative fixed discounting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Such a dual approach reflects the needs of the platform and/or the government’s policy-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Specifically, the platform might require the fixed discounting route because that may be easier and more straightforward to implement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' on the other hand, the government may think it is justified to give different trips different discounts depending on the conditions at the time, and the increased complexity that may come with implementing such a system is not a major concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='1 Variable discounting: FairRide We first propose a new pricing mechanism called FairRide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' The intuition behind FairRide is that a pricing mechanism should take different riders’ differing ability to afford rides into account in order to reduce disparate impact on the most disadvantaged in society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' To that effect, we propose a mechanism which prices rides for EDA-regions separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We focus on EDA-trips, and run a multiple regression model to determine pricing only for rides that begin or end in an EDA (EDA-trips).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Considering only trips by EDA residents leads to rides to be priced in accordance with the riders’ ability to afford them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In Section 4 we compare FairRide with machine learning models commonly employed for pricing in the literature [9,21,15,16,29,1], and find that looking only at EDA- trips leads to most models resulting in more trips for EDA residents than the current pricing mechanism, and lead to a higher relative rideability (R2), and FairRide outperforms them all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' A naive baseline of simply applying a $5 discount on all rides is also implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='2 Fixed discounting: FixedFairRide The second fair pricing mechanism, called FixedFairRide, solves an optimization problem to maximize EDA-trips while maintaining or increasing revenue in a manner that offers a consistent, fixed, discount (δ) to all EDA-trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In other words, we determine a fixed amount to discount EDA-trips by to ensure a fixed discounting policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Our overarching goal is to determine the optimal value, δ, for discounting EDA-trips such that relative rideability increases and the number of EDA trips is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' This can be mathematically formulated as: η = max δ � t �� l N(δ ∗ p) � (5) where η represents the total number of rides, t denotes a particular time period, l represents EDA location pairs, δ denotes the discount for EDA-trips, p is the price of the EDA trip, and N is the number of EDA trips at that price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Now we introduce two constraints to this optimization according to two pos- sible scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In the first setting, the government covers the discount for EDA residents as a subsidy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Thus, the platform and the driver receive the same amount 8 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Saxena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' of revenue they receive currently from a ride, but the EDA resident would get a discount from the original price that will be topped up by the government subsidy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Here we do not need to consider revenue, since the price charged by the platform does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Nor do we need to consider the feasibility of the trip for the driver, since the driver still receives the same exact fare as the pre-discount price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' However, the government may wish to set a ceiling for the discount they offer (say, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=', 30%), which can be a constraint for this setting: 0 < δ < n (6) where n is the maximum discount the government is willing to subsidize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In the second setting, we assume the platform offers to cover the discount themselves to help address the disparate impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' However, the platform would likely prefer the total revenue to not drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Thus, the alternative constraint is: Revenue ≥ r, (7) 0 < δ < 1 (8) where r denotes the total revenue by current pricing, while δ must be greater than 0 indicating the discount for EDA-trips is always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In addition, we observe that we can increase revenue beyond r simply by adding a tremendous amount of new, heavily discounted trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' However, below a certain price, a trip may not be worth it for the driver since they may not earn enough from it to cover their costs and make a profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' To consider the feasibility of the trip for the driver, we enforce another floor for revenue so that the driver is not in danger of not earning anything from the trip and the floored constraint thus becomes: Revenue ≥ max(r, η ∗ pmin) (9) where r is the total revenue with current pricing, η the total number of rides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' pmin is the minimum price for a trip which can be a function of factors like travel time, distance, demand, etc, deemed important by the platform for a trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' The discontinuities in the price due to surge levels make this a non-convex problem, and not straightforward to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We therefore employ grid search [12] while varying values for δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' 5 Experimental Evaluation In this section, we use the real-world data for the City of Chicago to run ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We use this dataset since this is, as far as we are aware, the only publicly available dataset that contains price (or fare) information for trips, which is necessary to address bias in pricing in ride-hailing [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' The data from all ride-hailing platforms that operate within the city of Chicago are aggregated, standardized, and anonymized before release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Which ride-hailing platform ser- viced a trip is not identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' For each trip, the dataset provides the trip start (or ‘pick-up’) time, trip end (or ‘drop-off’) time, each rounded off to the nearest 15 Unveiling and Mitigating Bias in Ride-Hailing Pricing 9 minutes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' trip pick-up and drop-off locations at the level of Chicago census tract or community area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' duration of the trip in seconds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' and the fare of trip, rounded to the nearest $2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Locations for pick-ups or drop-offs outside the city limits of Chicago are unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Drivers and riders are not identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We look at trips from January to October 2021, and focus our attention on trips that were not authorized as ‘shared.’ Since the Covid-19 pandemic began in early 2020, most ride-hailing services disabled the option for shared rides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Finally, we perform a spatial-join between the pick-up and drop-off coordi- nates of this dataset with a spatial dataset released by the City of Chicago that identifies EDA regions in Chicago, to help identify trips within our data that begin or end in EDA- or non-EDA-regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='1 The Profound Bias We use the metrics we proposed in Section 4 to unveil the intrinsic discrimination of the ride-hailing companies’ current pricing mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We find that R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Such a low R2 indicates the pervasiveness of real bias, and the extent of disparate impact it could cause on disadvantaged communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We also observe below a stark difference in consumer surplus and affordability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Next we look at affordability via consumer surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We first use RANSAC regression to estimate when surge pricing was in effect, and the surge levels for rides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' The general trend can be observed in Figure 1a: generally, as the surge multiplier increases, the number of trips that occur decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' While we do not have ground truth for surge levels to measure how accurate the predictions from our model are, the trend we observe is in line with the surge level trends in [6], who had access to ground truth as the work was in conjunction with Uber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We can also make the following observations from Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' First, a greater share of EDA-trips take place at lower surge levels as compared to non-EDA-trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Second, the number of EDA-trips falls below 1,000 at surge level 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='4x, when the average trip price is $62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' But for non-EDA-trips, it takes until surge level 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='9x for the number of trips to fall below 1,000 trips, at an average price of $97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Now that we know the surge levels for trips, we use the equations from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='2 to calculate price elasticities and consumer surplus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Price elasticity is negative for both EDA-trips and non-EDA-trips, except in cases when surge levels are very high (> 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='0x) and with very few rides (typically <100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' This indicates that as prices increase, demand typically decreases, and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We find that the consumer surplus of non-EDA-trips is much higher than that of EDA-trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Examining only surge levels with a reasonable number of rides (at the highest surge levels we observe less than a 100 rides at each level, so we do not include these in the following figures), the total consumer surplus for non-EDA- trips is $17,972,629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='829, while the surplus for EDA-trips is $2,326,555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='876 over the 10 month period we look at (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' On dividing by the total number of rides serviced in each category, we get an average consumer surplus of $67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='11 for non-EDA-trips, and an average consumer surplus of $36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='76 for EDA-trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In 2016, [6] estimated that the consumer surplus for UberX across the entire United States in the year 2015 was $6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='8 billion, thus one day’s consumer surplus across all cities in the US by their estimates would be $18 million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Further, these calculations are for 7 years ago, and they would likely be higher now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Compared 10 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Saxena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' (a) Share of trips at surge levels for the first ten months of 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' (b) Consumer Surplus for trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' 1: Surge levels and affordability (∆c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' to these figures, and considering that Chicago is a major city in the United States, our estimates are likely lower than the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Thus riders beginning or ending trips in non-EDA areas are able to afford ride-hailing services far more easily than riders that begin or end trips in EDA neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='2 Pricing Mechanisms FairRide We compare FairRide to machine learning models used for pricing [9,21,15,16,29,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We also compare against a naive baseline, which is a discount of $5 applied to all EDA rides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We find that training on data organized by region leads to lower prices for approximately 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='7% of EDA-trips, and a 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='59% increase in rides for the period we examine in 2021 for rides beginning or ending in an EDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In other words, as rides become more affordable, there are an additional 1,803,514 rides that begin or end in an EDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' As a result, with FairRide Relative Rideability (R2) increases by 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='6% to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='457, and affordability (∆c) increases by 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='5% to $2,850,552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' FixedFairRide We then test FixedFairRide with different values of pmin (the minimum price for a trip) and observe that as pmin increases, the value of δ (the trip discount) decreases (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Thus, as the minimum price per trip increases, the lower the discount will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Each of these values of pmin results in more EDA-trips than the current pricing mechanism, all also lead to more trips than FairRide up till pmin = 15, which results in 6,824,375 trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' There is a significant increase in Relative Rideability (R2) as well as affordability (∆c or consumer surplus) for EDA residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' In the dataset we use for our experiments the lowest fare value is $2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='50, and the highest proportion of trips (approximately 14%) happen around $10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Thus it is likely a minimum trip price much higher than $10 will not be practical in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' EDA trips Non-EDA trips 20 15 f Trips Percentage 5 0 Surge level20,000,000 15,000,000 (in USD) 10,000,000 Value ( 5,000,000 0 EDA trips Non-EDA tripsUnveiling and Mitigating Bias in Ride-Hailing Pricing 11 Table 1: Number of trips, R2, and affordability from different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Model η (% change) R2 (% change) ∆c (% change) Original 5,066,849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='337 2,326,555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='87 FairRide 6,870,363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='457 (+35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='60) 2,850,552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='77 (+22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='52) Mohd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' (2020) [16] 6,867,535 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='456 (+35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='31) 2,722,461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='33 (+17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='01) Miao (2017) [15] 6,868,152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='456 (+35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='31) 2,818,648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='50 (+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='15) Wolk (2020) [29] 6,864,923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='456 (+35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='31) 2,534,351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='94 (+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='93) Baseline (-$5) 6,618,494 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='440 (+30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='56) 2,739,784.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='93 (+17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='76) Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' (2020) [9] 6,843,205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='455 (+35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='01) 2,612,239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='28 (+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='27) Rathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' (2019) [21] 6,385,843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='424 (+25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='81) 2,667,079.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='84 (+14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='63) Alkhatib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' (2013) [1] 6,608,789 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='439 (+30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='26) 2,476,128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='68 (+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='42) FixedFairRide(pmin = 12) 8,142,519 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='541 (+60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='53) 3,201,963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='73 (+37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='62) Table 2: Discount δ different minimum trip price pmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' pmin 5 6 7 8 9 10 11 12 13 14 15 δ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content='24 6 Conclusion Although a number of studies in the recent past have explored pricing for ride- hailing services, including looking at fair compensation for drivers, and insurance policy as a safeguard against ride-hailing using personal information for pricing, none have looked at fairer pricing to reduce disparate impact on disadvantaged residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' We proposed applicable fairness metrics to unveil the intrinsic bias in ride-hailing along with a flexible pricing mechanism to price rides more fairly and make trips more affordable for disadvantaged residents who are under-served by public transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' The empirical experiments reveal the profound bias caused by existing ride-hailing pricing, and show pricing trips by our mechanism leads to more affordable and equitable ride-hailing services which could assist government policy-making for fair ride-hailing policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=' Alkhatib, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E1T4oBgHgl3EQf3gUP/content/2301.03489v1.pdf'} +page_content=', Najadat, H.' 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b/udE_T4oBgHgl3EQf-Bxu/content/tmp_files/2301.08384v1.pdf.txt @@ -0,0 +1,1745 @@ +GENERAL RIGIDITY PRINCIPLES FOR STABLE AND +MINIMAL ELASTIC CURVES +TATSUYA MIURA AND KENSUKE YOSHIZAWA +Abstract. For a wide class of curvature energy functionals defined for planar +curves under the fixed-length constraint, we obtain optimal necessary con- +ditions for global and local minimizers. +Our results extend Maddocks’ and +Sachkov’s rigidity principles for Euler’s elastica by a totally different approach, +and in particular lead to complete classification of stable closed p-elasticae for +all p ∈ (1, ∞) and of stable pinned p-elasticae for p ∈ (1, 2]. Our proof is based +on a simple but robust ‘cut-and-paste’ trick without computing the energy +nor its second variation, which works well for planar periodic curves but also +extend to some non-periodic or non-planar cases. +Contents +1. +Introduction +1 +2. +Main results +3 +3. +Preliminary +10 +4. +Rigidity under the clamped boundary condition +11 +5. +Rigidity under the pinned boundary condition +14 +6. +Planar p-elasticae +18 +7. +Spatial elasticae +22 +References +25 +1. Introduction +In variational theory it is commonly important to detect locally or globally min- +imal critical points in order to obtain practical solutions or effective inequalities. +Here and hereafter, as usual, an element x ∈ X is called global minimizer (resp. +local minimizer) of a functional F : X → [0, ∞] if F(x) ≤ F(x′) holds for all x′ ∈ X +(resp. x′ ∈ U, where U is some neighborhood of x in X). We also often refer to +global and local minimality as minimality and stability, respectively. +In this paper we focus on variational problems involving curvature of planar +curves. More precisely we consider an energy functional of the form +F(γ) := +� L +0 +f +� +|k| +� +ds +Date: January 23, 2023. +2020 Mathematics Subject Classification. 49Q10, 53A04. +Key words and phrases. Stability; elastica; p-elastica; boundary value problem; bending +energy. +1 +arXiv:2301.08384v1 [math.DG] 20 Jan 2023 + +2 +TATSUYA MIURA AND KENSUKE YOSHIZAWA +defined for planar curves γ : [0, L] → R2, where s denotes the arclength parameter, +k the signed curvature, and f : [0, ∞) → [0, ∞] a nonnegative Borel function with +additional properties specified later. +Our class of functionals will cover at least the standard bending energy f(x) = x2 +as well as the p-bending energy f(x) = xp for p ∈ (1, ∞). The bending energy is +introduced by D. Bernoulli and studied by L. Euler in the 18th century (see [16, 40] +for the history and [14, 2] for physical backgrounds) but still studied extensively +from mathematical points of view; see e.g. [39, 34, 23, 22] and references therein. +The p-bending energy has interesting analytic and geometric features in its own +right and also appears in various contexts; see e.g. [41, 10, 17, 28, 29, 31, 38, 4, 5, +18, 32, 30, 7, 12, 26, 27]. The case of polynomial f was also studied [1, 13] (see also +[11]). A relevant model is used for analyzing DNA cyclization [8]. Throughout this +paper, we will impose the fixed-length constraint on curves, which is indispensable +in the standard theory of bending rods or plates. The type of boundary condition +is also important for stability and minimality. We will address not only the typical +clamped boundary condition, fixing the endpoints up to first order, but also the +zeroth-order counterpart called pinned boundary condition. +The typical cases to which our theory applies are thus the classical problem of +Euler’s elastica [9] (p = 2) as well as its Lp-counterpart called p-elastica [41, 18, +26, 27], which is defined as a fixed-length critical point of the p-bending energy +Bp[γ] := +� L +0 +|k|pds. +On the level of critical points, if p = 2, all solutions are classified by Euler and later +parameterized in terms of Jacobian elliptic integrals and functions by Saalsch¨utz +(cf. [16, 19]). The authors recently extended this classification to planar p-elasticae +for a general power p ∈ (1, ∞) in terms of suitable p-elliptic integrals and functions +[26]. Going into boundary value problems, one usually aims at (i) finding all critical +points, (ii) finding all global minimizers, and (iii) finding all local minimizers for +given boundary data. These three problems involve additional difficulties, which +are generally independent. For some well-prepared boundary data, the problems +(i) and (ii) are completely solved even for all p ∈ (1, ∞); see [26] for closed p- +elasticae and [27] for pinned p-elasticae. Concerning (ii) when p = 2, see also [23] +for straightened boundary data and [25] for the cuspidal case. On the other hand, to +the authors’ knowledge, the problem (iii) is solved only for closed elasticae (p = 2) +[15, 35, 3]. +In this paper we first reveal general rigidity principles (necessary conditions) +induced by stability and minimality for a wide class of functionals, including the +p-bending energy as a special example. In particular, those results together with +our previous work [26, 27] lead to complete classification of the stability of closed +p-elasticae for p ∈ (1, ∞) and pinned p-elasticae for p ∈ (1, 2], thus solving the +problem (iii) above, and also an effective rigidity result even for pinned p-elasticae +with p ∈ (2, ∞). To the authors’ knowledge, our study provides the first stability +analysis for p-elastica. (See [12] for instability of spherical closed free p-elasticae +with p ∈ (0, 1), where the length is unconstrained.) Even for the p-bending energy, +if p ̸= 2, then the combination of the lack of quadraticity and the presence of +the fixed-length constraint leads to significant methodological challenges. Here we +propose a very simple ‘cut-and-paste’ trick, which turns out to be surprisingly +robust. The contents of this paper were briefly announced in [24]. + +STABLE AND MINIMAL ELASTIC CURVES +3 +This paper is organized as follows: In Section 2 we state our main results, both +in terms of general principles and applications to p-elasticae, and explain the key +idea of our proof, while mentioning some previous studies more precisely. After +short preliminaries in Section 3, we prove our general principles in the clamped +case (Theorems 2.1 and 2.3) in Section 4, and in the pinned case (Theorems 2.7 +and 2.8) in Section 5. In Section 6 we discuss applications to planar p-elasticae. +Finally we apply our method to some spatial elasticae in Section 7. +2. Main results +We first very briefly review some of relevant previous studies in order to motivate +our main results. +In 1906, Born developed stability theory for Euler’s elastica [6]. Among other +results, he found the general principle that if a clamped planar elastica has no +zero of the curvature (i.e., locally convex), then it is stable with respect to smooth +perturbations. Born also addresses more general elasticae, albeit with the help of +numerical computations, and gave the first detailed comparisons with experiments. +Maddocks [20, 21] developed linear stability analysis for Euler’s elastica (with +possibly nonuniform bending rigidity), which explains physically natural (in)stability +for various boundary conditions, extending some previous results cited therein. For +Euler’s elastica, it is shown that ‘higher modes’ (with many inflection points) are +basically unstable for both clamped and pinned boundary conditions. In particular, +the results for the pinned boundary condition are summarized as follows: +(M1) The pinned first mode loses its stability when the endpoints meet. +(M2) The pinned higher modes are always unstable. +A schematic diagram for (M1) is given in Figure 1. The higher modes in (M2) +correspond to the curves γN +arc, γN +loop with N ≥ 2 in Figure 2. In fact, it is rigorously +known (cf. [19, 42]) that all the pinned elasticae are classified as in Figure 2. Com- +bined with this classification, Maddocks’ results imply that the embedded convex +arc γ1 +arc is the only stable one (in the sense of linear stability). Langer–Singer [15] +developed a different approach, which not only covers spatial elasticae with special +symmetry but also recovers some of Maddocks’ results. +Stable (pinned) +Unstable (pinned) +Stable (clamped) +Figure 1. Stability of convex arcs and loops. + +4 +TATSUYA MIURA AND KENSUKE YOSHIZAWA +γ1 +arc +γ1 +loop +γ2 +arc +γ3 +arc +γ2 +loop +γ3 +loop +Figure 2. Pinned elasticae of arc type (left) and of loop type (right). +In a recent series of papers [34, 33, 36, 35, 37], focusing on planar elasticae subject +to the clamped boundary condition, Sachkov and his coauthors developed a more +detailed theory of not only stability but also minimality by an optimal control +approach. In particular, Sachkov found optimal rigidity principles in terms of the +natural periodic structure of elasticae, cf. statements (1) and (3.3) (or (3.5)) in [33, +Theorem 5.1]. Here the periodicity means that the tangent direction is periodic; +the curve itself is generically quasi-periodic, cf. Figure 3. Sachkov’s principles are +roughly summarized as follows: +(S1) If a clamped elastica is minimal, then it does not exceed one period. +(S2) If a clamped elastica is stable, then it does not contain three inflection +points in its interior. +Principle (S1) is trivial for closed elasticae in view of scaling — any closed curve has +less bending energy than its multiple covering of same length — but is not trivial +for non-closed elasticae since simple scaling arguments do not work. Principle (S2) +is in line with Maddocks’ instability result for higher modes. Both (S1) and (S2) are +optimal (cf. Remarks 4.1 and 6.2) and very useful for detecting (local) minimizers. +Wavelike elastica +Borderline elastica +Orbitlike elastica +Inflection points +Periodicity +Examples +Circular elastica +Figure 3. Basic patterns of elasticae. +Our main results extend all (M1), (M2), (S1), (S2) to a wide class of energy func- +tionals (in the natural Sobolev framework). Such extensions are highly nontrivial +even for p-elasticae, since both Maddocks’ and Sachkov’s methods importantly use +the fact that the curvature term is quadratic. Indeed, Sachkov’s argument depends + +STABLE AND MINIMAL ELASTIC CURVES +5 +on explicit computations only valid for the standard bending energy; Maddocks’ +argument is based on a rather ‘representation-free’ linear stability analysis (which +makes it possible to address non-uniform cases) but the rigorous application of lin- +ear stability analysis on the nonlinear level would be significantly delicate even for +p-elasticae with p ̸= 2 due to the lack of Hilbert structures, which in particular +leads to a generic loss of regularity. +The rest of this section proceeds as follows: We first present general principles +in the clamped case corresponding to (S1) and (S2) in Section 2.1, because of the +simplicity of the statements and also their applicability to any boundary condition, +cf. Remark 2.5. Then we turn to the pinned case corresponding to (M1) and (M2) +in Section 2.2. In Section 2.3, we collect some typical consequences in p-elastica +theory. Finally we explain the key idea of our proof in Section 2.4. +2.1. General rigidity principles: Clamped case. Let p ∈ (1, ∞), L > 0 and +P0, P1, V0, V1 ∈ R2 with |P0 − P1| < L and |V0| = |V1| = 1. Define +W 2,p +arc (0, L; R2) := +� +γ ∈ W 2,p(0, L; R2) +�� |γ′| ≡ 1 +� +, +the set of planar arclength parameterized Sobolev curves of length L, and then +Aclamp := {γ ∈ W 2,p +arc (0, L; R2) | γ(0) = P0, γ(L) = P1, γ′(0) = V0, γ′(L) = V1}, +the admissible set subject to the clamped boundary condition, equipped with the +relative topology induced by the W 2,p-norm. +The first result extends (S1) in a contrapositive form. Our key hypothesis is the +following property of regularity-improvement: +(H1) +If γ is a global minimizer of F in Aclamp, then γ ∈ C2([0, L]; R2). +Hereafter we often abbreviate γ ∈ C2([0, L]; R2) as γ ∈ C2. +This property is +naturally expected whenever f is not too wild, since any minimizer weakly solves +the Euler–Lagrange equation. Under this hypothesis we obtain +Theorem 2.1 (Rigidity of clamped global minimizers). Suppose (H1) holds. If +γ ∈ Aclamp has two points 0 ≤ s1 < s2 ≤ L with s2 − s1 < L such that +γ′(s1) = γ′(s2) and +− k(s1) ̸= k(s2), +(2.1) +then γ is not a global minimizer of F in Aclamp. +This result indeed extends (S1) because after one period (s2 = s1 + period) one +has γ′(s1) = γ′(s2) and k(s1) = k(s2) so that (2.1) holds unless the curvature +vanishes there. +The next result extends (S2). To this end we will similarly suppose: +(H1’) +If γ is a local minimizer of F in Aclamp, then γ ∈ C2. +In addition, we need to use a certain well-periodic structure. +Definition 2.2 (Well-periodic curve). We call an arclength parameterized planar +curve γ : [0, L] → R2 well-periodic if γ has continuous signed curvature k ∈ C([0, L]) +of the form k(s) = Φ(s − s0), where s0 ∈ R, and Φ ∈ C(R) is an antiperiodic odd +function with antiperiod T > 0 such that { s ∈ R | Φ(s) = 0 } = TZ. We also call +T the antiperiod of the well-periodic curve γ. +This kind of periodicity naturally appears in the objective critical points thanks +to the invariance of F with respect to the change of the sign of curvature as well +as the orientation of parameter. For this well-periodic class we obtain + +6 +TATSUYA MIURA AND KENSUKE YOSHIZAWA +Theorem 2.3 (Rigidity of clamped local minimizers). Suppose (H1’) holds. +If +γ ∈ Aclamp is well-periodic, and has three points 0 ≤ s1 < s2 < s3 ≤ L with +s3 − s1 < L such that +k(s1) = k(s2) = k(s3) = 0, +(2.2) +then γ is not a local minimizer of F in Aclamp. +This result directly extends (S2), and gives an effective rigidity for ‘inflectional’ +critical points. +Remark 2.4. Theorem 2.1 (resp. Theorem 2.3) is optimal in the sense that if s2 − +s1 = L (resp. s3 − s1 = L) then the assertion may fail even for classical elasticae, +cf. Remark 4.1 (resp. Remark 6.2). +Remark 2.5. Our principles immediately extend to more general cases. For ex- +ample, regardless the choice of boundary conditions, the same results hold true +away from the endpoints; more precisely, for a given minimal (resp. stable) curve +γ : [0, L] → R2 subject to any boundary condition, the restriction γ|[δ1,L−δ2] with +δ1, δ2 > 0 must be minimal (resp. stable) under the clamped boundary condition +and hence satisfy the above assertion. +In addition, our principles also directly +work in the case of length-penalized problems (i.e., critical points of the functional +F +λL among length-unconstrained admissible curves, where λ ∈ R and L denotes +the length) as well as the case of higher codimensions (i.e., planar curves in Rn +with any n ≥ 2). +2.2. General rigidity principles: Pinned case. Now we turn to the pinned +boundary condition. More precisely, for L > 0 and P0, P1 ∈ R2 such that |P0−P1| < +L, we consider the admissible class +Apin := {γ ∈ W 2,p +arc (0, L; R2) | γ(0) = P0, γ(L) = P1}. +In this class, a critical point often satisfies the so-called natural boundary condition +that the curvature vanishes at the endpoints, thanks to the arbitrariness of first- +order variations at the endpoints. In view of this condition, it is natural to consider +the following subclass of well-periodic curves, which we call +m +2 -fold curves; the +number m corresponds to the superscripts in Figure 2, and roughly speaking counts +the modes with the convention that ‘1-fold’ means the one-period of the curvature +(i.e., twice the antiperiod). +Definition 2.6 ( m +2 -Fold well-periodic curve). For m ∈ N, we say that a well- +periodic curve γ : [0, L] → R2 is m +2 -fold if k(0) = 0 and T = L/m hold, where T is +the antiperiod of γ defined in Definition 2.2. +We first state our instability result for higher modes, extending (M2). Here we +will suppose both the regularity-improvement and the natural boundary condition: +If γ is a local minimizer of F in Apin, then γ ∈ C2 and k(0) = k(L) = 0. +(H2) +Theorem 2.7. Suppose (H2) holds. Let γ ∈ Apin be a 1-fold well-periodic curve. +Then γ is not a local minimizer of F in Apin. +We then address the first mode, extending (M1). In fact, our proof of this part +will be based on a different mechanism from all the previous cases, and thus we will +suppose a rather different type of hypothesis: +f is strictly convex on [0, ∞). +(H3) + +STABLE AND MINIMAL ELASTIC CURVES +7 +Under this hypothesis we show that if a curve has a specific loop structure as in +the left part of Figure 1 or such as γ1 +loop in Figure 2, then it is unstable. +Theorem 2.8. Suppose (H3) holds. Let γ ∈ Apin be a 1 +2-fold well-periodic curve. +If γ is not injective on (0, L) and if γ′(0) · (P1 − P0) > 0, then γ is not a local +minimizer of F in Apin. +Notice that the assumption automatically rules out the case that P0 = P1. +2.3. Applications to p-elastica. Now we discuss some concrete applications to +p-elasticae under some well-chosen boundary conditions. +Our previous work ensures that for all p ∈ (1, ∞) the p-bending energy Bp +satisfies hypotheses (H1), (H1’), (H2), (H3); see [26] for C2-regularity in (H1), +(H1’), (H2), and see [27] for the natural boundary condition in (H2). In addition, +all wavelike p-elasticae obtained in [26] belong to the class of well-periodic curves. +We first address the case of closed curves, which can be regarded as a special +case of the clamped boundary condition by taking P0 = P1 and V0 = V1. It is +shown in [26] that any closed planar p-elastica is either a circle or a figure-eight +p-elastica, possibly multiply covered. It is easy to deduce from H¨older’s inequality +and Fenchel’s theorem that a global minimizer is a once covered circle. Here we +apply Theorem 2.3 to extend the known classification of stability for p = 2 [15, 35, 3] +to a general power p ∈ (1, ∞). +Theorem 2.9 (Stability of closed p-elasticae). Let p ∈ (1, ∞) and γ ∈ Aclamp with +P0 = P1 and V0 = V1. Then γ is a local minimizer of Bp in Aclamp if and only if γ +is either a possibly multiply covered circle or a once covered figure-eight p-elastica. +Since the C1-regular homotopy classes of closed planar curves are exactly clas- +sified by the rotation number, and since the above local minimizers have different +rotation numbers, we reach a uniqueness theorem `a la Langer–Singer in R2 [15]. +Corollary 2.10. For each C1-regular homotopy class of immersed circles in R2, +stable closed planar p-elasticae are unique up to similarity and reparameterization. +We then turn to the pinned boundary condition. We call a critical point of Bp +in Apin pinned p-elastica. In [27] the authors have classified all pinned p-elasticae +and proved uniqueness of global minimizers. In general, there are infinitely many +pinned p-elasticae, which are either of wavelike or flat-core type. +We first completely classify the stability of wavelike pinned p-elasticae. +It is +shown in [27] that any wavelike pinned p-elasticae is classified as an m +2 -fold well- +periodic curve as in Figure 2, to which Theorems 2.7 and 2.8 are applicable. It +turns out that all but the global minimizer is unstable. +Theorem 2.11 (Stability of wavelike pinned p-elasticae). Let p ∈ (1, ∞). Suppose +that γ ∈ Apin is a wavelike p-elastica. Then γ is a local minimizer of Bp in Apin if +and only if γ is a global minimizer (convex arc). +The above wavelike result already yields strong rigidity. Indeed, it is also shown +in [27] that if |P0−P1| +L +< +1 +p−1, then any pinned p-elastica is wavelike. Hence we reach +the following uniqueness of stable pinned p-elasticae. +Corollary 2.12. Suppose either that p ∈ (1, 2], or that p ∈ (2, ∞) and |P0−P1| +L +< +1 +p−1. Then stable pinned planar p-elasticae in Apin are unique up to isometries. + +8 +TATSUYA MIURA AND KENSUKE YOSHIZAWA +For p ∈ (1, 2], our result covers the full range of pinned boundary data. This +not only extends Maddocks’ linear stability analysis as in Figure 1 from p = 2 to +p ∈ (1, 2], but also directly justifies it on the nonlinear level (even for p = 2). In +particular, our result could possibly be the first explicit statement on the uniqueness +of stable pinned elasticae. At any rate, the result is completely new for p ̸= 2. +We further make progress by addressing flat-core pinned p-elasticae, the only +remaining case. +Flat-core p-elasticae appear only in the degenerate case p > 2 +and have peculiar non-periodicity, cf. Figure 4, thus representing the substantial +difference between the structures of p-elasticae and of the classical elasticae. In +particular, our previous rigidity results are based on periodicity and not directly +applicable to flat-core p-elasticae in general. However, the same machinery enables +us to develop rather ad-hoc arguments to prove the following new instability: +Theorem 2.13 (Unstable flat-core pinned p-elasticae). A flat-core pinned planar +p-elastica is unstable either if it contains two adjacent loops in opposite directions, +or if a loop touches an endpoint. +Figure 4. Flat-core p-elasticae. The left two curves are unstable +(Theorem 2.13). The right curve is classified as a quasi-alternating +flat-core p-elastica (Definition 6.5). +In particular, invoking the classification in [27], we obtain the following di- +chotomy theorem for stable pinned planar p-elasticae without any assumption on +the exponent p nor the boundary condition. Here we call that a flat-core pinned +planar p-elastica is quasi-alternating if the curve is not ruled out by the above the- +orem, or in other words given by a certain alternating concatenation of segments +and multi-loops as in Figure 4 (see Definition 6.5 for details). +Corollary 2.14. Any stable pinned planar p-elasticae is either a global minimizer +(convex arc) or a quasi-alternating flat-core p-elastica. +It is an interesting open problem whether there indeed exists a stable quasi- +alternating flat-core p-elastica. If it were true, then it would give an interesting +example of a stabilization phenomenon due to degeneracy. +2.4. A trick. Finally we explain the main idea that underlies most of our argu- +ments. All the aforementioned previous studies [6, 20, 21, 15, 34, 33, 36, 35, 37] +involve explicit computations on Jacobian elliptic functions or some functional an- +alytic structures, which are based on the quadraticity to some degree. On the other +hand, Avvakmov–Karpenkov–Sossinsky [3] gave a new proof that multiply-covered + +STABLE AND MINIMAL ELASTIC CURVES +9 +figure-eight elasticae are unstable in the plane, by directly constructing an energy- +decreasing perturbation. This method has the strong advantage that it is almost +purely based on the geometric structure of the curve. However, their construc- +tion relies on the very special geometry of the figure-eight elastica. A particularly +important point is that their construction involves rescaling and hence does not +directly extend to some non-closed curves. In fact, such a direct construction of an +energy-decreasing perturbation is often obstructed by the fixed-length constraint. +A trick we propose here to circumvent the above obstruction is a very simple +indirect argument. The key idea is to construct an equal-energy competitor that +lies in the energy space but is not regular enough to be minimal or stable, implying +a contradiction to our natural regularity hypothesis. It turns out that the construc- +tions of equal-energy competitors are reduced to surprisingly simple ‘cut-and-paste’ +arguments, with the help of intrinsic periodicity and symmetry of objective criti- +cal points. We demonstrate the proof of Theorem 2.1 for a special example of an +elastica (but in fact the idea is completely same in the general case). As in Figure +5, for a given ‘more than one period’ orbitlike elastica we can rotate a ‘one-period’ +part to construct a competitor which is of class W 2,2 and has the unchanged length +and bending energy but cannot be a global minimizer since the discontinuity of the +curvature at the cut points contradicts the C2-regularity, completing the proof. To +construct local perturbations we need more structures as assumed in Theorem 2.3 +since the perturbations need to be close to the original curve. +s1 +s2 +γ +¯γ +π +Figure 5. An original curve γ with two points s1, s2 satisfying +γ′(s1) = γ′(s2) and k(s1) = k(s2) ̸= 0 (left) and the competitor ¯γ +with equal energy to γ (right). +One of the main discoveries in our study would be the fact that this simple trick +is quite robust. Although the new results presented above already provide general +and optimal rigidity results for planar critical points, we also expect more. As an +example, in Section 7, we apply our trick to elasticae in R3 to obtain old and new +spatial rigidity results in a unified manner. + +10 +TATSUYA MIURA AND KENSUKE YOSHIZAWA +3. Preliminary +3.1. Concatenation. We prepare a notation on a concatenation of curves: For +γj : [aj, bj] → R2 with Lj := bj − aj ≥ 0, we define γ1 ⊕ γ2 : [0, L1 + L2] → R2 by +(γ1 ⊕ γ2)(s) := +� +γ1(s + a1), +s ∈ [0, L1], +γ2(s + a2 − L1) + γ1(b1) − γ2(a2), +s ∈ [L1, L1 + L2], +and inductively define γ1 ⊕ · · · ⊕ γN := (γ1 ⊕ · · · ⊕ γN−1) ⊕ γN. We also define +N +� +j=1 +γj := γ1 ⊕ · · · ⊕ γN. +This notation will be useful for our cut-and-paste procedures. Similarly, for a point +P ∈ R2 and a curve γ : [0, L] → R2, we define +(P ⊕ γ)(s) := γ(s) − γ(0) + P, +s ∈ [0, L]. +3.2. Well-periodic curves. Here we collect some fundamental properties of well- +periodic curves in Definition 2.2. +Recall that a function Φ is said to be antiperiodic if there is T > 0 such that +−Φ(x) = Φ(x + T) holds for all x ∈ R, and this T is called antiperiod. +The +antiperiod T of a well-periodic curve in Definition 2.2 is uniquely determined as it +is also related with the zero set of the curvature. We stress that we have imposed +continuity on the curvature (and Φ) of a well-periodic curve; this is natural in view +of our regularity hypothesis. This continuity implies that Φ does not change the +sign on each connected component of R \ TZ. In addition, the combination of odd +symmetry and periodicity implies the symmetry Φ( T +2 +s) = Φ( T +2 −s), and also the +sign-changing property Φ(s)Φ(s + T) < 0 for s ∈ R \ TZ. +The following lemma exhibits some properties of well-periodic curves which we +will use later. The proof is elementary and safely omitted. +Lemma 3.1 (Symmetry of well-periodic curves). Let γ : [0, L] → R2 be a well- +periodic planar curve and k be the signed curvature of γ. +If k has three points +0 ≤ s1 < s2 < s3 ≤ L such that +k(si) = 0 (i = 1, 2, 3), +and k(s) ̸= 0 for s ∈ (s1, s2) ∪ (s2, s3), +(3.1) +then we have +γ′(s1 + σ) = γ′(s3 + σ) holds whenever s1 + σ, s3 + σ ∈ [0, L], +(3.2) +k(σ + s1) = −k(s3 − σ) holds whenever σ + s1, s3 − σ ∈ [0, L]. +In addition, if a well-periodic curve γ : [0, L] → R2 is m +2 -fold in the sense of +Definition 2.6, its signed curvature k satisfies that +k(s) = 0 +⇐⇒ +s = 0, +1 +mL, +2 +mL, . . . , L. +(3.3) +In particular, we may assume that k = Φ without loss of generality. +Remark 3.2. The definition of ‘ m +2 -fold’ here is in line with the terminology of ‘ m +2 - +fold figure-eight p-elastica’ used in [26, Definition 5.3]. Note that (p, m, r)-arcs and +(p, m, r)-loops defined in [27, Definition 3.2] are also m +2 -fold well-periodic curves +(see Figure 2 for p = 2 and m = 1, 2, 3). + +STABLE AND MINIMAL ELASTIC CURVES +11 +4. Rigidity under the clamped boundary condition +In this section we prove Theorems 2.1 and 2.3. +4.1. Rigidity for minimality. We first prove Theorem 2.1. The key idea is al- +ready mentioned in Section 2.4 and Figure 5. +Proof of Theorem 2.1. We argue by contradiction. Suppose that γ is a global min- +imizer of F in Aclamp, and has two points s1, s2 ∈ [0, L] satisfying (2.1). Up to +reversing the parameterization, we may assume that s1 may be equal to 0 but +s2 < L. Let us define a continuous function ¯γ : [0, L] → R2 by +¯γ := γ|[0,s1] ⊕ R +� +γ|[s1,s2] +� +⊕ γ|[s2,L], +where R denotes the affine transformation describing the rotation through 180 +degrees around the point +c := 1 +2 +� +γ(s1) + γ(s2) +� +. +In particular, we see that +¯γ(s) = −γ(−s + s1 + s2) + 2c, +s ∈ [s1, s2]. +By (2.1) and by definition of ¯γ we have ¯γ ∈ C1([0, L]; R2), particularly noting that +lim +s↓s1 ¯γ′(s) = γ′(s2) = γ′(s1), +lim +s↑s2 ¯γ′(s) = γ′(s1) = γ′(s2). +We also notice that the signed curvature ¯k of ¯γ is +¯k(s) = +� +−k(−s + s1 + s2), +s ∈ (s1, s2), +k(s), +s ∈ [0, L] \ [s1, s2], +which is bounded and hence ¯γ ∈ W 2,∞(0, L; R2) ⊂ W 2,p(0, L; R2). Since ¯γ satisfies +the same clamped boundary condition as γ (even if s1 = 0), we have ¯γ ∈ Aclamp. +Moreover, the above formula for ¯k implies F(¯γ) = F(γ). Hence ¯γ is also a global +minimizer of F in Aclamp. Then it follows from (H1) that ¯γ ∈ C2([0, L]; R2). On +the other hand, we see that by definition of ¯k, +lim +s↑s2 +¯k(s) = −k(s1), +lim +s↓s2 +¯k(s) = k(s2). +This together with (2.1) implies that ¯γ ̸∈ C2(0, L; R2). This is a contradiction. +□ +Remark 4.1. Theorem 2.1 is optimal in the sense that if s2 − s1 = L then the +assertion may fail. In fact, in the case of P0 = P1 and V0 = V1 (i.e., closed curves), +a one-fold circle is a global minimizer of B2 in Aclamp but its endpoints satisfy (2.1). +4.2. Rigidity for stability. Now we turn to the proof of Theorem 2.3. We first +give an abstract criterion based on a contradiction argument, which clarifies how +to deduce instability. As in the proof of Theorem 2.1, a key point is that we allow +equality in condition (ii) below. +Lemma 4.2. Suppose (H1’) holds. Let γ ∈ Aclamp and suppose that there exists a +sequence {γj}j∈N ⊂ Aclamp satisfying the following three conditions: +� +� +� +� +� +(i) γj → γ +in +W 2,p(0, L; R2) as j → ∞, +(ii) F(γj) ≤ F(γ) +for all (large) +j ∈ N, +(iii) γj /∈ C2(0, L; R2) +for all (large) +j ∈ N. +(C) + +12 +TATSUYA MIURA AND KENSUKE YOSHIZAWA +Then γ is not a local minimizer of F in Aclamp. +Proof. Let γ ∈ Aclamp satisfy the three conditions in (C). We argue by contradic- +tion, so suppose that γ is a local minimizer of F in Aclamp. Then there is δ > 0 +such that +F(γ) ≤ F(ξ) +for +ξ ∈ Aclamp with ∥γ − ξ∥W 2,p < δ. +(4.1) +By (C)-(i), there exists j0 ∈ N such that ∥γ −γj∥W 2,p < δ/2 for j ≥ j0. Henceforth +we consider only j ≥ j0. It follows from (C)-(ii) and (4.1) that +F(γj) ≤ F(γ) ≤ F(ξ) +for +ξ ∈ Aclamp with ∥γj − ξ∥W 2,p < δ +2. +This implies that γj are also local minimizers of F in Aclamp for all j ≥ j0. There- +fore, γj ∈ C2([0, L]; R2) follows from (H1’), which contradicts (C)-(iii). This com- +pletes the proof. +□ +In order to show Theorem 2.3, we now apply the above criterion to prove the +instability of any well-periodic curve γ that ‘strictly’ contains three inflection points +s1 < s2 < s3. Our perturbation γj here is constructed as follows: Noting that γ +has symmetry centered at s2 as in Figure 6 (1), we first cut the curve γ out at the +perturbed points s1 + 1 +j and s3 + 1 +j (with large j) and then rotate it through 180 +degrees around the middle of the cut points as in Figure 6 (2). By symmetry the +perturbed curve γj is close to the original one γ. It turns out that by periodicity +and our rotation procedure, the perturbed curve γj has discontinuous curvature at +the cut points. +s1 +s2 +s3 +s1+ 1 +j +s3+ 1 +j +π +(1) +(2) +(3) +cj +Figure 6. (1) An original well-periodic curve γ. (2) Construction +of a perturbation γj. (3) Convergence as j → ∞. +Proof of Theorem 2.3. Let γ ∈ Aclamp be a well-periodic curve and have three +points satisfying (2.2). Up to reversing the parameterization and taking the first +three inflection points, we may assume that s1, s2, s3 satisfy (3.1), in particular +s3 < L. In addition, up to reflection, we may assume that k > 0 in (s1, s2). Then +since γ is well-periodic, k < 0 in (s2, s3) and k > 0 in (s3, s3 + δ) with some small +δ ∈ (0, L − s3). For each integer j ≥ j0 with some j0 such that 1/j0 < δ, we define +a continuous function γj : [0, L] → R2 by +γj := γ|[0,s1+ 1 +j ] ⊕ Rj +� +γ|[s1+ 1 +j ,s3+ 1 +j ] +� +⊕ γ|[s3+ 1 +j ,L], +(4.2) + +STABLE AND MINIMAL ELASTIC CURVES +13 +where Rj denotes the rotation through 180 degrees around +cj := 1 +2 +� +γ(s1 + 1 +j ) + γ(s3 + 1 +j ) +� +. +In particular, we see that +γj(s) = −γ(−s + s1 + s3 + 2 +j ) + 2cj, +s ∈ Ij := (s1 + 1 +j , s3 + 1 +j ). +In fact γj is still a unit-speed C1-curve by construction and by (3.2) in Lemma 3.1; +for example, around s1 + 1 +j , by computing the one-sided limits independently and +using (3.2), we deduce that γ′ +j(s1 + 1 +j ) = γ′(s1 + 1 +j ), and also the derivative γ′ +j is +continuous by (3.2), e.g. in view of +lim +s↓s1+ 1 +j +γ′ +j(s) = +lim +s↓s1+ 1 +j +γ′(−s + s1 + s3 + 2 +j ) = γ′(s3 + 1 +j ) = γ′(s1 + 1 +j ); +the same argument works for s3 + 1 +j . Hence γj ∈ C1([0, L]; R2) for any j ≥ j0. In +addition, by (4.2) the signed curvature kj of γj is given by +kj(s) = +� +−k(−s + s1 + s3 + 2 +j ), +s ∈ Ij +k(s), +s ∈ [0, L] \ ¯Ij, +(4.3) +where ¯Ij := [s1 + 1 +j , s3 + 1 +j ]. Since k ∈ C([0, L]) by Definition 2.2, we have kj ∈ +L∞(0, L) and hence γj ∈ W 2,∞(0, L; R2) ⊂ W 2,p(0, L; R2) for all j ≥ j0. Moreover, +by (4.2) and the fact that s1 + 1 +j > 0 and s3 + 1 +j < L, it is clear that the curve γj +satisfies the same boundary condition as γ. Thus we have {γj}j≥j0 ⊂ Aclamp. +Now we prove that the sequence {γj}j≥j0 satisfies all the conditions in (C). +We first check (C)-(i). Recall that by Lemma 3.1, +k(s) = −k(−s + s1 + s3), +s ∈ Ij. +This together with (4.3) implies that kj → k a.e. in (0, L). In addition, noting that +∥kj∥L∞ ≤ ∥k∥L∞, we obtain kj → k in Lp(0, L). From this convergence and the +fact that γ(0) = γj(0) and γ′(0) = γ′ +j(0), we deduce that γj → γ in W 2,p(0, L; R2). +Next we check (C)-(ii). Since +� s3+ 1 +j +s1+ 1 +j +f +� +|kj(s)| +� +ds = +� s3+ 1 +j +s1+ 1 +j +f +� +|k(−s + s1 + s3 + 2 +j )| +� +ds += +� s1+ 1 +j +s3+ 1 +j +−f +� +|k(s)| +� +ds = +� s3+ 1 +j +s1+ 1 +j +f +� +|k(s)| +� +ds +and since γ and γj agree elsewhere, we have F(γj) = F(γ). +Finally, the discontinuity of curvature in (C)-(iii) follows since for any large j, +thanks to k|(s1,s2) > 0, we have +lim +s↑s1+ 1 +j +kj(s) = k(s1 + 1 +j ) > 0, +lim +s↓s1+ 1 +j +kj(s) = −k(s3 + 1 +j ) = −k(s1 + 1 +j ) < 0, +and hence kj is not continuous at s = s1 + 1 +j . The proof is complete. +□ +Remark 4.3. Since Aclamp ⊂ Apin, Theorem 2.1 (resp. Theorem 2.3) holds true +with Aclamp replaced by Apin whenever hypothesis (H1) (resp. (H1’)) holds true for +all unit vectors V0, V1. + +14 +TATSUYA MIURA AND KENSUKE YOSHIZAWA +5. Rigidity under the pinned boundary condition +This section is devoted to the proofs of Theorems 2.7 and 2.8. +5.1. Instability of one-fold waves. We first prove Theorem 2.7. In the same +spirit as Lemma 4.2, it is sufficient to construct {γj}j∈N ⊂ Apin satisfying +� +� +� +� +� +(i) γj → γ +in +W 2,p(0, L; R2) as j → ∞, +(ii) F(γj) ≤ F(γ) +for all (large) +j ∈ N, +(iii) the curvature kj of γj satisfies kj(0) ̸= 0 +for all (large) +j ∈ N. +(Cp) +Here we construct such a perturbation γj by extending the original curve by using +its intrinsic periodicity and shift the domain as in Figure 7 so that the curvature +does not vanish at the endpoints: +s = 1 +j +(1) +(2) +(3) +Figure 7. (1) A 1-fold well-periodic curve γ. (2) Construction of +a perturbation γj. (3) Convergence as j → ∞. +Proof of Theorem 2.7. Recalling (3.3), we infer that the curvature k of γ satisfies +k(s) = 0 +⇐⇒ +s = 0, +L +2 , L. +(5.1) +For each integer j ≥ j0 with some j0 such that 1/j0 < L, we define a continuous +map γj : [0, L] → R2 by +γj := P0 ⊕ γ|[ 1 +j ,L] ⊕ γ|[0, 1 +j ]. +(5.2) +Then we infer from Lemma 3.1 that γ′(0) = γ′(L) and hence +lim +s↓L− 1 +j +γ′ +j(s) = γ′(0) = γ′(L) = +lim +s↑L− 1 +j +γ′ +j(s), +which implies that γj ∈ C1([0, L]; R2) for each j ≥ j0. The signed curvature kj of +γj is +kj(s) := +� +k(s + 1 +j ) +s ∈ [0, L − 1 +j ), +k(s − L + 1 +j ) +s ∈ (L − 1 +j , L]. +(5.3) +Since lims↑L− 1 +j kj(s) = lims↓L− 1 +j kj(s) = 0 holds by (5.1), kj is continuous in [0, L], +and hence γj ∈ C2([0, L]; R2). By definition we have γj(0) = P0, γj(L) = P1, and +L[γj] = L. Thus we have {γj}j≥j0 ⊂ Apin. +Now we prove that the sequence {γj}j≥j0 satisfies all the conditions in (Cp). + +STABLE AND MINIMAL ELASTIC CURVES +15 +First, we check (Cp)-(i). It follows from (5.3) that kj → k a.e. in (0, L). Com- +bining this with the fact that ∥kj∥L∞ = ∥k∥L∞, we see that kj → k in Lp(0, L). +Since γj(0) = γ(0) and γ′ +j(0) = γ′( 1 +j ) → γ′(0), we have γj → γ in W 2,p(0, L; R2). +Next, we check (Cp)-(ii). It follows that +F(γj) = +� L− 1 +j +0 +f +� +|kj(s)| +� +ds + +� L +L− 1 +j +f +� +|kj(s)| +� +ds += +� L− 1 +j +0 +f +� +|k(s + 1 +j )| +� +ds + +� L +L− 1 +j +f +� +|k(s − L + 1 +j )| +� +ds = F(γ), +and hence {γj}j≥j0 satisfies F(γj) = F(γ). +Finally, we show (Cp)-(iii), i.e., the curvature at an endpoint does not vanish. +In fact, it follows from (5.1) and (5.2) that kj(0) = k( 1 +j ) ̸= 0 for any j ≥ j0. +Thus {γj}j≥j0 satisfies all the conditions in (Cp). Therefore, if γ were a local +minimizer, then so were all γj with large j, but this would contradict (H2). The +proof is complete. +□ +5.2. Instability of loops. Here we prove Theorem 2.8. Throughout this subsec- +tion, for notational simplicity, without loss of generality we assume that +(5.4) +P0 = (0, 0) and P1 = (l, 0), where l := |P0 − P1| ∈ [0, L). +To begin with, we note here that the antiperiodicity of the curvature of well-periodic +curves yields the following symmetry of curves; this is closely related with the +symmetry Φ( T +2 + s) = Φ( T +2 − s) which we have already observed in Section 3. +Lemma 5.1. Let γ = (x, y) : [0, L] → R2 be a +1 +2-fold well-periodic curve. +If +γ(0) = (0, 0) and γ(L) = (l, 0) with some l ̸= 0, then +x(L − s) + x(s) = l, +y(L − s) = y(s), +for s ∈ [0, L]. +Proof. This easily follows from elementary differential geometry with the fact that +k(s) = Φ(s) = −Φ(s − L) = Φ(L − s) = k(L − s) +for any s ∈ [0, L]. +(5.5) +The assumption l ̸= 0 is used for forcing the reflection axis to be vertical. +□ +In addition, again for notational simplicity, we also assume that +f(0) = 0 +by replacing f(t) with f(t)−f(0) if necessary; this does not lose generality since the +functional +� +γ f(|k|) ds is (locally) minimized if and only if so does +� +γ(f(|k|)−f(0)) ds +under the fixed-length constraint. Note that if f satisfies (H3) and f(0) = 0, then +one easily verifies that +λf(λ−1t) < f(t) for any t ∈ (0, ∞) and λ > 1, +(5.6) +since f((1 − λ−1)0 + λ−1t) < (1 − λ−1)f(0) + λ−1f(t). +In what follows we will construct an energy-decreasing perturbation. The key +idea here is to perform an odd extension of the loop and shift the domain so that the +symmetric loop is asymmetrically perturbed as in Figure 8 (3), and finally rescale +the loop as in Figure 8 (4) in order to increase the shortened length and recover the +admissibility. All these procedures decrease the energy. The convexity hypothesis +will be used in the last rescaling step. + +16 +TATSUYA MIURA AND KENSUKE YOSHIZAWA +(1) +s = − cj +s = L− 1 +j +(2) +(3) +(4) +0 +l +0 +l +0 +l +l +Figure 8. +(1) An original curve with a loop. (2) Shifting the end- +points. (3) Rigid motion for the boundary condition. (4) Rescaling +the loop for the length constraint. +Proof of Theorem 2.8. We first note that by assumption (5.4) with l > 0 we have +the reflection symmetry in Lemma 5.1, and in addition by γ′(0) · (P1 − P0) > 0 we +have +(5.7) +x′(0) > 0. +We also note that the non-injectivity assumption precisely means that +there are distinct a, b ∈ (0, L) such that γ(a) = γ(b). +(5.8) +In the proof below, we shall construct a family of curves {γj}j≥j0 ⊂ Apin satisfying +F(γj) < F(γ) for any j ≥ j0 +and +γj → γ in W 2,p(0, L; R2). +(5.9) +First, we define an odd extension of γ to the domain [−L, L] by +� +γ(s) +s ∈ [0, L], +−γ(−s) +s ∈ [−L, 0), +(5.10) +which is also denoted by γ : [−L, L] → R2. Then γ ∈ C1([−L, L]; R2) holds since +γ is of class C1 around s = 0. For j ≥ 1 +L, we have y( 1 +j ) = y(L − 1 +j ) by Lemma 5.1. +In addition, since x′(0) > 0 by (5.7) and also x′(L) > 0 by Lemma 5.1, for all large +j we have x(L − 1 +j ) − x( 1 +j ) < x(L) − x(0) = l so that +��γ(L − 1 +j ) − γ(0) +�� < l. +(5.11) +On the other hand, we infer from Lemma 5.1 that +x(L − 1 +j ) − x(− 1 +j ) = x(L − 1 +j ) + x( 1 +j ) = l, +y(L − 1 +j ) = y( 1 +j ), +and since |k(s)| ̸= 0 for any small s > 0, we have y( 1 +j ) ̸= 0 for all large j. Hence +��γ(L − 1 +j ) − γ(− 1 +j ) +�� = +� +l2 + 4y( 1 +j )2 > l +(5.12) +for all j ≥ j0 with sufficiently large j0. By (5.11) and (5.12), for each j ≥ j0 there +is +cj ∈ (− 1 +j , 1 +j ) +such that +���γ(L − 1 +j ) − γ(−cj) +��� = l. +(5.13) +(In fact we can show cj > 0 but this is not used.) Define ¯γj : [−cj, L − 1 +j ] → R2 by +¯γj(u) := Qj +� +γ(u) − γ(−cj) +� +, +u ∈ [−cj, L − 1 +j ], + +STABLE AND MINIMAL ELASTIC CURVES +17 +where Qj is a rotation matrix such that Qj(γ(L− 1 +j )−γ(−cj)) = (l, 0) and Qj → Id +(by (5.13), such Qj exists). Then we see that +¯γj(−cj) = (0, 0), +¯γj(L − 1 +j ) = (l, 0), +¯γj ∈ W 2,p(−cj, L − 1 +j ; R2) +(5.14) +but we still have L[¯γj] < L. Now we normalize the length. Let a, b ∈ (0, L) satisfy +(5.8). Then ¯γj(a) = ¯γj(b) also holds for all j ≥ j0 with j0 so large that +1 +j0 < a < +b < L − 1 +j0 . For j ≥ j0 we define a continuous function ¯Γj : [−cj, L − 1 +j ] → R2 by +¯Γj := γj|[−cj,a] ⊕ λjγj|[a,b] ⊕ γj|[b,L− 1 +j ], +λj := +b − a + 1 +j − cj +b − a +> 1. +Note that L[¯Γj] = L follows by the choice of the scaling factor λj. +Hereafter we let γj denote the arclength parameterization of ¯Γj. +We have +{γj}j≥j0 ⊂ C([0, L]; R2) by definition. +Since ¯Γj is defined only by rescaling at +a self-intersection point, we also have {γj} ⊂ C1([0, L]; R2). We also notice that +the signed curvature kj of γj is given by +kj(s) = +� +� +� +� +� +k(s − cj), +0 ≤ s < a + cj, +λ−1 +j k(λ−1 +j (s − a − cj) + a), +a + cj < s < b + 1 +j , +k(s − 1 +j ), +b + 1 +j < u ≤ L, +(5.15) +where k is the signed curvature of γ. Now we check that {γj}j≥j0 ⊂ Apin. We +have k ∈ C([0, L]) by Definition 2.2, and hence by (5.15) we get {γj}j≥j0 ⊂ +W 2,∞(0, L; R2) ⊂ W 2,p(0, L; R2). +Moreover, γj(0) = (0, 0) and γj(L) = (l, 0) +by definition and (5.14). Recalling L[γj] = L, we have {γj}j≥j0 ⊂ Apin. +Henceforth we show that the family {γj}j≥j0 satisfies (5.9). We first show the +strict inequality in (5.9). It follows from (5.15) and the change of variables that +F(γj) = +� a+cj +0 +f +� +|kj(s)| +� +ds + +� b+ 1 +j +a+cj +f +� +|kj(s)| +� +ds + +� L +b+ 1 +j +f +� +|kj(s)| +� +ds += +� a +−cj +f +� +|k(s)| +� +ds + +� b +a +f +� +λ−1 +j |k(s)| +� +λj ds + +� L− 1 +j +b +f +� +|k(s)| +� +ds. +(5.16) +By (5.6) and λj > 1, and the fact that k ̸= 0 in (0, L) since γ is 1 +2-fold, we obtain +� b +a +f +� +λ−1 +j |k(s)| +� +λj ds < +� b +a +f +� +|k(s)| +� +ds. +By (5.10) we have |k(s)| = |k(−s)| for s ∈ [− 1 +j , 1 +j ], and hence by |cj| < 1 +j and f ≥ 0, +� a +−cj +f +� +|k(s)| +� +ds ≤ +� |cj| +0 +f +� +|k(s)| +� +ds + +� a +0 +f +� +|k(s)| +� +ds +≤ +� +1 +j +0 +f +� +|k(s)| +� +ds + +� a +0 +f +� +|k(s)| +� +ds += +� L +L− 1 +j +f +� +|k(s)| +� +ds + +� a +0 +f +� +|k(s)| +� +ds, +where in the last equality we used k(L − s) = k(s), cf. (5.5). Therefore, by (5.16), +F(γj) < +� a +0 +f +� +|k(s)| +� +ds + +� b +a +f +� +|k(s)| +� +ds + +� L +b +f +� +|k(s)| +� +ds = F(γ). + +18 +TATSUYA MIURA AND KENSUKE YOSHIZAWA +It remains to show that γj → γ in W 2,p(0, L; R2). By (5.15) and the fact that +λj → 1, cj → 0, we see that kj → k a.e. in (0, L). In addition, since ∥kj∥L∞ ≤ +∥k∥L∞ holds by λj > 1, it follows that kj → k in Lp(0, L). Noting also the fact that +γj(0) = γ(0) and γ′ +j(0) = Qjγ′( 1 +j ) → γ′(0), we obtain γj → γ in W 2,p(0, L; R2). +The proof is complete. +□ +Remark 5.2. Theorem 2.8 will be used for showing instability of a (p, r, 1)-loop +with r := |P0−P1| +L +> 0, but does not cover the case of r = 0. In fact, if r = 0, then +we can regard a (p, r, 1)-loop as a half-fold figure-eight p-elastica, which is a global +minimizer of Bp in Apin [27, Theorem 1.3] and hence obviously stable. Therefore it +is necessary to assume that P0 ̸= P1 at least. +6. Planar p-elasticae +In this section we discuss the stability of closed and pinned planar p-elasticae. +6.1. Closed p-elastica. We first apply Theorem 2.3 to prove Theorem 2.9, thus +classifying the stability of closed planar p-elasticae. +Let Aclosed denote the set Aclamp in the special case that P0 = P1 and V0 = V1. +Recall from [26, Theorem 5.6] that any closed planar p-elastica is either a circle or +a figure-eight p-elastica, possibly multiply covered. +To begin with, we observe that an m-fold circle and a 1-fold figure-eight p-elastica +(in the sense of [26]) are indeed stable: +Proposition 6.1. If γ is an m-fold circle, where m ∈ N, or a 1-fold figure-eight +p-elastica, then γ is a local minimizer of Bp in Aclosed. +Proof. For m ∈ N ∪ {0}, let Zm ⊂ Aclosed denote the subset of fixed rotation +number m: +Zm := +� +γ ∈ Aclosed +��� N[γ] := +1 +2π +� L +0 k ds = m +� +. +Since the functional N is constant in a small W 2,p-neighborhood of any element of +Aclosed, if γ is a minimizer of Bp in Zm, then γ is a local minimizer in Aclosed. +It suffices to show that an m-fold circle (resp. a 1-fold figure-eight p-elastica) is a +minimizer of Bp in Zm if m ≥ 1 (resp. m = 0). The existence of a minimizer ¯γm in +Zm follows from the standard direct method (cf. [27, Proposition 4.1]). Then, ¯γm +is also a p-elastica by the Lagrange multiplier method [27]. On the other hand, by +the classification for closed p-elasticae [26, Theorem 5.6], in the case of m ≥ 1, ¯γm +must be an m-fold circle. In the remaining case of m = 0, the same classification +also implies that any p-elastica with rotation number 0 is an n-fold figure-eight +p-elastica for n ∈ N, and comparing their p-bending energy, we find that ¯γ0 must +be a 1-fold figure-eight p-elastica. +□ +Remark 6.2. Here is a good position to observe that Theorem 2.3 is optimal in the +sense that if s3 − s1 = L then the assertion may fail. In fact, a 1-fold figure-eight +p-elastica is a local minimizer of F = Bp in Aclosed by the above proposition, while +it is well-periodic and satisfies (2.2) at s = 0, L +2 , L if the endpoints are arranged to +be located at the crossing of the figure-eight (inflection points). +We are now ready to classify stable p-elasticae among closed curves. + +STABLE AND MINIMAL ELASTIC CURVES +19 +Proof of Theorem 2.9. In view of [26, Theorem 5.6] and Proposition 6.1 it suffices +to prove instability of m-fold figure-eight p-elasticae for m ≥ 2. If m ≥ 2, then +m-fold figure-eight p-elasticae are well-periodic and have at least three inflection +points satisfying (2.2), and hence they are unstable in Aclosed by Theorem 2.3. +□ +Proof of Corollary 2.10. As discussed in the proof of Proposition 6.1, if m ≥ 1, +then any global minimizer with rotation number m is an m-fold circle, which is +stable in Aclosed. +If m = 0, then it follows from Theorem 2.9 that any stable +zero-rotation-number planar closed curve is a 1-fold figure-eight p-elastica. +□ +6.2. Wavelike pinned p-elastica. Next we prove Theorem 2.11 and Corollary +2.12, which are now almost direct consequences of Theorems 2.3, 2.7, and 2.8, +combined with our previous results. +Proof of Theorem 2.11. Let γ be a wavelike pinned p-elastica and let r := |P0−P1| +L +. +Then it follows from [27, Theorem 1.1] that there exists n ∈ N such that γ is either +a (p, r, n)-arc or a (p, r, n)-loop, for which we write γn +arc or γn +loop, respectively. By +[26, Theorem 1.7], every pinned p-elastica is of class C2, and by [26, Theorems +1.2 and 1.3], the signed curvature of a wavelike p-elastica can be expressed by +the so-called p-elliptic function cnp, which is an odd, antiperiodic, and continuous +function. Therefore, γn +arc and γn +loop are well-periodic curves. Furthermore, +• γn +arc and γn +loop (n ≥ 3) are n +2 -fold well-periodic curves and contain three +inflection points satisfying (3.1), +• γ2 +arc and γ2 +loop are 1-fold well-periodic curves, +• γ1 +loop is a 1 +2-fold well-periodic curve satisfying (5.7) and (5.8) if |P0−P1| > 0 +(cf. [27, Lemma 3.8] for existence of a loop). +Hence, γn +arc and γn +loop with n ≥ 3 are unstable by Theorem 2.3 with Remark 4.3, +while so are γ2 +arc and γ2 +loop by Theorem 2.7. Moreover, if |P0 − P1| > 0, then γ1 +loop +is unstable by Theorem 2.8. Therefore, in any case, the only remaining candidate +for stable wavelike pinned p-elasticae is the global minimizer γ1 +arc. The proof is now +complete. +□ +Proof of Corollary 2.12. If p ∈ (1, 2] or p ∈ (2, ∞) and |P0−P1| +L +< +1 +p−1, then any +pinned p-elastica is a wavelike p-elastica (cf. [27, Theorem 1.1]). This together with +Theorem 2.11 asserts that any stable wavelike pinned p-elastica is a (p, r, 1)-arc. +□ +6.3. Flat-core p-elastica. Now we discuss some instability criteria for flat-core +p-elasticae, and prove Theorem 2.13. Flat-core p-elasticae appear as special exam- +ples of planar p-elasticae, and they are obtained by concatenating certain ‘loops’ +and ‘segments’ with some arbitrariness (see [41, 26]). In view of the natural bound- +ary condition induced by the pinned boundary condition, we focus on flat-core +p-elasticae each of whose endpoints is an endpoint of a segment or of a loop. More +precisely, we consider a curve γ : [0, L] → R2 represented by, up to similarity and +reparameterization, +γ = γL0 +seg ⊕ +� +N +� +j=1 +γσj +loop ⊕ γLj +seg +� +(F1) +for some N ∈ N, {σj}N +j=1 ⊂ {+, −}, and L1, . . . , LN ≥ 0. Here γLj +seg(s) := (s, 0) for +s ∈ [0, Lj], γ+ +loop denotes a certain loop, and γ− +loop denotes the reflection of γ+ +loop with + +20 +TATSUYA MIURA AND KENSUKE YOSHIZAWA +respect to the e1-axis. Although the explicit parameterization of γ+ +loop is known, +here we only use the following special properties: the curve γ+ +loop = (X, Y ) : [0, 1] → +R2 is an arclength parameterized curve such that the tangential vectors at both +the endpoints are rightward, i.e., +(γ+ +loop)′(0) = (γ+ +loop)′(1) = (1, 0), +(F2) +the signed curvature kloop of γ+ +loop satisfies +kloop > 0 in (0, 1), +kloop(0) = kloop(1) = 0, +(F3) +and γ+ +loop is reflectionally symmetric in the sense that +X(s) + X(1 − s) = 2X( 1 +2), +Y (s) = Y (1 − s) for s ∈ [0, 1]. +(F4) +If γ is a pinned planar flat-core p-elastica, then these properties directly follow by +our previous classification [27, Theorem 1.1] and explicit formulae [26, Theorem +1.3] (see also Figure 4). In addition, combining (F1) with (F2), we see that any +curve γ of the form (F1) satisfies +γ′(0) = γ′(L) = (1, 0). +(6.1) +The first criterion ensures that all loops need to lie ‘strictly inside’ for stability +under the pinned boundary condition. +Proposition 6.3. Let γ ∈ Apin be a flat-core p-elastica. +If a loop touches an +endpoint, then γ is not a local minimizer of Bp in Apin. +Proof. Without loss of generality, we may assume that γ(0) is an endpoint of a +loop. For each j ≥ j0 with some 1/j0 < L, we define γj : [0, L] → R2 by +γj := P0 ⊕ γ|[ 1 +j ,L] ⊕ γ|[0, 1 +j ], +cf. Figure 9. Then it is clear that γj ∈ C([0, L]; R2). Property (6.1) ensures that +γj ∈ C1([0, L]; R2). +Since γ ∈ C2([0, L]; R2), the signed curvature kj of γj is +bounded, in particular, γj ∈ W 2,∞(0, L; R2) ⊂ W 2,p(0, L; R2). By definition we +have γj(0) = P0, γj(L) = P1, and L[γj] = L. Thus {γj}j≥j0 ⊂ Apin holds. +As in the proof of Theorem 2.7, we see that {γj}j≥j0 satisfies the conditions +(Cp)-(i) and (Cp)-(ii). By property (F3), the signed curvature kj of γj satisfies +kj(0) ̸= 0, and hence the condition (Cp)-(iii) is satisfied. Thus {γj}j≥j0 satisfies all +the conditions in (Cp) and hence γ is unstable. +□ +s = 1 +j +(1) +(2) +Figure 9. (1) A flat-core p-elastica with a loop touching an end- +point. (2) Construction of a perturbation γj. + +STABLE AND MINIMAL ELASTIC CURVES +21 +The next criterion ensures that a positive gap is necessary between two loops in +opposite directions for stability even under the clamped boundary condition. +Proposition 6.4. Let γ ∈ Aclamp be a flat-core p-elastica. If γ contains one loop +and a part of a loop in the opposite direction with no segment between the two loops, +then γ is not a local minimizer of Bp in Aclamp. +Proof. Let γ ∈ Aclamp be a flat-core p-elastica satisfying the assumption. +This +means that there is some a ∈ (0, 1] such that if we define +Γ := γ+ +loop ⊕ γ− +loop|[0,a], +then up to similarity and orientation of the parameter, the curve γ can be decom- +posed as +γ = γ|[0,s1] ⊕ Γ ⊕ γ|[s2,L] +for some s1, s2 ∈ [0, L] (where L denotes the length after rescaling). For each j ≥ j0 +with some j0 such that 1/j0 < a, we define +Γj := γ− +loop|[0, 1 +j ] ⊕ γ+ +loop|[1− 1 +j ,1] ⊕ γ+ +loop|[0,1− 1 +j ] ⊕ γ− +loop|[ 1 +j ,a], +cf. Figure 10. Note that Γj is a unit-speed C1-curve by properties (F2), (F4), and +(6.1). On the other hand, by property (F3), the curvature of Γj is discontinuous at +s = 1/j and hence Γj is not of class C2. +We define γj := γ|[0,s1] ⊕ Γj ⊕ γ|[s2,L]. Now we prove that the sequence {γj}j≥j0 +satisfies all the conditions in (C). +We first check {γj}j≥j0 ⊂ Aclamp. By definition, γj is a C1-curve and L[γj] = L +holds. Since γ is a C2-curve, the signed curvature of γj is of bounded. Therefore +{γj}j≥j0 is a W 2,p-curve. By construction, γj satisfies the same boundary condition +as γ. Thus we have {γj}j≥j0 ⊂ Aclamp. +The condition (C)-(i) follows by straightforward calculations combined with the +fact that the signed curvature of γ+ +loop is uniformly continuous. The condition (C)- +(ii) follows since Bp[Γ] = Bp[Γj]. The condition (C)-(iii) follows from the fact that +Γj /∈ C2. Consequently, by Lemma 4.2 γ is unstable in Aclamp. +□ +(1) +(2) +Figure 10. (1) A flat-core p-elastica with adjacent opposite loops. +(2) Construction of a perturbation γj. +The proof of Theorem 2.13 is now already complete. +Proof of Theorem 2.13. It directly follows by Propositions 6.3 and 6.4. +□ + +22 +TATSUYA MIURA AND KENSUKE YOSHIZAWA +Now we introduce a new class of flat-core p-elasticae, which are not covered by +the previous two instability results (and presumably stable). Let us first call a +flat-core planar p-elastica γ alternating if up to similarity and reparameterization, +γ is given as in (F1) with strictly positive L0, . . . , LN > 0. The strict positivity +particularly means that the segments and the loops appear alternately. By relaxing +the alternating property, we now define the following class: +Definition 6.5 (Quasi-alternating). Let γ be a flat-core planar p-elastica of the +form (F1) with some N ∈ N, σ1, . . . , σN ∈ {+, −}, and L0, . . . , LN ≥ 0. We call γ +quasi-alternating if the following two conditions hold: +(i) L0, LN > 0. +(ii) For j ∈ {1, . . . , N − 1}, if Lj = 0, then σj = σj+1. +Proof of Corollary 2.14. Let γ be a pinned planar p-elastica. Then, by [27, The- +orem 1.1], γ is a wavelike p-elastica or a flat-core p-elastica. If γ is wavelike and +stable, then Theorem 2.11 implies that γ is a one-fold arc. On the other hand, +suppose that γ is flat-core and stable. By [27] any pinned flat-core p-elastica is of +the form (F1). Proposition 6.3 and Proposition 6.4 imply conditions (i) and (ii) in +Definition 6.5, respectively, and hence γ must be quasi-alternating. +□ +7. Spatial elasticae +Finally we discuss applications to spatial elasticae, i.e., critical points of the +bending energy B[γ] := +� +γ |κ|2ds, where κ := γss, among fixed-length W 2,2-curves +in R3. +Spatial elasticae are also smooth by a standard bootstrap argument so +that our trick is applicable well. A classification of spatial elasticae is obtained by +Langer–Singer (cf. [39]). +Here we obtain new rigidity principles due to the presence of spatial pertur- +bations. All the results are concerning the admissible space Aclamp of curves in +W 2,2 +arc (0, L; R3) subject to the standard clamped boundary condition. +Theorem 7.1 (Rigidity for spatially minimal elasticae). Let γ ∈ Aclamp. If there +are 0 ≤ s1 < s2 ≤ L with s2 −s1 < L such that all the vectors γ(s2)−γ(s1), γ′(s1), +γ′(s2) are orthogonal to a unit vector ω ∈ R3, and such that either γ′′(s1) · ω ̸= 0 +or γ′′(s2) · ω ̸= 0, then γ is not globally minimal. +Proof. Suppose on the contrary that γ is a global minimizer. Then γ is of class C2 +in particular. Without loss of generality we may assume that γ′′(s1) · ω ̸= 0. We +decompose γ = γ|[0,s1] ⊕ γ|[s1,s2] ⊕ γ[s2,L]. Let P be a unique plane passing through +γ(s1) and orthogonal to ω. By the assumption the plane P also passes through +γ(s2) and is parallel to γ′(s1) and γ′(s2). Let R denote the reflection about the +plane P, and let ˜γ := γ|[0,s1] ⊕ Rγ|[s1,s2] ⊕ γ[s2,L]. Then ˜γ ∈ Aclamp by the property +of P, and also B[˜γ] = B[γ], so that ˜γ is also a global minimizer and hence of class +C2. However ˜γ′′ has discontinuity at s1 since γ′′(s1) · ω ̸= 0 so that the vectors +˜γ′′(s1 − 0) = γ′′(s1) and ˜γ′′(s1 + 0) = γ′′(s1) − 2(γ′′(s1) · ω)ω do not coincide. This +is a contradiction. +□ +Theorem 7.2 (Rigidity for spatially stable elasticae). Let γ ∈ Aclamp. If there are +0 ≤ s1 < s2 ≤ L with s2 − s1 < L such that all the vectors γ(s2) − γ(s1), γ′(s1), +γ′(s2) are parallel, and such that either |γ′′(s1)| ̸= 0 or |γ′′(s2)| ̸= 0, then γ is not +locally minimal. + +STABLE AND MINIMAL ELASTIC CURVES +23 +Proof. Suppose on the contrary that γ is a local minimizer. Then γ is of class C2 +in particular. Without loss of generality we may assume that |γ′′(s1)| ̸= 0. We +decompose γ = γ|[0,s1] ⊕ γ|[s1,s2] ⊕ γ[s2,L]. Let L be a unique line passing through +γ(s1) and parallel to γ′(s1). By the assumption the line L also passes through γ(s2) +and is parallel to γ′(s2). Let Rθ denote a rotation about the axis L through angle +θ, and let γθ := γ = γ|[0,s1] ⊕Rθγ|[s1,s2] ⊕γ[s2,L]. Then γθ ∈ Aclamp by the property +of L, and also γθ → γ in W 2,2 as θ → 0 with B[γθ] = B[γ], so that for any small +θ the curve γθ is also a local minimizer and hence of class C2. However γ′′ +θ has +discontinuity at s1 (whenever θ ̸= 0) since γ′′(s1) is orthogonal to γ′(s1) and hence +to L, so that the non-zero vectors γ′′ +θ (s1 − 0) and γ′′ +θ (s1 + 0) form the angle θ ̸= 0 +due to Rθ. This is a contradiction. +□ +Below we provide some concrete examples of applications of the above results. +The first example is about the known fact that all closed planar elasticae except +for a one-fold circle are unstable in R3. This fact is shown by Langer–Signer [15] +and also follows by Maddocks’ analysis [20, 21], both through explicit computations +of second variations. Theorem 7.2 provides a geometric proof of this fact. +Corollary 7.3. A more than two-fold circular elastica is unstable in the set of +closed curves of fixed length in R3 (cf. Figure 11). +Figure 11. A more than two-fold circular elastica (left) and its +perturbation for instability (right). +Corollary 7.4. A one-fold figure-eight elastica is unstable in the set of closed curves +of fixed length in R3 (cf. Figure 12). +Figure 12. A one-fold figure-eight elastica (left) and its pertur- +bation for instability (right). + +24 +TATSUYA MIURA AND KENSUKE YOSHIZAWA +In the same way we can produce various examples of planar elasticae (both +orbitlike and wavelike) which are stable in the plane but unstable in the space, cf. +Figure 13. Some of such phenomena are also treated by Maddocks [20, 21]. +Figure 13. A wavelike elastica, to which Theorem 7.2 is applicable. +The final example is about a helix, an example of a purely spatial elastica. By +Langer–Singer [15] it is shown that if a helix has more than two turn, then it is +unstable. Theorem 7.1 gives a new rigidity for minimality. +Corollary 7.5. A helical elastica with more than one turn is not minimal in the +set of clamped curves of fixed length in R3 (cf. Figure 14). +Figure 14. A helix (left) and its deformation for non-minimality (right). +We can also deduce by Theorem 7.1 that a spatial elastica is not minimal if it +has a self-intersection (in its interior) at which one of the two curvature vectors are +linearly independent from both the two tangent vectors. +We expect that our method also works for more general spatial (p-)elasticae or +even other ambient spaces under some symmetry. +Acknowledgements. The first author is supported by JSPS KAKENHI Grant +Numbers 18H03670, 20K14341, and 21H00990, and by Grant for Basic Science +Research Projects from The Sumitomo Foundation. The second author is supported +by JSPS KAKENHI Grant Number 22K20339. + +Maple 2022 (Intel x86-64) +輪集 +表示 +捶入 +書式 +一 +^儿 +11月23日(水)13:37 +... +*221119_helix.mw +T>三 +夕 +? +索 +S+< +于+又非夷行MathMath +C Text +Times New Roman +12 > B I 三三 T 三 +》 +氮入 +絶对轴 +微横分 +表の +一般的記号 +夕儿 + 变数 +ジ术儿 +变数 +值 +色 +90 +透明度 +光况度 +行列 +視点角度 +单位 +照明 +座標轴 +字 +卜儿 +操作 +2 +待機 +集可能Maple才卜/Users/k-yoshizawa/Documents/MAPLE書類2021--/MY安定性/2022年—:1×：54.18M時：0.25s-么夕:100%卜一 +zoom +LINE +』 +ATEXISTABLE AND MINIMAL ELASTIC CURVES +25 +References +[1] J. Arroyo, O. J. Garay, and J. J. Menc´ıa. Closed generalized elastic curves in S2(1). J. Geom. +Phys., 48(2-3):339–353, 2003. +[2] B. Audoly and Y. Pomeau. Elasticity and geometry: From Hair Curls to the Non-Linear +Response of Shells. Oxford University Press, Oxford, 2010. +[3] S. Avvakumov, O. Karpenkov, and A. Sossinsky. Euler elasticae in the plane and the Whitney- +Graustein theorem. Russ. J. Math. Phys., 20(3):257–267, 2013. +[4] S. Blatt, C. Hopper, and N. 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In Curvature and variational modeling +in physics and biophysics, volume 1002 of AIP Conf. Proc., pages 3–32. Amer. Inst. Phys., +Melville, NY, 2008. +[40] C. Truesdell. The influence of elasticity on analysis: the classic heritage. Bull. Amer. Math. +Soc. (N.S.), 9(3):293–310, 1983. +[41] K. Watanabe. Planar p-elastic curves and related generalized complete elliptic integrals. Kodai +Math. J., 37(2):453–474, 2014. +[42] K. Yoshizawa. The critical points of the elastic energy among curves pinned at endpoints. +Discrete Contin. Dyn. Syst., 42(1):403–423, 2022. +(T. Miura) Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, +Meguro-ku, Tokyo 152-8551, Japan +Email address: miura@math.titech.ac.jp +(K. Yoshizawa) Institute of Mathematics for Industry, Kyushu University, 744 Mo- +tooka, Nishi-ku, Fukuoka 819-0395, Japan +Email address: k-yoshizawa@imi.kyushu-u.ac.jp + diff --git a/udE_T4oBgHgl3EQf-Bxu/content/tmp_files/load_file.txt b/udE_T4oBgHgl3EQf-Bxu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e13a89558a76e0055ccfaeeaf6a44bdc148b654 --- /dev/null +++ b/udE_T4oBgHgl3EQf-Bxu/content/tmp_files/load_file.txt @@ -0,0 +1,1259 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf,len=1258 +page_content='GENERAL RIGIDITY PRINCIPLES FOR STABLE AND MINIMAL ELASTIC CURVES TATSUYA MIURA AND KENSUKE YOSHIZAWA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' For a wide class of curvature energy functionals defined for planar curves under the fixed-length constraint, we obtain optimal necessary con- ditions for global and local minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Our results extend Maddocks’ and Sachkov’s rigidity principles for Euler’s elastica by a totally different approach, and in particular lead to complete classification of stable closed p-elasticae for all p ∈ (1, ∞) and of stable pinned p-elasticae for p ∈ (1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Our proof is based on a simple but robust ‘cut-and-paste’ trick without computing the energy nor its second variation, which works well for planar periodic curves but also extend to some non-periodic or non-planar cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Main results 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Preliminary 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Rigidity under the clamped boundary condition 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Rigidity under the pinned boundary condition 14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Planar p-elasticae 18 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Spatial elasticae 22 References 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Introduction In variational theory it is commonly important to detect locally or globally min- imal critical points in order to obtain practical solutions or effective inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Here and hereafter, as usual, an element x ∈ X is called global minimizer (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' local minimizer) of a functional F : X → [0, ∞] if F(x) ≤ F(x′) holds for all x′ ∈ X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' x′ ∈ U, where U is some neighborhood of x in X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We also often refer to global and local minimality as minimality and stability, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In this paper we focus on variational problems involving curvature of planar curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' More precisely we consider an energy functional of the form F(γ) := � L 0 f � |k| � ds Date: January 23, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 49Q10, 53A04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Stability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' elastica;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' p-elastica;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' boundary value problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' bending energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='08384v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='DG] 20 Jan 2023 2 TATSUYA MIURA AND KENSUKE YOSHIZAWA defined for planar curves γ : [0, L] → R2, where s denotes the arclength parameter, k the signed curvature, and f : [0, ∞) → [0, ∞] a nonnegative Borel function with additional properties specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Our class of functionals will cover at least the standard bending energy f(x) = x2 as well as the p-bending energy f(x) = xp for p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The bending energy is introduced by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Bernoulli and studied by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Euler in the 18th century (see [16, 40] for the history and [14, 2] for physical backgrounds) but still studied extensively from mathematical points of view;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' [39, 34, 23, 22] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The p-bending energy has interesting analytic and geometric features in its own right and also appears in various contexts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' [41, 10, 17, 28, 29, 31, 38, 4, 5, 18, 32, 30, 7, 12, 26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The case of polynomial f was also studied [1, 13] (see also [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A relevant model is used for analyzing DNA cyclization [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Throughout this paper, we will impose the fixed-length constraint on curves, which is indispensable in the standard theory of bending rods or plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The type of boundary condition is also important for stability and minimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We will address not only the typical clamped boundary condition, fixing the endpoints up to first order, but also the zeroth-order counterpart called pinned boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The typical cases to which our theory applies are thus the classical problem of Euler’s elastica [9] (p = 2) as well as its Lp-counterpart called p-elastica [41, 18, 26, 27], which is defined as a fixed-length critical point of the p-bending energy Bp[γ] := � L 0 |k|pds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' On the level of critical points, if p = 2, all solutions are classified by Euler and later parameterized in terms of Jacobian elliptic integrals and functions by Saalsch¨utz (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' [16, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The authors recently extended this classification to planar p-elasticae for a general power p ∈ (1, ∞) in terms of suitable p-elliptic integrals and functions [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Going into boundary value problems, one usually aims at (i) finding all critical points, (ii) finding all global minimizers, and (iii) finding all local minimizers for given boundary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' These three problems involve additional difficulties, which are generally independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' For some well-prepared boundary data, the problems (i) and (ii) are completely solved even for all p ∈ (1, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' see [26] for closed p- elasticae and [27] for pinned p-elasticae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Concerning (ii) when p = 2, see also [23] for straightened boundary data and [25] for the cuspidal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' On the other hand, to the authors’ knowledge, the problem (iii) is solved only for closed elasticae (p = 2) [15, 35, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In this paper we first reveal general rigidity principles (necessary conditions) induced by stability and minimality for a wide class of functionals, including the p-bending energy as a special example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In particular, those results together with our previous work [26, 27] lead to complete classification of the stability of closed p-elasticae for p ∈ (1, ∞) and pinned p-elasticae for p ∈ (1, 2], thus solving the problem (iii) above, and also an effective rigidity result even for pinned p-elasticae with p ∈ (2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' To the authors’ knowledge, our study provides the first stability analysis for p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (See [12] for instability of spherical closed free p-elasticae with p ∈ (0, 1), where the length is unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=') Even for the p-bending energy, if p ̸= 2, then the combination of the lack of quadraticity and the presence of the fixed-length constraint leads to significant methodological challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Here we propose a very simple ‘cut-and-paste’ trick, which turns out to be surprisingly robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The contents of this paper were briefly announced in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' STABLE AND MINIMAL ELASTIC CURVES 3 This paper is organized as follows: In Section 2 we state our main results, both in terms of general principles and applications to p-elasticae, and explain the key idea of our proof, while mentioning some previous studies more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' After short preliminaries in Section 3, we prove our general principles in the clamped case (Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3) in Section 4, and in the pinned case (Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='8) in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In Section 6 we discuss applications to planar p-elasticae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Finally we apply our method to some spatial elasticae in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Main results We first very briefly review some of relevant previous studies in order to motivate our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In 1906, Born developed stability theory for Euler’s elastica [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Among other results, he found the general principle that if a clamped planar elastica has no zero of the curvature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=', locally convex), then it is stable with respect to smooth perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Born also addresses more general elasticae, albeit with the help of numerical computations, and gave the first detailed comparisons with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Maddocks [20, 21] developed linear stability analysis for Euler’s elastica (with possibly nonuniform bending rigidity), which explains physically natural (in)stability for various boundary conditions, extending some previous results cited therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' For Euler’s elastica, it is shown that ‘higher modes’ (with many inflection points) are basically unstable for both clamped and pinned boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In particular, the results for the pinned boundary condition are summarized as follows: (M1) The pinned first mode loses its stability when the endpoints meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (M2) The pinned higher modes are always unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A schematic diagram for (M1) is given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The higher modes in (M2) correspond to the curves γN arc, γN loop with N ≥ 2 in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In fact, it is rigorously known (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' [19, 42]) that all the pinned elasticae are classified as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Com- bined with this classification, Maddocks’ results imply that the embedded convex arc γ1 arc is the only stable one (in the sense of linear stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Langer–Singer [15] developed a different approach, which not only covers spatial elasticae with special symmetry but also recovers some of Maddocks’ results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Stable (pinned) Unstable (pinned) Stable (clamped) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Stability of convex arcs and loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 4 TATSUYA MIURA AND KENSUKE YOSHIZAWA γ1 arc γ1 loop γ2 arc γ3 arc γ2 loop γ3 loop Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Pinned elasticae of arc type (left) and of loop type (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In a recent series of papers [34, 33, 36, 35, 37], focusing on planar elasticae subject to the clamped boundary condition, Sachkov and his coauthors developed a more detailed theory of not only stability but also minimality by an optimal control approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In particular, Sachkov found optimal rigidity principles in terms of the natural periodic structure of elasticae, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' statements (1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3) (or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='5)) in [33, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Here the periodicity means that the tangent direction is periodic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' the curve itself is generically quasi-periodic, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Sachkov’s principles are roughly summarized as follows: (S1) If a clamped elastica is minimal, then it does not exceed one period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (S2) If a clamped elastica is stable, then it does not contain three inflection points in its interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Principle (S1) is trivial for closed elasticae in view of scaling — any closed curve has less bending energy than its multiple covering of same length — but is not trivial for non-closed elasticae since simple scaling arguments do not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Principle (S2) is in line with Maddocks’ instability result for higher modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Both (S1) and (S2) are optimal (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Remarks 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2) and very useful for detecting (local) minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Wavelike elastica Borderline elastica Orbitlike elastica Inflection points Periodicity Examples Circular elastica Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Basic patterns of elasticae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Our main results extend all (M1), (M2), (S1), (S2) to a wide class of energy func- tionals (in the natural Sobolev framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Such extensions are highly nontrivial even for p-elasticae, since both Maddocks’ and Sachkov’s methods importantly use the fact that the curvature term is quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Indeed, Sachkov’s argument depends STABLE AND MINIMAL ELASTIC CURVES 5 on explicit computations only valid for the standard bending energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Maddocks’ argument is based on a rather ‘representation-free’ linear stability analysis (which makes it possible to address non-uniform cases) but the rigorous application of lin- ear stability analysis on the nonlinear level would be significantly delicate even for p-elasticae with p ̸= 2 due to the lack of Hilbert structures, which in particular leads to a generic loss of regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The rest of this section proceeds as follows: We first present general principles in the clamped case corresponding to (S1) and (S2) in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1, because of the simplicity of the statements and also their applicability to any boundary condition, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then we turn to the pinned case corresponding to (M1) and (M2) in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3, we collect some typical consequences in p-elastica theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Finally we explain the key idea of our proof in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' General rigidity principles: Clamped case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let p ∈ (1, ∞), L > 0 and P0, P1, V0, V1 ∈ R2 with |P0 − P1| < L and |V0| = |V1| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Define W 2,p arc (0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) := � γ ∈ W 2,p(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) �� |γ′| ≡ 1 � , the set of planar arclength parameterized Sobolev curves of length L, and then Aclamp := {γ ∈ W 2,p arc (0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) | γ(0) = P0, γ(L) = P1, γ′(0) = V0, γ′(L) = V1}, the admissible set subject to the clamped boundary condition, equipped with the relative topology induced by the W 2,p-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The first result extends (S1) in a contrapositive form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Our key hypothesis is the following property of regularity-improvement: (H1) If γ is a global minimizer of F in Aclamp, then γ ∈ C2([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Hereafter we often abbreviate γ ∈ C2([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) as γ ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This property is naturally expected whenever f is not too wild, since any minimizer weakly solves the Euler–Lagrange equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Under this hypothesis we obtain Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 (Rigidity of clamped global minimizers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Suppose (H1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If γ ∈ Aclamp has two points 0 ≤ s1 < s2 ≤ L with s2 − s1 < L such that γ′(s1) = γ′(s2) and − k(s1) ̸= k(s2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1) then γ is not a global minimizer of F in Aclamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This result indeed extends (S1) because after one period (s2 = s1 + period) one has γ′(s1) = γ′(s2) and k(s1) = k(s2) so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1) holds unless the curvature vanishes there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The next result extends (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' To this end we will similarly suppose: (H1’) If γ is a local minimizer of F in Aclamp, then γ ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In addition, we need to use a certain well-periodic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2 (Well-periodic curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We call an arclength parameterized planar curve γ : [0, L] → R2 well-periodic if γ has continuous signed curvature k ∈ C([0, L]) of the form k(s) = Φ(s − s0), where s0 ∈ R, and Φ ∈ C(R) is an antiperiodic odd function with antiperiod T > 0 such that { s ∈ R | Φ(s) = 0 } = TZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We also call T the antiperiod of the well-periodic curve γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This kind of periodicity naturally appears in the objective critical points thanks to the invariance of F with respect to the change of the sign of curvature as well as the orientation of parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' For this well-periodic class we obtain 6 TATSUYA MIURA AND KENSUKE YOSHIZAWA Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3 (Rigidity of clamped local minimizers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Suppose (H1’) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If γ ∈ Aclamp is well-periodic, and has three points 0 ≤ s1 < s2 < s3 ≤ L with s3 − s1 < L such that k(s1) = k(s2) = k(s3) = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2) then γ is not a local minimizer of F in Aclamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This result directly extends (S2), and gives an effective rigidity for ‘inflectional’ critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3) is optimal in the sense that if s2 − s1 = L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' s3 − s1 = L) then the assertion may fail even for classical elasticae, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Our principles immediately extend to more general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' For ex- ample, regardless the choice of boundary conditions, the same results hold true away from the endpoints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' more precisely, for a given minimal (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' stable) curve γ : [0, L] → R2 subject to any boundary condition, the restriction γ|[δ1,L−δ2] with δ1, δ2 > 0 must be minimal (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' stable) under the clamped boundary condition and hence satisfy the above assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In addition, our principles also directly work in the case of length-penalized problems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=', critical points of the functional F +λL among length-unconstrained admissible curves, where λ ∈ R and L denotes the length) as well as the case of higher codimensions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=', planar curves in Rn with any n ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' General rigidity principles: Pinned case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Now we turn to the pinned boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' More precisely, for L > 0 and P0, P1 ∈ R2 such that |P0−P1| < L, we consider the admissible class Apin := {γ ∈ W 2,p arc (0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) | γ(0) = P0, γ(L) = P1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In this class, a critical point often satisfies the so-called natural boundary condition that the curvature vanishes at the endpoints, thanks to the arbitrariness of first- order variations at the endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In view of this condition, it is natural to consider the following subclass of well-periodic curves, which we call m 2 -fold curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' the number m corresponds to the superscripts in Figure 2, and roughly speaking counts the modes with the convention that ‘1-fold’ means the one-period of the curvature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=', twice the antiperiod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='6 ( m 2 -Fold well-periodic curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' For m ∈ N, we say that a well- periodic curve γ : [0, L] → R2 is m 2 -fold if k(0) = 0 and T = L/m hold, where T is the antiperiod of γ defined in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We first state our instability result for higher modes, extending (M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Here we will suppose both the regularity-improvement and the natural boundary condition: If γ is a local minimizer of F in Apin, then γ ∈ C2 and k(0) = k(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (H2) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Suppose (H2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ ∈ Apin be a 1-fold well-periodic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then γ is not a local minimizer of F in Apin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We then address the first mode, extending (M1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In fact, our proof of this part will be based on a different mechanism from all the previous cases, and thus we will suppose a rather different type of hypothesis: f is strictly convex on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (H3) STABLE AND MINIMAL ELASTIC CURVES 7 Under this hypothesis we show that if a curve has a specific loop structure as in the left part of Figure 1 or such as γ1 loop in Figure 2, then it is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Suppose (H3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ ∈ Apin be a 1 2-fold well-periodic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If γ is not injective on (0, L) and if γ′(0) · (P1 − P0) > 0, then γ is not a local minimizer of F in Apin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Notice that the assumption automatically rules out the case that P0 = P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Applications to p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Now we discuss some concrete applications to p-elasticae under some well-chosen boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Our previous work ensures that for all p ∈ (1, ∞) the p-bending energy Bp satisfies hypotheses (H1), (H1’), (H2), (H3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' see [26] for C2-regularity in (H1), (H1’), (H2), and see [27] for the natural boundary condition in (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In addition, all wavelike p-elasticae obtained in [26] belong to the class of well-periodic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We first address the case of closed curves, which can be regarded as a special case of the clamped boundary condition by taking P0 = P1 and V0 = V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' It is shown in [26] that any closed planar p-elastica is either a circle or a figure-eight p-elastica, possibly multiply covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' It is easy to deduce from H¨older’s inequality and Fenchel’s theorem that a global minimizer is a once covered circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Here we apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3 to extend the known classification of stability for p = 2 [15, 35, 3] to a general power p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='9 (Stability of closed p-elasticae).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let p ∈ (1, ∞) and γ ∈ Aclamp with P0 = P1 and V0 = V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then γ is a local minimizer of Bp in Aclamp if and only if γ is either a possibly multiply covered circle or a once covered figure-eight p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Since the C1-regular homotopy classes of closed planar curves are exactly clas- sified by the rotation number, and since the above local minimizers have different rotation numbers, we reach a uniqueness theorem `a la Langer–Singer in R2 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' For each C1-regular homotopy class of immersed circles in R2, stable closed planar p-elasticae are unique up to similarity and reparameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We then turn to the pinned boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We call a critical point of Bp in Apin pinned p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In [27] the authors have classified all pinned p-elasticae and proved uniqueness of global minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In general, there are infinitely many pinned p-elasticae, which are either of wavelike or flat-core type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We first completely classify the stability of wavelike pinned p-elasticae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' It is shown in [27] that any wavelike pinned p-elasticae is classified as an m 2 -fold well- periodic curve as in Figure 2, to which Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='8 are applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' It turns out that all but the global minimizer is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='11 (Stability of wavelike pinned p-elasticae).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Suppose that γ ∈ Apin is a wavelike p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then γ is a local minimizer of Bp in Apin if and only if γ is a global minimizer (convex arc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The above wavelike result already yields strong rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Indeed, it is also shown in [27] that if |P0−P1| L < 1 p−1, then any pinned p-elastica is wavelike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Hence we reach the following uniqueness of stable pinned p-elasticae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Suppose either that p ∈ (1, 2], or that p ∈ (2, ∞) and |P0−P1| L < 1 p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then stable pinned planar p-elasticae in Apin are unique up to isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 8 TATSUYA MIURA AND KENSUKE YOSHIZAWA For p ∈ (1, 2], our result covers the full range of pinned boundary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This not only extends Maddocks’ linear stability analysis as in Figure 1 from p = 2 to p ∈ (1, 2], but also directly justifies it on the nonlinear level (even for p = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In particular, our result could possibly be the first explicit statement on the uniqueness of stable pinned elasticae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' At any rate, the result is completely new for p ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We further make progress by addressing flat-core pinned p-elasticae, the only remaining case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Flat-core p-elasticae appear only in the degenerate case p > 2 and have peculiar non-periodicity, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Figure 4, thus representing the substantial difference between the structures of p-elasticae and of the classical elasticae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In particular, our previous rigidity results are based on periodicity and not directly applicable to flat-core p-elasticae in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' However, the same machinery enables us to develop rather ad-hoc arguments to prove the following new instability: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='13 (Unstable flat-core pinned p-elasticae).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A flat-core pinned planar p-elastica is unstable either if it contains two adjacent loops in opposite directions, or if a loop touches an endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Flat-core p-elasticae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The left two curves are unstable (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The right curve is classified as a quasi-alternating flat-core p-elastica (Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In particular, invoking the classification in [27], we obtain the following di- chotomy theorem for stable pinned planar p-elasticae without any assumption on the exponent p nor the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Here we call that a flat-core pinned planar p-elastica is quasi-alternating if the curve is not ruled out by the above the- orem, or in other words given by a certain alternating concatenation of segments and multi-loops as in Figure 4 (see Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='5 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Any stable pinned planar p-elasticae is either a global minimizer (convex arc) or a quasi-alternating flat-core p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' It is an interesting open problem whether there indeed exists a stable quasi- alternating flat-core p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If it were true, then it would give an interesting example of a stabilization phenomenon due to degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Finally we explain the main idea that underlies most of our argu- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' All the aforementioned previous studies [6, 20, 21, 15, 34, 33, 36, 35, 37] involve explicit computations on Jacobian elliptic functions or some functional an- alytic structures, which are based on the quadraticity to some degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' On the other hand, Avvakmov–Karpenkov–Sossinsky [3] gave a new proof that multiply-covered STABLE AND MINIMAL ELASTIC CURVES 9 figure-eight elasticae are unstable in the plane, by directly constructing an energy- decreasing perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This method has the strong advantage that it is almost purely based on the geometric structure of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' However, their construc- tion relies on the very special geometry of the figure-eight elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A particularly important point is that their construction involves rescaling and hence does not directly extend to some non-closed curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In fact, such a direct construction of an energy-decreasing perturbation is often obstructed by the fixed-length constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A trick we propose here to circumvent the above obstruction is a very simple indirect argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The key idea is to construct an equal-energy competitor that lies in the energy space but is not regular enough to be minimal or stable, implying a contradiction to our natural regularity hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' It turns out that the construc- tions of equal-energy competitors are reduced to surprisingly simple ‘cut-and-paste’ arguments, with the help of intrinsic periodicity and symmetry of objective criti- cal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We demonstrate the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 for a special example of an elastica (but in fact the idea is completely same in the general case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' As in Figure 5, for a given ‘more than one period’ orbitlike elastica we can rotate a ‘one-period’ part to construct a competitor which is of class W 2,2 and has the unchanged length and bending energy but cannot be a global minimizer since the discontinuity of the curvature at the cut points contradicts the C2-regularity, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' To construct local perturbations we need more structures as assumed in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3 since the perturbations need to be close to the original curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' s1 s2 γ ¯γ π Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' An original curve γ with two points s1, s2 satisfying γ′(s1) = γ′(s2) and k(s1) = k(s2) ̸= 0 (left) and the competitor ¯γ with equal energy to γ (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' One of the main discoveries in our study would be the fact that this simple trick is quite robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Although the new results presented above already provide general and optimal rigidity results for planar critical points, we also expect more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' As an example, in Section 7, we apply our trick to elasticae in R3 to obtain old and new spatial rigidity results in a unified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 10 TATSUYA MIURA AND KENSUKE YOSHIZAWA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Preliminary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We prepare a notation on a concatenation of curves: For γj : [aj, bj] → R2 with Lj := bj − aj ≥ 0, we define γ1 ⊕ γ2 : [0, L1 + L2] → R2 by (γ1 ⊕ γ2)(s) := � γ1(s + a1), s ∈ [0, L1], γ2(s + a2 − L1) + γ1(b1) − γ2(a2), s ∈ [L1, L1 + L2], and inductively define γ1 ⊕ · · · ⊕ γN := (γ1 ⊕ · · · ⊕ γN−1) ⊕ γN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We also define N � j=1 γj := γ1 ⊕ · · · ⊕ γN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This notation will be useful for our cut-and-paste procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Similarly, for a point P ∈ R2 and a curve γ : [0, L] → R2, we define (P ⊕ γ)(s) := γ(s) − γ(0) + P, s ∈ [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Well-periodic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Here we collect some fundamental properties of well- periodic curves in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Recall that a function Φ is said to be antiperiodic if there is T > 0 such that −Φ(x) = Φ(x + T) holds for all x ∈ R, and this T is called antiperiod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The antiperiod T of a well-periodic curve in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2 is uniquely determined as it is also related with the zero set of the curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We stress that we have imposed continuity on the curvature (and Φ) of a well-periodic curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' this is natural in view of our regularity hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This continuity implies that Φ does not change the sign on each connected component of R \\ TZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In addition, the combination of odd symmetry and periodicity implies the symmetry Φ( T 2 +s) = Φ( T 2 −s), and also the sign-changing property Φ(s)Φ(s + T) < 0 for s ∈ R \\ TZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The following lemma exhibits some properties of well-periodic curves which we will use later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The proof is elementary and safely omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 (Symmetry of well-periodic curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ : [0, L] → R2 be a well- periodic planar curve and k be the signed curvature of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If k has three points 0 ≤ s1 < s2 < s3 ≤ L such that k(si) = 0 (i = 1, 2, 3), and k(s) ̸= 0 for s ∈ (s1, s2) ∪ (s2, s3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1) then we have γ′(s1 + σ) = γ′(s3 + σ) holds whenever s1 + σ, s3 + σ ∈ [0, L], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2) k(σ + s1) = −k(s3 − σ) holds whenever σ + s1, s3 − σ ∈ [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In addition, if a well-periodic curve γ : [0, L] → R2 is m 2 -fold in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='6, its signed curvature k satisfies that k(s) = 0 ⇐⇒ s = 0, 1 mL, 2 mL, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3) In particular, we may assume that k = Φ without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The definition of ‘ m 2 -fold’ here is in line with the terminology of ‘ m 2 - fold figure-eight p-elastica’ used in [26, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Note that (p, m, r)-arcs and (p, m, r)-loops defined in [27, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2] are also m 2 -fold well-periodic curves (see Figure 2 for p = 2 and m = 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' STABLE AND MINIMAL ELASTIC CURVES 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Rigidity under the clamped boundary condition In this section we prove Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Rigidity for minimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We first prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The key idea is al- ready mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='4 and Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Suppose that γ is a global min- imizer of F in Aclamp, and has two points s1, s2 ∈ [0, L] satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Up to reversing the parameterization, we may assume that s1 may be equal to 0 but s2 < L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let us define a continuous function ¯γ : [0, L] → R2 by ¯γ := γ|[0,s1] ⊕ R � γ|[s1,s2] � ⊕ γ|[s2,L], where R denotes the affine transformation describing the rotation through 180 degrees around the point c := 1 2 � γ(s1) + γ(s2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In particular, we see that ¯γ(s) = −γ(−s + s1 + s2) + 2c, s ∈ [s1, s2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1) and by definition of ¯γ we have ¯γ ∈ C1([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2), particularly noting that lim s↓s1 ¯γ′(s) = γ′(s2) = γ′(s1), lim s↑s2 ¯γ′(s) = γ′(s1) = γ′(s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We also notice that the signed curvature ¯k of ¯γ is ¯k(s) = � −k(−s + s1 + s2), s ∈ (s1, s2), k(s), s ∈ [0, L] \\ [s1, s2], which is bounded and hence ¯γ ∈ W 2,∞(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) ⊂ W 2,p(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Since ¯γ satisfies the same clamped boundary condition as γ (even if s1 = 0), we have ¯γ ∈ Aclamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Moreover, the above formula for ¯k implies F(¯γ) = F(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Hence ¯γ is also a global minimizer of F in Aclamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then it follows from (H1) that ¯γ ∈ C2([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' On the other hand, we see that by definition of ¯k, lim s↑s2 ¯k(s) = −k(s1), lim s↓s2 ¯k(s) = k(s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1) implies that ¯γ ̸∈ C2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 is optimal in the sense that if s2 − s1 = L then the assertion may fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In fact, in the case of P0 = P1 and V0 = V1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=', closed curves), a one-fold circle is a global minimizer of B2 in Aclamp but its endpoints satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Rigidity for stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Now we turn to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We first give an abstract criterion based on a contradiction argument, which clarifies how to deduce instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' As in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1, a key point is that we allow equality in condition (ii) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Suppose (H1’) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ ∈ Aclamp and suppose that there exists a sequence {γj}j∈N ⊂ Aclamp satisfying the following three conditions: � � � � � (i) γj → γ in W 2,p(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) as j → ∞, (ii) F(γj) ≤ F(γ) for all (large) j ∈ N, (iii) γj /∈ C2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) for all (large) j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (C) 12 TATSUYA MIURA AND KENSUKE YOSHIZAWA Then γ is not a local minimizer of F in Aclamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ ∈ Aclamp satisfy the three conditions in (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We argue by contradic- tion, so suppose that γ is a local minimizer of F in Aclamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then there is δ > 0 such that F(γ) ≤ F(ξ) for ξ ∈ Aclamp with ∥γ − ξ∥W 2,p < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1) By (C)-(i), there exists j0 ∈ N such that ∥γ −γj∥W 2,p < δ/2 for j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Henceforth we consider only j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' It follows from (C)-(ii) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1) that F(γj) ≤ F(γ) ≤ F(ξ) for ξ ∈ Aclamp with ∥γj − ξ∥W 2,p < δ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This implies that γj are also local minimizers of F in Aclamp for all j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' There- fore, γj ∈ C2([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) follows from (H1’), which contradicts (C)-(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This com- pletes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ In order to show Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3, we now apply the above criterion to prove the instability of any well-periodic curve γ that ‘strictly’ contains three inflection points s1 < s2 < s3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Our perturbation γj here is constructed as follows: Noting that γ has symmetry centered at s2 as in Figure 6 (1), we first cut the curve γ out at the perturbed points s1 + 1 j and s3 + 1 j (with large j) and then rotate it through 180 degrees around the middle of the cut points as in Figure 6 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By symmetry the perturbed curve γj is close to the original one γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' It turns out that by periodicity and our rotation procedure, the perturbed curve γj has discontinuous curvature at the cut points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' s1 s2 s3 s1+ 1 j s3+ 1 j π (1) (2) (3) cj Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (1) An original well-periodic curve γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (2) Construction of a perturbation γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (3) Convergence as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ ∈ Aclamp be a well-periodic curve and have three points satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Up to reversing the parameterization and taking the first three inflection points, we may assume that s1, s2, s3 satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1), in particular s3 < L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In addition, up to reflection, we may assume that k > 0 in (s1, s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then since γ is well-periodic, k < 0 in (s2, s3) and k > 0 in (s3, s3 + δ) with some small δ ∈ (0, L − s3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' For each integer j ≥ j0 with some j0 such that 1/j0 < δ, we define a continuous function γj : [0, L] → R2 by γj := γ|[0,s1+ 1 j ] ⊕ Rj � γ|[s1+ 1 j ,s3+ 1 j ] � ⊕ γ|[s3+ 1 j ,L], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2) STABLE AND MINIMAL ELASTIC CURVES 13 where Rj denotes the rotation through 180 degrees around cj := 1 2 � γ(s1 + 1 j ) + γ(s3 + 1 j ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In particular, we see that γj(s) = −γ(−s + s1 + s3 + 2 j ) + 2cj, s ∈ Ij := (s1 + 1 j , s3 + 1 j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In fact γj is still a unit-speed C1-curve by construction and by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' for example, around s1 + 1 j , by computing the one-sided limits independently and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2), we deduce that γ′ j(s1 + 1 j ) = γ′(s1 + 1 j ), and also the derivative γ′ j is continuous by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' in view of lim s↓s1+ 1 j γ′ j(s) = lim s↓s1+ 1 j γ′(−s + s1 + s3 + 2 j ) = γ′(s3 + 1 j ) = γ′(s1 + 1 j );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' the same argument works for s3 + 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Hence γj ∈ C1([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) for any j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In addition, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2) the signed curvature kj of γj is given by kj(s) = � −k(−s + s1 + s3 + 2 j ), s ∈ Ij k(s), s ∈ [0, L] \\ ¯Ij, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3) where ¯Ij := [s1 + 1 j , s3 + 1 j ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Since k ∈ C([0, L]) by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2, we have kj ∈ L∞(0, L) and hence γj ∈ W 2,∞(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) ⊂ W 2,p(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) for all j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Moreover, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2) and the fact that s1 + 1 j > 0 and s3 + 1 j < L, it is clear that the curve γj satisfies the same boundary condition as γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Thus we have {γj}j≥j0 ⊂ Aclamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Now we prove that the sequence {γj}j≥j0 satisfies all the conditions in (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We first check (C)-(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Recall that by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1, k(s) = −k(−s + s1 + s3), s ∈ Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3) implies that kj → k a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' in (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In addition, noting that ∥kj∥L∞ ≤ ∥k∥L∞, we obtain kj → k in Lp(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' From this convergence and the fact that γ(0) = γj(0) and γ′(0) = γ′ j(0), we deduce that γj → γ in W 2,p(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Next we check (C)-(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Since � s3+ 1 j s1+ 1 j f � |kj(s)| � ds = � s3+ 1 j s1+ 1 j f � |k(−s + s1 + s3 + 2 j )| � ds = � s1+ 1 j s3+ 1 j −f � |k(s)| � ds = � s3+ 1 j s1+ 1 j f � |k(s)| � ds and since γ and γj agree elsewhere, we have F(γj) = F(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Finally, the discontinuity of curvature in (C)-(iii) follows since for any large j, thanks to k|(s1,s2) > 0, we have lim s↑s1+ 1 j kj(s) = k(s1 + 1 j ) > 0, lim s↓s1+ 1 j kj(s) = −k(s3 + 1 j ) = −k(s1 + 1 j ) < 0, and hence kj is not continuous at s = s1 + 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Since Aclamp ⊂ Apin, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3) holds true with Aclamp replaced by Apin whenever hypothesis (H1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (H1’)) holds true for all unit vectors V0, V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 14 TATSUYA MIURA AND KENSUKE YOSHIZAWA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Rigidity under the pinned boundary condition This section is devoted to the proofs of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Instability of one-fold waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We first prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In the same spirit as Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2, it is sufficient to construct {γj}j∈N ⊂ Apin satisfying � � � � � (i) γj → γ in W 2,p(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) as j → ∞, (ii) F(γj) ≤ F(γ) for all (large) j ∈ N, (iii) the curvature kj of γj satisfies kj(0) ̸= 0 for all (large) j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (Cp) Here we construct such a perturbation γj by extending the original curve by using its intrinsic periodicity and shift the domain as in Figure 7 so that the curvature does not vanish at the endpoints: s = 1 j (1) (2) (3) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (1) A 1-fold well-periodic curve γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (2) Construction of a perturbation γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (3) Convergence as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3), we infer that the curvature k of γ satisfies k(s) = 0 ⇐⇒ s = 0, L 2 , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1) For each integer j ≥ j0 with some j0 such that 1/j0 < L, we define a continuous map γj : [0, L] → R2 by γj := P0 ⊕ γ|[ 1 j ,L] ⊕ γ|[0, 1 j ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2) Then we infer from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 that γ′(0) = γ′(L) and hence lim s↓L− 1 j γ′ j(s) = γ′(0) = γ′(L) = lim s↑L− 1 j γ′ j(s), which implies that γj ∈ C1([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) for each j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The signed curvature kj of γj is kj(s) := � k(s + 1 j ) s ∈ [0, L − 1 j ), k(s − L + 1 j ) s ∈ (L − 1 j , L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3) Since lims↑L− 1 j kj(s) = lims↓L− 1 j kj(s) = 0 holds by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1), kj is continuous in [0, L], and hence γj ∈ C2([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By definition we have γj(0) = P0, γj(L) = P1, and L[γj] = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Thus we have {γj}j≥j0 ⊂ Apin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Now we prove that the sequence {γj}j≥j0 satisfies all the conditions in (Cp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' STABLE AND MINIMAL ELASTIC CURVES 15 First, we check (Cp)-(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3) that kj → k a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' in (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Com- bining this with the fact that ∥kj∥L∞ = ∥k∥L∞, we see that kj → k in Lp(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Since γj(0) = γ(0) and γ′ j(0) = γ′( 1 j ) → γ′(0), we have γj → γ in W 2,p(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Next, we check (Cp)-(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' It follows that F(γj) = � L− 1 j 0 f � |kj(s)| � ds + � L L− 1 j f � |kj(s)| � ds = � L− 1 j 0 f � |k(s + 1 j )| � ds + � L L− 1 j f � |k(s − L + 1 j )| � ds = F(γ), and hence {γj}j≥j0 satisfies F(γj) = F(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Finally, we show (Cp)-(iii), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=', the curvature at an endpoint does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In fact, it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2) that kj(0) = k( 1 j ) ̸= 0 for any j ≥ j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Thus {γj}j≥j0 satisfies all the conditions in (Cp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Therefore, if γ were a local minimizer, then so were all γj with large j, but this would contradict (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Instability of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Here we prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Throughout this subsec- tion, for notational simplicity, without loss of generality we assume that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='4) P0 = (0, 0) and P1 = (l, 0), where l := |P0 − P1| ∈ [0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' To begin with, we note here that the antiperiodicity of the curvature of well-periodic curves yields the following symmetry of curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' this is closely related with the symmetry Φ( T 2 + s) = Φ( T 2 − s) which we have already observed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ = (x, y) : [0, L] → R2 be a 1 2-fold well-periodic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If γ(0) = (0, 0) and γ(L) = (l, 0) with some l ̸= 0, then x(L − s) + x(s) = l, y(L − s) = y(s), for s ∈ [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This easily follows from elementary differential geometry with the fact that k(s) = Φ(s) = −Φ(s − L) = Φ(L − s) = k(L − s) for any s ∈ [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='5) The assumption l ̸= 0 is used for forcing the reflection axis to be vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ In addition, again for notational simplicity, we also assume that f(0) = 0 by replacing f(t) with f(t)−f(0) if necessary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' this does not lose generality since the functional � γ f(|k|) ds is (locally) minimized if and only if so does � γ(f(|k|)−f(0)) ds under the fixed-length constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Note that if f satisfies (H3) and f(0) = 0, then one easily verifies that λf(λ−1t) < f(t) for any t ∈ (0, ∞) and λ > 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='6) since f((1 − λ−1)0 + λ−1t) < (1 − λ−1)f(0) + λ−1f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In what follows we will construct an energy-decreasing perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The key idea here is to perform an odd extension of the loop and shift the domain so that the symmetric loop is asymmetrically perturbed as in Figure 8 (3), and finally rescale the loop as in Figure 8 (4) in order to increase the shortened length and recover the admissibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' All these procedures decrease the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The convexity hypothesis will be used in the last rescaling step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 16 TATSUYA MIURA AND KENSUKE YOSHIZAWA (1) s = − cj s = L− 1 j (2) (3) (4) 0 l 0 l 0 l l Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (1) An original curve with a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (2) Shifting the end- points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (3) Rigid motion for the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (4) Rescaling the loop for the length constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We first note that by assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='4) with l > 0 we have the reflection symmetry in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1, and in addition by γ′(0) · (P1 − P0) > 0 we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='7) x′(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We also note that the non-injectivity assumption precisely means that there are distinct a, b ∈ (0, L) such that γ(a) = γ(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='8) In the proof below, we shall construct a family of curves {γj}j≥j0 ⊂ Apin satisfying F(γj) < F(γ) for any j ≥ j0 and γj → γ in W 2,p(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='9) First, we define an odd extension of γ to the domain [−L, L] by � γ(s) s ∈ [0, L], −γ(−s) s ∈ [−L, 0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='10) which is also denoted by γ : [−L, L] → R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then γ ∈ C1([−L, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) holds since γ is of class C1 around s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' For j ≥ 1 L, we have y( 1 j ) = y(L − 1 j ) by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In addition, since x′(0) > 0 by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='7) and also x′(L) > 0 by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1, for all large j we have x(L − 1 j ) − x( 1 j ) < x(L) − x(0) = l so that ��γ(L − 1 j ) − γ(0) �� < l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='11) On the other hand, we infer from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 that x(L − 1 j ) − x(− 1 j ) = x(L − 1 j ) + x( 1 j ) = l, y(L − 1 j ) = y( 1 j ), and since |k(s)| ̸= 0 for any small s > 0, we have y( 1 j ) ̸= 0 for all large j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Hence ��γ(L − 1 j ) − γ(− 1 j ) �� = � l2 + 4y( 1 j )2 > l (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='12) for all j ≥ j0 with sufficiently large j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='11) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='12), for each j ≥ j0 there is cj ∈ (− 1 j , 1 j ) such that ���γ(L − 1 j ) − γ(−cj) ��� = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='13) (In fact we can show cj > 0 but this is not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=') Define ¯γj : [−cj, L − 1 j ] → R2 by ¯γj(u) := Qj � γ(u) − γ(−cj) � , u ∈ [−cj, L − 1 j ], STABLE AND MINIMAL ELASTIC CURVES 17 where Qj is a rotation matrix such that Qj(γ(L− 1 j )−γ(−cj)) = (l, 0) and Qj → Id (by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='13), such Qj exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then we see that ¯γj(−cj) = (0, 0), ¯γj(L − 1 j ) = (l, 0), ¯γj ∈ W 2,p(−cj, L − 1 j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='14) but we still have L[¯γj] < L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Now we normalize the length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let a, b ∈ (0, L) satisfy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then ¯γj(a) = ¯γj(b) also holds for all j ≥ j0 with j0 so large that 1 j0 < a < b < L − 1 j0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' For j ≥ j0 we define a continuous function ¯Γj : [−cj, L − 1 j ] → R2 by ¯Γj := γj|[−cj,a] ⊕ λjγj|[a,b] ⊕ γj|[b,L− 1 j ], λj := b − a + 1 j − cj b − a > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Note that L[¯Γj] = L follows by the choice of the scaling factor λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Hereafter we let γj denote the arclength parameterization of ¯Γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We have {γj}j≥j0 ⊂ C([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Since ¯Γj is defined only by rescaling at a self-intersection point, we also have {γj} ⊂ C1([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We also notice that the signed curvature kj of γj is given by kj(s) = � � � � � k(s − cj), 0 ≤ s < a + cj, λ−1 j k(λ−1 j (s − a − cj) + a), a + cj < s < b + 1 j , k(s − 1 j ), b + 1 j < u ≤ L, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='15) where k is the signed curvature of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Now we check that {γj}j≥j0 ⊂ Apin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We have k ∈ C([0, L]) by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2, and hence by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='15) we get {γj}j≥j0 ⊂ W 2,∞(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) ⊂ W 2,p(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Moreover, γj(0) = (0, 0) and γj(L) = (l, 0) by definition and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Recalling L[γj] = L, we have {γj}j≥j0 ⊂ Apin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Henceforth we show that the family {γj}j≥j0 satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We first show the strict inequality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='15) and the change of variables that F(γj) = � a+cj 0 f � |kj(s)| � ds + � b+ 1 j a+cj f � |kj(s)| � ds + � L b+ 1 j f � |kj(s)| � ds = � a −cj f � |k(s)| � ds + � b a f � λ−1 j |k(s)| � λj ds + � L− 1 j b f � |k(s)| � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='16) By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='6) and λj > 1, and the fact that k ̸= 0 in (0, L) since γ is 1 2-fold, we obtain � b a f � λ−1 j |k(s)| � λj ds < � b a f � |k(s)| � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='10) we have |k(s)| = |k(−s)| for s ∈ [− 1 j , 1 j ], and hence by |cj| < 1 j and f ≥ 0, � a −cj f � |k(s)| � ds ≤ � |cj| 0 f � |k(s)| � ds + � a 0 f � |k(s)| � ds ≤ � 1 j 0 f � |k(s)| � ds + � a 0 f � |k(s)| � ds = � L L− 1 j f � |k(s)| � ds + � a 0 f � |k(s)| � ds, where in the last equality we used k(L − s) = k(s), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Therefore, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='16), F(γj) < � a 0 f � |k(s)| � ds + � b a f � |k(s)| � ds + � L b f � |k(s)| � ds = F(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 18 TATSUYA MIURA AND KENSUKE YOSHIZAWA It remains to show that γj → γ in W 2,p(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='15) and the fact that λj → 1, cj → 0, we see that kj → k a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' in (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In addition, since ∥kj∥L∞ ≤ ∥k∥L∞ holds by λj > 1, it follows that kj → k in Lp(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Noting also the fact that γj(0) = γ(0) and γ′ j(0) = Qjγ′( 1 j ) → γ′(0), we obtain γj → γ in W 2,p(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='8 will be used for showing instability of a (p, r, 1)-loop with r := |P0−P1| L > 0, but does not cover the case of r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In fact, if r = 0, then we can regard a (p, r, 1)-loop as a half-fold figure-eight p-elastica, which is a global minimizer of Bp in Apin [27, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3] and hence obviously stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Therefore it is necessary to assume that P0 ̸= P1 at least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Planar p-elasticae In this section we discuss the stability of closed and pinned planar p-elasticae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Closed p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We first apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3 to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='9, thus classifying the stability of closed planar p-elasticae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let Aclosed denote the set Aclamp in the special case that P0 = P1 and V0 = V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Recall from [26, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='6] that any closed planar p-elastica is either a circle or a figure-eight p-elastica, possibly multiply covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' To begin with, we observe that an m-fold circle and a 1-fold figure-eight p-elastica (in the sense of [26]) are indeed stable: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If γ is an m-fold circle, where m ∈ N, or a 1-fold figure-eight p-elastica, then γ is a local minimizer of Bp in Aclosed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' For m ∈ N ∪ {0}, let Zm ⊂ Aclosed denote the subset of fixed rotation number m: Zm := � γ ∈ Aclosed ��� N[γ] := 1 2π � L 0 k ds = m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Since the functional N is constant in a small W 2,p-neighborhood of any element of Aclosed, if γ is a minimizer of Bp in Zm, then γ is a local minimizer in Aclosed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' It suffices to show that an m-fold circle (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' a 1-fold figure-eight p-elastica) is a minimizer of Bp in Zm if m ≥ 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' m = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The existence of a minimizer ¯γm in Zm follows from the standard direct method (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' [27, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then, ¯γm is also a p-elastica by the Lagrange multiplier method [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' On the other hand, by the classification for closed p-elasticae [26, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='6], in the case of m ≥ 1, ¯γm must be an m-fold circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In the remaining case of m = 0, the same classification also implies that any p-elastica with rotation number 0 is an n-fold figure-eight p-elastica for n ∈ N, and comparing their p-bending energy, we find that ¯γ0 must be a 1-fold figure-eight p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Here is a good position to observe that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3 is optimal in the sense that if s3 − s1 = L then the assertion may fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In fact, a 1-fold figure-eight p-elastica is a local minimizer of F = Bp in Aclosed by the above proposition, while it is well-periodic and satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2) at s = 0, L 2 , L if the endpoints are arranged to be located at the crossing of the figure-eight (inflection points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We are now ready to classify stable p-elasticae among closed curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' STABLE AND MINIMAL ELASTIC CURVES 19 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In view of [26, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='6] and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 it suffices to prove instability of m-fold figure-eight p-elasticae for m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If m ≥ 2, then m-fold figure-eight p-elasticae are well-periodic and have at least three inflection points satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2), and hence they are unstable in Aclosed by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' As discussed in the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1, if m ≥ 1, then any global minimizer with rotation number m is an m-fold circle, which is stable in Aclosed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If m = 0, then it follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='9 that any stable zero-rotation-number planar closed curve is a 1-fold figure-eight p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Wavelike pinned p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Next we prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='11 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='12, which are now almost direct consequences of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='7, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='8, combined with our previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ be a wavelike pinned p-elastica and let r := |P0−P1| L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then it follows from [27, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1] that there exists n ∈ N such that γ is either a (p, r, n)-arc or a (p, r, n)-loop, for which we write γn arc or γn loop, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By [26, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='7], every pinned p-elastica is of class C2, and by [26, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3], the signed curvature of a wavelike p-elastica can be expressed by the so-called p-elliptic function cnp, which is an odd, antiperiodic, and continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Therefore, γn arc and γn loop are well-periodic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Furthermore, γn arc and γn loop (n ≥ 3) are n 2 -fold well-periodic curves and contain three inflection points satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1), γ2 arc and γ2 loop are 1-fold well-periodic curves, γ1 loop is a 1 2-fold well-periodic curve satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='7) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='8) if |P0−P1| > 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' [27, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='8] for existence of a loop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Hence, γn arc and γn loop with n ≥ 3 are unstable by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3 with Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3, while so are γ2 arc and γ2 loop by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Moreover, if |P0 − P1| > 0, then γ1 loop is unstable by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Therefore, in any case, the only remaining candidate for stable wavelike pinned p-elasticae is the global minimizer γ1 arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The proof is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If p ∈ (1, 2] or p ∈ (2, ∞) and |P0−P1| L < 1 p−1, then any pinned p-elastica is a wavelike p-elastica (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' [27, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This together with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='11 asserts that any stable wavelike pinned p-elastica is a (p, r, 1)-arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Flat-core p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Now we discuss some instability criteria for flat-core p-elasticae, and prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Flat-core p-elasticae appear as special exam- ples of planar p-elasticae, and they are obtained by concatenating certain ‘loops’ and ‘segments’ with some arbitrariness (see [41, 26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In view of the natural bound- ary condition induced by the pinned boundary condition, we focus on flat-core p-elasticae each of whose endpoints is an endpoint of a segment or of a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' More precisely, we consider a curve γ : [0, L] → R2 represented by, up to similarity and reparameterization, γ = γL0 seg ⊕ � N � j=1 γσj loop ⊕ γLj seg � (F1) for some N ∈ N, {σj}N j=1 ⊂ {+, −}, and L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' , LN ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Here γLj seg(s) := (s, 0) for s ∈ [0, Lj], γ+ loop denotes a certain loop, and γ− loop denotes the reflection of γ+ loop with 20 TATSUYA MIURA AND KENSUKE YOSHIZAWA respect to the e1-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Although the explicit parameterization of γ+ loop is known, here we only use the following special properties: the curve γ+ loop = (X, Y ) : [0, 1] → R2 is an arclength parameterized curve such that the tangential vectors at both the endpoints are rightward, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=', (γ+ loop)′(0) = (γ+ loop)′(1) = (1, 0), (F2) the signed curvature kloop of γ+ loop satisfies kloop > 0 in (0, 1), kloop(0) = kloop(1) = 0, (F3) and γ+ loop is reflectionally symmetric in the sense that X(s) + X(1 − s) = 2X( 1 2), Y (s) = Y (1 − s) for s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (F4) If γ is a pinned planar flat-core p-elastica, then these properties directly follow by our previous classification [27, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1] and explicit formulae [26, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3] (see also Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' In addition, combining (F1) with (F2), we see that any curve γ of the form (F1) satisfies γ′(0) = γ′(L) = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1) The first criterion ensures that all loops need to lie ‘strictly inside’ for stability under the pinned boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ ∈ Apin be a flat-core p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If a loop touches an endpoint, then γ is not a local minimizer of Bp in Apin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Without loss of generality, we may assume that γ(0) is an endpoint of a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' For each j ≥ j0 with some 1/j0 < L, we define γj : [0, L] → R2 by γj := P0 ⊕ γ|[ 1 j ,L] ⊕ γ|[0, 1 j ], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then it is clear that γj ∈ C([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Property (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1) ensures that γj ∈ C1([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Since γ ∈ C2([0, L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2), the signed curvature kj of γj is bounded, in particular, γj ∈ W 2,∞(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2) ⊂ W 2,p(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By definition we have γj(0) = P0, γj(L) = P1, and L[γj] = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Thus {γj}j≥j0 ⊂ Apin holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' As in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='7, we see that {γj}j≥j0 satisfies the conditions (Cp)-(i) and (Cp)-(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By property (F3), the signed curvature kj of γj satisfies kj(0) ̸= 0, and hence the condition (Cp)-(iii) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Thus {γj}j≥j0 satisfies all the conditions in (Cp) and hence γ is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ s = 1 j (1) (2) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (1) A flat-core p-elastica with a loop touching an end- point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (2) Construction of a perturbation γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' STABLE AND MINIMAL ELASTIC CURVES 21 The next criterion ensures that a positive gap is necessary between two loops in opposite directions for stability even under the clamped boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ ∈ Aclamp be a flat-core p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If γ contains one loop and a part of a loop in the opposite direction with no segment between the two loops, then γ is not a local minimizer of Bp in Aclamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ ∈ Aclamp be a flat-core p-elastica satisfying the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This means that there is some a ∈ (0, 1] such that if we define Γ := γ+ loop ⊕ γ− loop|[0,a], then up to similarity and orientation of the parameter, the curve γ can be decom- posed as γ = γ|[0,s1] ⊕ Γ ⊕ γ|[s2,L] for some s1, s2 ∈ [0, L] (where L denotes the length after rescaling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' For each j ≥ j0 with some j0 such that 1/j0 < a, we define Γj := γ− loop|[0, 1 j ] ⊕ γ+ loop|[1− 1 j ,1] ⊕ γ+ loop|[0,1− 1 j ] ⊕ γ− loop|[ 1 j ,a], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Note that Γj is a unit-speed C1-curve by properties (F2), (F4), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' On the other hand, by property (F3), the curvature of Γj is discontinuous at s = 1/j and hence Γj is not of class C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We define γj := γ|[0,s1] ⊕ Γj ⊕ γ|[s2,L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Now we prove that the sequence {γj}j≥j0 satisfies all the conditions in (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We first check {γj}j≥j0 ⊂ Aclamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By definition, γj is a C1-curve and L[γj] = L holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Since γ is a C2-curve, the signed curvature of γj is of bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Therefore {γj}j≥j0 is a W 2,p-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By construction, γj satisfies the same boundary condition as γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Thus we have {γj}j≥j0 ⊂ Aclamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The condition (C)-(i) follows by straightforward calculations combined with the fact that the signed curvature of γ+ loop is uniformly continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The condition (C)- (ii) follows since Bp[Γ] = Bp[Γj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The condition (C)-(iii) follows from the fact that Γj /∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Consequently, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2 γ is unstable in Aclamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ (1) (2) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (1) A flat-core p-elastica with adjacent opposite loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (2) Construction of a perturbation γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='13 is now already complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' It directly follows by Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ 22 TATSUYA MIURA AND KENSUKE YOSHIZAWA Now we introduce a new class of flat-core p-elasticae, which are not covered by the previous two instability results (and presumably stable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let us first call a flat-core planar p-elastica γ alternating if up to similarity and reparameterization, γ is given as in (F1) with strictly positive L0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' , LN > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The strict positivity particularly means that the segments and the loops appear alternately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By relaxing the alternating property, we now define the following class: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='5 (Quasi-alternating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ be a flat-core planar p-elastica of the form (F1) with some N ∈ N, σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' , σN ∈ {+, −}, and L0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' , LN ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We call γ quasi-alternating if the following two conditions hold: (i) L0, LN > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (ii) For j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' , N − 1}, if Lj = 0, then σj = σj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ be a pinned planar p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then, by [27, The- orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1], γ is a wavelike p-elastica or a flat-core p-elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If γ is wavelike and stable, then Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='11 implies that γ is a one-fold arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' On the other hand, suppose that γ is flat-core and stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By [27] any pinned flat-core p-elastica is of the form (F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='4 imply conditions (i) and (ii) in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='5, respectively, and hence γ must be quasi-alternating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Spatial elasticae Finally we discuss applications to spatial elasticae, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=', critical points of the bending energy B[γ] := � γ |κ|2ds, where κ := γss, among fixed-length W 2,2-curves in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Spatial elasticae are also smooth by a standard bootstrap argument so that our trick is applicable well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A classification of spatial elasticae is obtained by Langer–Singer (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Here we obtain new rigidity principles due to the presence of spatial pertur- bations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' All the results are concerning the admissible space Aclamp of curves in W 2,2 arc (0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' R3) subject to the standard clamped boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 (Rigidity for spatially minimal elasticae).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ ∈ Aclamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If there are 0 ≤ s1 < s2 ≤ L with s2 −s1 < L such that all the vectors γ(s2)−γ(s1), γ′(s1), γ′(s2) are orthogonal to a unit vector ω ∈ R3, and such that either γ′′(s1) · ω ̸= 0 or γ′′(s2) · ω ̸= 0, then γ is not globally minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Suppose on the contrary that γ is a global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then γ is of class C2 in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Without loss of generality we may assume that γ′′(s1) · ω ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We decompose γ = γ|[0,s1] ⊕ γ|[s1,s2] ⊕ γ[s2,L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let P be a unique plane passing through γ(s1) and orthogonal to ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By the assumption the plane P also passes through γ(s2) and is parallel to γ′(s1) and γ′(s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let R denote the reflection about the plane P, and let ˜γ := γ|[0,s1] ⊕ Rγ|[s1,s2] ⊕ γ[s2,L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then ˜γ ∈ Aclamp by the property of P, and also B[˜γ] = B[γ], so that ˜γ is also a global minimizer and hence of class C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' However ˜γ′′ has discontinuity at s1 since γ′′(s1) · ω ̸= 0 so that the vectors ˜γ′′(s1 − 0) = γ′′(s1) and ˜γ′′(s1 + 0) = γ′′(s1) − 2(γ′′(s1) · ω)ω do not coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2 (Rigidity for spatially stable elasticae).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let γ ∈ Aclamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' If there are 0 ≤ s1 < s2 ≤ L with s2 − s1 < L such that all the vectors γ(s2) − γ(s1), γ′(s1), γ′(s2) are parallel, and such that either |γ′′(s1)| ̸= 0 or |γ′′(s2)| ̸= 0, then γ is not locally minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' STABLE AND MINIMAL ELASTIC CURVES 23 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Suppose on the contrary that γ is a local minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then γ is of class C2 in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Without loss of generality we may assume that |γ′′(s1)| ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We decompose γ = γ|[0,s1] ⊕ γ|[s1,s2] ⊕ γ[s2,L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let L be a unique line passing through γ(s1) and parallel to γ′(s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By the assumption the line L also passes through γ(s2) and is parallel to γ′(s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Let Rθ denote a rotation about the axis L through angle θ, and let γθ := γ = γ|[0,s1] ⊕Rθγ|[s1,s2] ⊕γ[s2,L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Then γθ ∈ Aclamp by the property of L, and also γθ → γ in W 2,2 as θ → 0 with B[γθ] = B[γ], so that for any small θ the curve γθ is also a local minimizer and hence of class C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' However γ′′ θ has discontinuity at s1 (whenever θ ̸= 0) since γ′′(s1) is orthogonal to γ′(s1) and hence to L, so that the non-zero vectors γ′′ θ (s1 − 0) and γ′′ θ (s1 + 0) form the angle θ ̸= 0 due to Rθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' □ Below we provide some concrete examples of applications of the above results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The first example is about the known fact that all closed planar elasticae except for a one-fold circle are unstable in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' This fact is shown by Langer–Signer [15] and also follows by Maddocks’ analysis [20, 21], both through explicit computations of second variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2 provides a geometric proof of this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A more than two-fold circular elastica is unstable in the set of closed curves of fixed length in R3 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A more than two-fold circular elastica (left) and its perturbation for instability (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A one-fold figure-eight elastica is unstable in the set of closed curves of fixed length in R3 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A one-fold figure-eight elastica (left) and its pertur- bation for instability (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 24 TATSUYA MIURA AND KENSUKE YOSHIZAWA In the same way we can produce various examples of planar elasticae (both orbitlike and wavelike) which are stable in the plane but unstable in the space, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Some of such phenomena are also treated by Maddocks [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A wavelike elastica, to which Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='2 is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The final example is about a helix, an example of a purely spatial elastica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' By Langer–Singer [15] it is shown that if a helix has more than two turn, then it is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 gives a new rigidity for minimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A helical elastica with more than one turn is not minimal in the set of clamped curves of fixed length in R3 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' A helix (left) and its deformation for non-minimality (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We can also deduce by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='1 that a spatial elastica is not minimal if it has a self-intersection (in its interior) at which one of the two curvature vectors are linearly independent from both the two tangent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' We expect that our method also works for more general spatial (p-)elasticae or even other ambient spaces under some symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The first author is supported by JSPS KAKENHI Grant Numbers 18H03670, 20K14341, and 21H00990, and by Grant for Basic Science Research Projects from The Sumitomo Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' The second author is supported by JSPS KAKENHI Grant Number 22K20339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Maple 2022 (Intel x86-64) 輪集 表示 捶入 書式 一 ^儿 11月23日(水)13:37 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 221119_helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='mw T>三 夕 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' 索 S+< 于+又非夷行MathMath C Text Times New Roman 12 > B I 三三 T 三 》 氮入 絶对轴 微横分 表の 一般的記号 夕儿 变数 ジ术儿 变数 值 色 90 透明度 光况度 行列 視点角度 单位 照明 座標轴 字 卜儿 操作 2 待機 集可能Maple才卜/Users/k-yoshizawa/Documents/MAPLE書類2021--/MY安定性/2022年—:1×：54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='18M時：0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='25s-么夕:100%卜一 zoom LINE 』 ATEXISTABLE AND MINIMAL ELASTIC CURVES 25 References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Arroyo, O.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Miura) Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan Email address: miura@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='titech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='jp (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content=' Yoshizawa) Institute of Mathematics for Industry, Kyushu University, 744 Mo- tooka, Nishi-ku, Fukuoka 819-0395, Japan Email address: k-yoshizawa@imi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='kyushu-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} +page_content='jp' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE_T4oBgHgl3EQf-Bxu/content/2301.08384v1.pdf'} diff --git a/utE3T4oBgHgl3EQfNwlp/content/tmp_files/2301.04386v1.pdf.txt b/utE3T4oBgHgl3EQfNwlp/content/tmp_files/2301.04386v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..13f1f74134d15f2ead759d35e00e15699ee60509 --- /dev/null +++ b/utE3T4oBgHgl3EQfNwlp/content/tmp_files/2301.04386v1.pdf.txt @@ -0,0 +1,2154 @@ +1 +Decentralized iLQR for Cooperative Trajectory +Planning of Connected Autonomous Vehicles via +Dual Consensus ADMM +Zhenmin Huang, Shaojie Shen, and Jun Ma +Abstract—Developments in cooperative trajectory planning of +connected autonomous vehicles (CAVs) have gathered consider- +able momentum and research attention. Generally, such problems +present strong non-linearity and non-convexity, rendering great +difficulties in finding the optimal solution. Existing methods typ- +ically suffer from low computational efficiency, and this hinders +the appropriate applications in large-scale scenarios involving +an increasing number of vehicles. To tackle this problem, we +propose a novel decentralized iterative linear quadratic regulator +(iLQR) algorithm by leveraging the dual consensus alternating +direction method of multipliers (ADMM). First, the original non- +convex optimization problem is reformulated into a series of +convex optimization problems through iterative neighbourhood +approximation. Then, the dual of each convex optimization +problem is shown to have a consensus structure, which facilitates +the use of consensus ADMM to solve for the dual solution +in a fully decentralized and parallel architecture. Finally, the +primal solution corresponding to the trajectory of each vehicle +is recovered by solving a linear quadratic regulator (LQR) +problem iteratively, and a novel trajectory update strategy is +proposed to ensure the dynamic feasibility of vehicles. With the +proposed development, the computation burden is significantly +alleviated such that real-time performance is attainable. Two +traffic scenarios are presented to validate the proposed algorithm, +and thorough comparisons between our proposed method and +baseline methods (including centralized iLQR, IPOPT, and SQP) +are conducted to demonstrate the scalability of the proposed +approach. +Index Terms—Autonomous driving, multi-agent system, it- +erative quadratic regulator (iLQR), differential dynamic pro- +gramming (DDP), alternating direction method of multipliers +(ADMM), connected autonomous vehicles, cooperative trajectory +planning, non-convex optimization. +I. INTRODUCTION +The continuous increase in the number of on-road vehicles +has imposed a heavy burden on the traffic system, resulting in +severe congestion and safety issues. Meanwhile, the generally +non-cooperative nature of human drivers leads to intense com- +petition for limited traffic resources and even worsens the situ- +ation. The existing issues, together with the rapid development +of autonomous driving technologies and information technolo- +gies, prompt the emergence of connected autonomous vehicles +(CAVs), which provide a cooperative driving solution that +Zhenmin Huang, Shaojie Shen, and Jun Ma are with the Department of +Electronic and Computer Engineering, The Hong Kong University of Science +and Technology, Hong Kong SAR, China (email: zhuangdf@connect.ust.hk; +eeshaojie@ust.hk; jun.ma@ust.hk) +This work has been submitted to the IEEE for possible publication. +Copyright may be transferred without notice, after which this version may +no longer be accessible. +can potentially alleviate the problem [1]. CAVs are generally +equipped with one or more communication devices that enable +the information exchange with other on-road vehicles (vehicle- +to-vehicle), roadside infrastructure (vehicle-to-infrastructure), +and cloud device (vehicle-to-cloud) [2]. Such communication +conveys the driving intention between CAVs and forms the +physical basis of cooperative driving. However, from the +perspective of algorithm development, the problem of cooper- +ative trajectory planning between CAVs still remains largely +unsolved. The main target of cooperative trajectory planning is +to make joint decisions and generate cooperative trajectories +for all involved CAVs that satisfy vehicle dynamics, traffic +rules, safety conditions, and other pertinent constraints while +maximizing the overall efficiency [3]–[5]. Although various +methods are proposed, the complexity of traffic scenarios +and the inherent coupling nature of cooperative trajectory +planning render this kind of problem hard to solve. Moreover, +with the increasing number of participant CAVs, the scale +of the cooperative trajectory planning problem also expands, +resulting in deteriorated time performance. These issues pose +great difficulties in finding the optimal solution in real-time, +especially under limited computational resources and for large- +scale scenarios, making cooperative trajectory planning a long- +standing challenge in the scope of autonomous driving. +Currently, two categories of methods are intensively in- +vestigated to tackle the cooperative trajectory planning prob- +lem. The learning-based methods provide a feasible way +to leverage real-world driving data and capture complicated +driving behaviours to handle difficult traffic scenarios [6]– +[8]. However, the requirement for an excessive amount of +real-world data and the lack of interpretability of neural +networks hinder their wide applications. On the other hand, +the optimization-based methods are more mature and well- +developed with predictable results and good interpretability. +Such methods formulate the cooperative trajectory planning +problem in the form of constrained optimization, which in- +corporates prescribed constraints concerning safety conditions +and control limits, and try to minimize pertinent objective +functions such as control costs and deviation from reference +track, etc. Within the family of optimization-based methods, +both centralized approaches and decentralized approaches are +well researched. The centralized approaches generally adopt +a central coordinator that perceives complete information +and makes decisions for all CAVs [9], [10]. A hierarchical +centralized coordination scheme is proposed in [11], which +is performed by a central traffic coordinator that handles the +arXiv:2301.04386v1 [cs.RO] 11 Jan 2023 + +2 +traffics at the intersection. Although centralized methods serve +as a natural solution to the cooperative trajectory planning +problem, one of the shortcomings is the low computational +efficiency caused by the increasing number of CAVs and the +limited computing power of the central coordinator. Such poor +scalability with respect to the number of vehicles prevents +their application to scenarios of large scale. On the contrary, +decentralized methods require every CAV to make its own +decision based on local information [12]–[14]. Therefore, the +computation load is naturally distributed among all vehicles, +which results in better scalability. Nevertheless, incomplete +information perceived by each CAV essentially complicates +the optimization problem; and apparently, more efforts on +algorithm development are required. +In general, the optimization problems formulated by +optimization-based +methods +involve +strong +non-linearity. +Therefore, nonlinear programming is typically required; and in +this sense, the efficiency of which relies heavily on the solver. +Sequential quadratic programming (SQP) and interior point +optimizer (IPOPT) are widely used nonlinear programming +solvers for solving such optimization problems, but they +suffer from low efficiency in most scenarios. Compared with +SQP and IPOPT, differential dynamic programming (DDP) +presents desirable features such as low memory consumption +and significant improvement in computational efficiency, and +therefore triggers wide applications to various kinds of optimal +control problems [15]–[17]. Meanwhile, the iterative linear +quadratic regulator (iLQR) provides a simplified version of +DDP by eliminating the second-order terms of system dy- +namics to further enhance the efficiency. The main obstacle +that prevents the usage of the original DDP/iLQR algorithm +in trajectory planning for vehicles is that it cannot handle +constraints other than system dynamics. To solve this problem, +control-limited DDP [18] incorporates the control limits into +the DDP framework, while CiLQR [19] combines iLQR with +log barrier function to further incorporate general forms of +constraints. These works enable the application of DDP/iLQR +to trajectory planning. The recent application of iLQR to +cooperative trajectory planning is inspired by [20], which +formulates the interactive trajectory planning problem as a +potential game and obtains the corresponding Nash equilib- +rium by solving a single optimal control problem with the +iLQR algorithm. Although being one of the most efficient and +scalable methods in the scope of optimal control and trajectory +planning, the direct application of DDP/iLQR onto high- +dimensional dense systems in a centralized fashion still results +in suboptimal performance. The development of a distributed +version of DDP/iLQR is desirable to further enhance the +computational efficiency and scalability. +With its distributed nature, the alternating direction method +of multipliers (ADMM) [21] is widely deployed for solving +optimization problems in various domains [22]–[25]. The basic +idea of ADMM is to decompose the original problem into +smaller, manageable ones such that they can be solved in a +parallel and distributed manner. In [26], a trajectory planning +algorithm based on ADMM is presented to separate the vehicle +dynamic constraints from other constraints such that they can +be dealt with respectively; and superior performance in terms +of computational efficiency over SQP and IPOPT is demon- +strated. Meanwhile, consensus ADMM is proposed [27] for +solving the consensus optimization problem over a connected +undirected graph. Due to the broad existence of systems with +a communication network structure, it has been deployed in +various domains such as resource allocation [28] and model +predictive control [29]. This is followed by dual consensus +ADMM [30], [31], which extends the consensus ADMM to +an optimization problem whose dual problem is a consensus +optimization problem. The recent applications of ADMM to +cooperative trajectory planning are exemplified by [32]–[35]. +Particularly in [32]–[34], ADMM is applied to separate the +independent constraints from the coupling constraints of the +multi-agent system, such that the former ones can be handled +in a parallel manner, yielding a partially decentralized solution. +Meanwhile, a fully decentralized optimization framework is +proposed in [35] to solve multi-robot cooperative trajectory +planning problems over a large scale. However, its perfor- +mance is still far away from being real-time, even with a +relatively small number of agents. +Inheriting both the efficiency of iLQR towards non-linearity +and the distributed nature of dual consensus ADMM, this +paper presents a novel fully decentralized and parallelizable +optimization framework for cooperative trajectory planning +of CAVs, based on a convex reformulation methodology to +tackle the inherent non-convexity residing in the corresponding +optimization problem. Through the optimization framework, +coupling constraints on safe distances between CAVs are +essentially decoupled and handled by all CAVs collaboratively. +Nonlinear vehicle dynamics and control limits are ensured to +be satisfied by a novel dynamically feasible trajectory update +strategy. The main contributions of this paper are listed as +follows: +• A convex reformulation methodology is proposed to +address the non-convexity in the original constrained +optimization problem for cooperative trajectory planning +of CAVs. +• A novel fully decentralized and parallelizable optimiza- +tion framework is introduced by inheriting the merits of +iLQR and dual consensus ADMM to split the original +optimization problem into sub-problems, such that the +computation load is distributed evenly among all CAVs +and the computational efficiency is enhanced signifi- +cantly. +• An innovative dynamically feasible trajectory update +strategy based on iLQR is introduced to guarantee the +satisfaction of system dynamics and prescribed control +limits of the CAVs. +• A fully parallel implementation of the proposed method +based on multi-processing is provided to reach real-time +performance. Since the proposed development is a unified +framework for trajectory planning of multi-agent systems, +the same implementation can be readily generalized to a +wide range of applications. +The remainder of this paper is organized as follows. Section +II presents the formulation of the cooperative trajectory plan- +ning problem for connected autonomous vehicles. Section III + +3 +presents the convex reformulation and the decentralized iLQR +algorithm based on dual consensus ADMM for solving the +problem. In Section IV, two traffic scenarios are provided to +demonstrate the effectiveness of the proposed algorithm, and +thorough discussions are also presented. At last, Section V +gives the conclusion of this work and several possible future +works. +Notations: Ra×b denotes the space of real matrices con- +taining a rows and b columns, and Ra to denote the space +of a-dimensional real column vectors. Similarly, we use Za×b +to denote integer matrices containing a rows and b columns +and Za to denote a-dimensional integer column vectors, re- +spectively. 0a×b represents an all-zero matrix with a rows +and b columns. In denotes an n by n identity matrix. A⊤ +and x⊤ represent the transpose of a matrix and a column +vector, respectively. The operator || · || denotes the Euclidean +norm of a vector. We also use blocdiag{A1, A2, ..., An} to +denote the block diagonal matrix with block diagonal entries +A1, A2, ..., An. (x1, x2, ..., xn) denotes the concatenated vec- +tor of the following sets of column vectors {x1, x2, ..., xn}, +namely (x1, x2, ..., xn) = [x⊤ +1 , x⊤ +2 , ..., x⊤ +n ]⊤. The proximal op- +erator is defined as Proxρ +f(x) := arg min +y +{f(y)+ ρ +2||x−y||2}. +II. PROBLEM FORMULATION +For a traffic scenario involving N CAVs, we use N = +{1, 2, ..., N} to denote the set containing the index of each +CAV. Considering the discrete-time setting, we use T += +{0, 1, ..., T − 1} to denote the time stamps. The system +dynamics for each vehicle i, i ∈ N, is given by the following +discrete-time non-linear equations +xi +τ+1 = f(xi +τ, ui +τ), +(1) +where xi +τ ∈ Rn and ui +τ ∈ Rm are the state vector and control +input vector of vehicle i at time τ for τ ∈ T , and xi +0 is given +and fixed. +We denote xτ = (x1 +τ, x2 +τ, ..., xN +τ ) and uτ = (u1 +τ, u2 +τ, ..., uN +τ ) +as the concatenated vector of all vehicles’ state variables +and control inputs at time τ, and furthermore, U += +(u0, u1..., uT −1) represents the concatenated vectors of all +vehicles’ inputs at all time. Following [20], the overall cost +can be given as +J(U) = +T −1 +� +τ=0 +Pτ(xτ, uτ) + PT (xT ) +(2) +where +Pτ(xτ, uτ) = +N +� +i=1 +Ci +τ(xi +τ, ui +τ) + +� +1≤i 0 +2: initialize for all i ∈ N: pi,0 = yi,0 = zi,0 = si,0 = 0 +3: repeat: for all i ∈ N +4: +Broadcast yi,k to all other vehicles +5: +pi,k+1 = pi,k + ρ � +j̸=i(yi,k − yj,k) +6: +si,k+1 = si,k + σ(yi,k − zi,k) +7: +yi,k+1 = arg minyi{θi(yi) + yi⊤(pi,k+1 + si,k+1) ++ σ +2 ||yi − zi,k||2 + ρ � +j̸=i ||yi − yi,k+yj,k +2 +||2} +8: +zi,k+1 = arg minzi{ξi(zi) − zi⊤si,k+1 ++ σ +2 ||zi − yi,k+1||2} +9: until termination criterion is satisfied +In particular, Step 7 of Algorithm 1 is performed by +yi,k+1 = +1 +σ + 2ρdi +(JiδXi,k+1 + ri,k+1), +(30) +where ri,k+1 = ρ � +j̸=i(yi,k +yj,k)+σzi,k −pi,k+1 −si,k+1. +δXi,k+1 = arg min +δXi +{F i(δXi) + ||JiδXi + ri,k+1||2 +2(σ + 2ρdi) +} (31) +is an auxiliary variable and converges to a minimizer of +(26) [31]. di is the degree of node i which equals to N − 1 in +our case. Meanwhile, Step 8 of Algorithm 1 is performed by +zi,k+1 = si,k+1 +σ ++yi,k+1− 1 +Nσ Prox +1 +Nσ +G (N(si,k+1+σyi,k+1)). +(32) +Based on the previous restatement of the existing dual +consensus ADMM method, we propose the following dual +update and the primal solution recovery method that are +specific to our optimization problem. In particular, (31) and +(32) require further discussion. +1) Dual Update: In (32), the first +1 +2N(N − 1)(T + 1) +elements and the last NTm elements of zi,k+1 correspond +to collision avoidance costs and box constraints, respectively, +and therefore they should be handled separately. The last term +is +Prox +1 +Nσ +G (N(si,k+1 + σyi,k+1)) += arg min +z +{G(z) + +1 +2Nσ ||z − N(si,k+1 + σyi,k+1)||2} += (arg min +z[1] +{||z[1] + l||2 ++ +1 +2Nσ ||z[1] − N(si,k+1 +[1] ++ σyi,k+1 +[1] +)||2}, +arg min +z[2] +{IX u(z[2]) + +1 +2Nσ ||z[2] − N(si,k+1 +[2] ++ σyi,k+1 +[2] +)||2}). +(33) +Thus, we decompose the problem into two parallel sub- +problems. The first sub-problem is a simple unconstrained +quadratic program, and the analytical solution is given as +z[1] = +N +2Nσ + 1(si,k+1 +[1] ++ σyi,k+1 +[1] +− 2σl). +(34) +Plug it back to (32) and we have +zi,k+1 +[1] += +2 +2Nσ + 1(Nsi,k+1 +[1] ++ Nσyi,k+1 +[1] ++ l) +(35) +which performs weighted sum of s, y, and l. +The second sub-problem is equivalent to the following +constrained optimization problem: +min +z[2] ||z[2] − N(si,k+1 +[2] ++ σyi,k+1 +[2] +)||2 +s.t. z[2] ∈ X u, +(36) +and z[2] is a minimizer of (36). To solve for z[2], we first +introduce the following definition of projection. +Definition 2. Given a closed convex set X in D, there exists +a unique minimizer for every v ∈ D to the following problem +minx{||x−v||2|x ∈ X} , which is called projection of v onto +X and denoted as ProjX (v). +From this definition, the solution to the second sub-problem +is +zi,k+1,∗ +[2] += ProjX u(N(si,k+1 +[2] ++ σyi,k+1 +[2] +)). +(37) +Since the convex set X u is constructed by imposing separate +box constraints on each element of the vector ub, the result +of this projection is simply given by confining each element +of N(si,k+1 +[2] ++σyi,k+1 +[2] +) into the set of X u respectively, which +can be performed by element-wise min-max operation. Plug +it back to (32) and we have +zi,k+1 +[2] += si,k+1 +σ ++ yi,k+1 − +1 +Nσ zi,k+1,∗ +[2] +. +(38) +Finally, we have zi,k+1 = (zi,k+1 +[1] +, zi,k+1 +[2] +). +Such results reveal that z is updated by element-wise addi- +tion and min-max operation, and therefore can be performed +efficiently. Equations (35), (37), (38), and (30) form the steps +of update for dual variables y and z. +2) Primal Solution Recovery: Perform expansion on the +square term of (31) yields +δXi,k+1 = arg min +δXi +{F i(δXi) ++ +1 +2(σ + 2ρdi)(δXi⊤Ji⊤JiδXi + 2ri,k+1⊤JiδXi)} +(39) +Utilizing the sparse structure of the Jacobian matrix Ji = +[ ˜J⊤ ˜O⊤]⊤, we can obtain the following result for second-order +terms of δXi +δXi⊤Ji⊤JiδXi = δXi⊤ ˜Ji⊤ ˜JiδXi + δXi⊤ ˜Oi⊤ ˜OiδXi += δXi⊤ ˜Ji⊤ ˜JiδXi + δui⊤δui += +T +� +τ=0 +δxi⊤ +τ Ji⊤ +τ Ji +τδxi +τ + +T −1 +� +τ=0 +δui⊤ +τ δui +τ. +(40) +Similarly, for first-order terms of δXi, we have +2ri,k+1⊤JiδXi = 2ri,k+1⊤ +[1] +˜JiδX + 2ri,k+1⊤ +[2] +˜OiδX += 2ri,k+1⊤ +[1] +˜JiδX + 2˜ri,k+1⊤ +[2] +δui += +T +� +τ=0 +2ri,k+1⊤ +[1],τ +Ji +τδxi +τ + +T −1 +� +τ=0 +2˜ri,k+1⊤ +[2],τ +δui +τ. +(41) +In particular, r[1],τ represents the piece of r[1] from row +number N(N −1)τ/2 to row number N(N −1)(τ +1)/2−1, +˜ri +[2] represents the piece of ri +[2] from row number (i − 1)Tm + +7 +to row number iTm−1, and ˜ri +[2],τ represents the piece of ˜ri +[2] +from row number τm to (τ + 1)m − 1, inclusively. +Plug (16), (40), and (41) back into (39), and we can show +that δXi,k+1 is the optimizer of the following minimization +problem +min +δxi +0,δui +0,...,δxi +T +T +� +τ=0 +1 +2δxi⊤ +τ (Ci +τ,xx + +Ji⊤ +τ Ji +τ +2(σ + 2ρdi))δxi +τ ++ δxi⊤ +τ (Ci +τ,x + +Ji⊤ +τ ri,k+1 +[1],τ +σ + 2ρdi +) ++ +T −1 +� +τ=0 +1 +2δui⊤ +τ (Ci +τ,uu + +I +2(σ + 2ρdi))δui +τ ++ δui⊤ +τ (Ci +τ,u + +˜ri,k+1 +[2],τ +σ + 2ρdi +) +s.t. δxi +τ+1 = Ai +τδxi +τ + Bi +τδui +τ. +(42) +Remarkably, (42) is a standard LQR optimal control problem +and can be solved efficiently via dynamic programming. +Remark 3. With the introduction of problem (42), the orig- +inally coupled optimization problem is essentially decoupled, +as each vehicle is solving an optimal control problem that +only involves its own state and input variables. The influence +of other vehicles on their own trajectory is reflected by the +additional terms containing J and r. Iteratively solving the +LQR problem (42) by each vehicle until ADMM termination +corresponds to a single backward pass in the centralized iLQR +solver, where state variables of all vehicles are involved. +Based on the iterative convex reformulation of the original +problem around current nominal trajectories, a decentralized +iLQR algorithm is introduced. +Based on the above discussion, we propose Algorithm 2, +which is a decentralized version of the iLQR algorithm that +solves problem (9). Each vehicle performs two loops: the outer +loop performs convex reformulation of problem (9), which +resembles the outer loop of a centralized iLQR solver. The +inner loop performs ADMM iterations to solve the introduced +convex problem in a decentralized and parallelizable manner, +during which each vehicle solves an LQR problem (42) as +well as performing updates of dual variables y, z, p, and s. +Remark 4. It is important to notice that during Step 7 of +Algorithm 2, only p and s are reset. The values of y and z +obtained by the previous ADMM iterations are carried through +to the next ADMM iterations and act as initialization. The +principle behind this is that the induced convex optimization +problem does not change a lot between consecutive convex re- +formulations: it is only drifting slowly. Therefore, the results of +dual variables y and z from the previous convex reformulation +loop serve as a good guess to the optimal y∗ and z∗ for the +next loop, thus fastening the ADMM convergence greatly. +C. Dynamically Feasible Trajectory Update +It should be noted that δXi for vehicle i obtained through +the solving of problem (42) is only feasible for the linearized +system dynamics but not feasible for the original nonlinear +Algorithm 2 Decentralized iLQR via Dual Consensus ADMM +(for vehicle i) +1: initialize {xi +τ, ui +τ}T +τ=0 +2: initialize for all i ∈ N: +pi,0 = yi,0 = zi,0 = si,0 = 0 +3: choose σ, ρ > 0 +4: repeat: +5: +Send {xi +τ}T +τ=1, receive {xj +τ}T +τ=1 from j ∈ N − {i} +6: +Compute l, Ji, {Ai +τ}T −1 +τ=0 , {Bi +τ}T −1 +τ=0 +7: +reset pi,0 = si,0 = 0 +8: +repeat: +9: +Send yi,k, receive yj,k from j ∈ N − {i} +10: +pi,k+1 = pi,k + ρ � +j̸=i(yi,k − yj,k) +11: +si,k+1 = si,k + σ(yi,k − zi,k) +12: +ri,k+1 = ρ � +j̸=i(yi,k + yj,k) ++σzi,k − pi,k+1 − si,k+1 +13: +Compute δXi,k+1 by solving LQR problem (42) +14: +Update yi,k+1 using (30) +15: +Update zi,k+1 using (35) and (37) +16: +until termination criterion is satisfied +17: +Update {xi +τ, ui +τ}T +τ=0 +18: until termination criterion is satisfied +system dynamics. Therefore, the direct addition of δXi and the +nominal trajectory causes a violation of the nonlinear dynamic +constraints. To handle this problem, we propose a dynamically +feasible update strategy, which corresponds to Step 17 of +Algorithm 2. +Solving problem (42) via dynamic programming yields a +series of feedback control matrices {ki +τ, Ki +τ}T −1 +τ=0 such that +δui +τ = ki +τ + Ki +τδxi +τ, ∀τ ∈ T . +(43) +Following the typical iLQR forward pass, we update the +nominal trajectory as +ui +τ = ˆui +τ + αki +τ + Ki +τ(xi +τ − ˆxi +τ) +ui +τ ← clip(ui +τ) +xi +τ+1 = f(xi +τ, ui +τ), +(44) +where α is the line search parameter. Clipping of inputs on +each time stamp is performed such that the inputs satisfy the +box constraints strictly. +Based on the previous discussion, we propose Algorithm +3 to perform dynamically feasible trajectory updates for each +vehicle. For better convergence, line search method is used. In +Algorithm 3, each vehicle iterates the line search parameter α +through the same list of candidate α and updates its trajectory +by (44) to generate a set of candidate trajectories. After that, +the candidate trajectories are broadcast to all other vehicles. +Then, each vehicle finds the trajectory corresponding to the +optimal α that produces the lowest overall cost, and uses it +to update the current trajectory. Noted that in Algorithm 3, +synchronization only takes place at Step 5, which causes a +minor impact on the overall algorithm. +D. Analysis of Complexity +Here, we give a brief discussion on the complexity of the +proposed algorithm Algorithm 2. In our simulations, the most + +8 +Algorithm 3 Dynamically Feasible Update with Line Search +(for vehicle i) +1: initialize t listi, c listi = empty +2: for α in α list: +3: +Compute traj with α using (44) +4: +Append traj to the t listi +5: Send t listi, receive t listj from j ∈ N − {i} +6: for i in len(α list): +7: +t set = empty +8: +for j in N: +9: +Append t listj[i] to t set +10: +Compute cost of t set +11: +Append cost to c list +12: index = arg min(c list) +13: current trajectory ← t listi[index] +computationally demanding part of Algorithm 2 is Step 13, +which requires solving an LQR optimal control problem. It +is obvious that for each vehicle, problem (42) involves only +state variables and control inputs local to that vehicle, and +therefore its scale does not grow with the number of CAVs. +In other words, we can conclude that solving problem (42) is +of O(1) complexity. +For comparison, solving problem (9) in a centralized manner +tackles an LQR optimal control problem with the state space +growing linearly with respect to the number of CAVs, which +results in poor scalability. Generally, O(N 3) complexity is +induced. +It should be noted that problem (42) needs to be solved +by each vehicle in an iterative manner. With the assumption +that the number of iterations for the outer loop is N1 and the +number of iterations for the inner loop is N2, we conclude that +if only Step 13 is considered, the complexity of Algorithm 2 +is O(N1 · N2). +Other steps such as Steps 10 and 12 have higher complexity +theoretically, but they only involve very simple operations +such as element-wise summation, and therefore only take up +a negligible amount of time in our simulations. +IV. SIMULATION RESULTS +A. Vehicle Model +We assume that all vehicles involved in our simulation +possess the same dynamics. Referring to [18], Single vehicle +dynamics is characterized by the following equations: +px,τ+1 = px,τ + fr(vτ, δτ) cos(θτ), +py,τ+1 = py,τ + fr(vτ, δτ) sin(θτ), +θτ+1 = θτ + arcsin(τsvτ sin(δτ) +b +), +vτ+1 = vτ + τsaτ. +(45) +(45) describes a discrete-time vehicle model with state vector +(px, py, θ, v) and input vector (δ, a). px and py represent the +global X and Y Cartesian coordinates of the vehicle center, θ is +the heading angle of the vehicle with respect to the positive X +axis of the global Cartesian coordinate system, v is the velocity +of the vehicle, δ is the steering angle of the front wheel, and +a is the acceleration. Moreover, we use the subscript [·]τ to +indicate variables of time stamp τ with τ ∈ T , and τs is the +time interval. The function fr(v, δ) is defined as +fr(v, δ) = b + τsv cos(δ) − +� +b2 − (τsv sin(δ))2 +(46) +where b denotes the vehicle wheelbase. +The system dynamics of all vehicles considered as a single +system can be obtained by trivially stacking all single vehicle +dynamics together, yielding a model with 4N state variables +and 2N control inputs. +B. General Settings +In this paper, we consider the same scenarios as in [33], +which includes a T-junction with 3 vehicles and an intersection +with 12 vehicles. We implement the proposed algorithm with +both a single-process version and a multi-process version with +the number of processes equals to the number of vehicles. We +also implement three baselines for comparison, including a +centralized iLQR solver with log barrier functions [19], an +IPOPT solver, and an SQP solver, with the last two solvers +provided by CasADi [36]. All algorithms are implemented in +Python 3.7, running on a server with 2× Intel(R) Xeon(R) +Gold 6348 CPU @ 2.60GHz. For multi-process implementa- +tion, each process is bound to a (logical) core on the CPU, +and the communication is realized via shared memory. +For details, we set the length and width of each vehicle +as 2.50 m and 1.60 m respectively. The input steering angle +is bounded between ±0.6 rad and the acceleration is within +[−3.0, 1.5] m/s2. We set the weighting matrices in the host +cost as Q = diag(1, 1, 0, 0) and R = diag(1, 1). The time +interval for discrete dynamics is set to be τs = 0.1 s and the +horizon length is T = 100. The safe distance for all collision +avoidance costs is dsafe = 5.5 m, and the scaling factor between +host cost and collision avoidance cost is β = 1.44. Trajectories +of all vehicles are initialized with zero inputs. The terminal +condition of the proposed method and the centralized iLQR +solver is set according to the change of overall cost. When +the absolute change of overall cost between two consecutive +iterations is less than 1, both algorithms terminate. +C. Main Results +1) Scenario 1: We first consider a T-junction scenario with +three vehicles. As is shown in Fig. 1(a), the vehicles are +represented by rectangles in different colors, and each dotted +line is the reference trajectory corresponding to the vehicle +of the same color, with each dot representing the reference +position of the vehicle center corresponding to a particular +time stamp. The vehicles are required to pass the T-function +following the reference trajectories as close as possible and +avoid collisions. +For Scenario 1, we set σ = 0.1 and ρ = 0.01. Due to +the initialization scheme described in Remark 4, the stopping +criterion of ADMM (Step 16 of Algorithm 2) can be set as +termination after a fixed, small number of iterations. Here, we +set the number of ADMM iterations to be 2. The results are +shown in Fig. 1(b)-(d). Each sub-figure corresponds to a partic- +ular time stamp, with the rectangles showing the poses of the + +9 +(a) τ = 0 +(b) τ = 50 +(c) τ = 75 +(d) τ = 99 +Fig. 1. Simulation results for Scenario 1 with the proposed method at time stamps τ = 0, τ = 50, τ = 75, and τ = 99. +Fig. 2. Steering angles and accelerations of 3 vehicles and minimal distance +between the center of vehicles in Scenario 1. +Fig. 3. Iterations of trajectories of 3 vehicles in Scenario 1. +vehicles and the dotted lines showing their past trajectories. All +three vehicles succeed in reaching their destination following +smooth, feasible trajectories while keeping safe distances from +each other to avoid collisions. Fig. 2 shows the inputs and the +minimal distance. Box constraints are satisfied and the minimal +distance between the center of vehicles is 3.92 m, which proves +that the trajectories are collision-free. Both the single-process +and the multi-process implementations yield numerically the +same results. +Our algorithm takes 7 iterations to stop. The drifting of +vehicle trajectories through iterations are shown in Fig. 3. +The group of trajectories of the same color corresponds to the +same vehicle, with more solid trajectories corresponding to a +Fig. 4. Variance of {yi} with respect to number of iterations in Scenario 1. +higher number of iterations. Fig. 3 qualitatively shows how +the trajectories converge to the optimal trajectories through +iterations. To further show the convergence of our algorithm, +we adopt the conclusion that the set of dual variables {yi} +should converge to the same vector y∗ as the algorithm +converges. Therefore, we compute the variance of each ele- +ment of {yi} over i and obtain the mean of those variances. +We plot the mean of variances with respect to the iteration +number. The result is given in Fig. 4 in log-scale, showing +monotonically decreasing of the variances of {yi}, which +clearly demonstrates the convergence of our algorithm. +The first row of Table I demonstrates the computation time +of all five implementations. The multi-process implementation +of our algorithm takes 0.035 s to finish, which is the fastest +of all. It reaches approximately 1.5× speed compared to the +centralized iLQR solver, 2.4× speed to the single-process +version of our method, 9.3× speed to the IPOPT solver, +and 23.4× speed to the SQP solver. It should be noted that +the single-process implementation of our proposed method is +slower than the centralized iLQR solver, which implies that +the proposed algorithm does not reduce the overall amount +of computation, but provides a decentralized optimization +framework such that parallel computing can be used to speed +up the optimization process. +Due to different termination criteria, the quality of the +final solution varies between solvers. Table III shows the +overall cost of each solution. The overall cost of the proposed +algorithm is nearly the same as the centralized iLQR solver +and only slightly larger than the IPOPT solver, which implies + +5.0 +2.5 +0.0 - +-2.5 +-5.0 - +-7.5 +-10.0 - +-12.5 +-10 +0 +5 +105.0 +2.5 +0.0: +-2.5 +-5.0 - +-7.5 +-10.0- +-12.5 +-5 +-10 +0 +5 +105.0- +2.5 +0.0 - +-2.5 +-5.0 - +-7.5 +-10.0 - +-12.5 +-10 +-5 +0 +5 +105.0 +2.5 +0.0 - +-2.5 +-5.0 - +-7.5 +-10.0 - +-12.5 +-10 +-5 +0 +5 +100.5 +(rad) +0.0 +-0.5 +0 +s +(m +-2 +a +Min. Distance (m) +10 +5 +min_dist=3.92m +0 +20 +40 +60 +80 +T5.0- +2.5 +0.0- +-2.5 +-5.0 - +-7.5 +-10.0 - +-12.5 +-10 +-5 +0 +5 +10100 +y Variance +10-1 +0 +1 +2 +3 +4 +5 +6 +Iteration Number10 +(a) τ = 0 +(b) τ = 25 +(c) τ = 50 +(d) τ = 60 +(e) τ = 74 +(f) τ = 99 +Fig. 5. Simulation results for Scenario 2 with the proposed method at time stamps τ = 0, τ = 25, τ = 50, τ = 60, τ = 74, and τ = 99. +Fig. 6. Steering angles and accelerations of 12 vehicles and minimal distance +between the center of vehicles in Scenario 2. +comparable solution quality. Meanwhile, the SQP solver fails +to obtain a collision-free solution, which is reflected by an +overall cost that is much larger than the rest. +2) Scenario 2: To further demonstrate the superiority in +computational efficiency of our proposed method, we consider +a scenario of a larger scale. As is shown in Fig. 5(a), Scenario +2 is an intersection with 12 vehicles passing simultaneously. +Fig. 7. Iterations of trajectories of 12 vehicles in Scenario 2. +We discover that for faster convergence, the parameters σ and +ρ should be reduced with the number of vehicles increasing. +Therefore, we scale down σ and ρ to σ = 0.01 and ρ = +0.001, and the number of ADMM iterations is set to be 3. The +simulation results are shown in Fig. 5(b)-(f). Similarly, smooth +trajectories are obtained. Fig. 6 shows that the inputs satisfy +box constraints and the minimal distance is 2.98 m, which also +guarantees safety. The proposed algorithm takes 24 iterations + +20 +10 +0 +-10 +-20 +-20 +-10 +0 +10 +2020 +10 +0 +-10 +-20 +-20 +-10 +0 +10 +2020 +10 +0 +-10 +-20 +-20 +-10 +0 +10 +2020 +10 +LCOO6 +0 +-10 +-20 +-20 +-10 +0 +10 +200.5 +(rad) +0.0 +-0.5 +0 +s +(m +Min. Distance (m) +8 +6 +4 +min_dist=2.98m. +0 +20 +40 +60 +80 +T20 +10 +0 +-10 +-20 : +-20 +-10 +0 +10 +2020 +10 +0 +-10 +-20 +-20 +-10 +0 +10 +2020 +10 +0 +-10 +-20 +-20 +-10 +0 +10 +2011 +TABLE I +COMPARISON OF COMPUTATION TIME BETWEEN THE PROPOSED METHOD, CENTRALIZED ILQR, IPOPT, AND SQP FOR SCENARIOS 1 AND 2 +Proposed method +Centralized iLQR +IPOPT +SQP +Single-process +Multi-process +Scenario 1 +0.084 s +0.035 s +0.052 s +0.327 s +0.818 s +Scenario 2 +1.691 s +0.250 s +1.186 s +11.140 s +—— +Fig. 8. Variance of {yi} with respect to number of iterations in Scenario 2. +to finish. The iteration of trajectories is shown in Fig. 7. Fig. +8 plots the variance of {yi} with respect to iteration number +in log-scale. Again, the near monotonically decreasing of the +variance of {yi} demonstrates the convergence. +The second row of Table I shows the computation time of +all implementations corresponding to Scenario 2. Again, the +multi-process version of the proposed method is the fastest, +taking 0.250s to finish, which is roughly 4.7× as fast as the +centralized iLQR solver, 6.8× as fast as the single-process +implementation of our method, and 44.6× as fast as the IPOPT +solver. For this scenario, the SQP solver fails to converge. +Compared to scenario 1 with 3 vehicles, the proposed method +obtains much higher folds of speed-up, which supports the +claim that it scales better with the number of participating +vehicles than baseline methods. +For the solution quality, the second row of Table III reveals +that the solution of our method converges to a smaller overall +cost than the centralized iLQR solver, and is comparable to the +one obtained by the IPOPT solver, with the overall cost being +slightly bigger. We consider that such a small compromise in +the solution quality for a significant amount of speed-up is +acceptable. +Although it is better to reduce the value of σ and ρ for the +scenario of larger scale, keeping the same σ and ρ as Scenario +1 still yields satisfactory results with only slight suboptimality +(see Table II). The overall iteration number increases from +24 to 28, which causes the computation time to increase by +roughly 20%. +D. Discussion +1) Scalabilty: To further show the scalability of the pro- +posed algorithm, we conduct simulations on Scenario 2 with +the number of vehicles varying from N = 4 to N = 12 +Fig. 9. Computation time with respect to different numbers of vehicles for +the centralized iLQR solver and the proposed method. +TABLE II +COMPARISON OF PERFORMANCE FOR SOLVING SCENARIO 2 WITH +PARAMETERS σ = 0.01, ρ = 0.001 AND σ = 0.1, ρ = 0.01 +Iteration num. +Cost +Time +σ = 0.01, ρ = 0.001 +24 +2303.7 +0.250 s +σ = 0.1, ρ = 0.01 +28 +2306.9 +0.303 s +while keeping all other parameters unchanged. For each case, +we apply both the centralized iLQR solver and the proposed +method with multi-process implementation and measure the +computation time respectively. The results are shown in Fig. +9. It is clear that when the number of vehicles is set to 4, our +method takes nearly the same time as the centralized iLQR +solver. However, as the number of vehicles increases, the +computation time of the centralized iLQR method increases +sharply and becomes much higher than our method. Quan- +titatively, when the number of CAVs increases by 3 times +from N = 4 to N = 12, the computation time of centralized +iLQR increases by 12.1×, which shows poor scalability. On +the contrary, the computation time of our method increases by +only 2.74×, which is slower than the increase of the number +of vehicles. +2) Safety: In the proposed method, collision avoidance +is achieved by adding a soft penalty to the overall cost +whenever the distance between two vehicles is smaller than +the given safe distance. Generally, such a formulation does not +guarantee to produce a collision-free solution. However, safety +conditions can still be satisfied by setting a large enough β to +scale up the collision avoidance penalty. Theoretically, when +β tends to infinity, the soft penalty essentially becomes hard +constraint as an infinite cost is induced whenever two vehicles +go within the safe distance, although an overly large β could +result in slow convergence. + +101 +100 + Variance +10-1 +10-2 +0 +5 +10 +15 +20 +Iteration Number1.2 +Proposed method +Centralized iLQR +1.0 +0.8 +0.6 +0.4 - +0.2 +0.0 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Number ofVehicles12 +TABLE III +COMPARISON OF FINAL COST BETWEEN THE PROPOSED METHOD, CENTRALIZED ILQR, IPOPT, AND SQP FOR SCENARIOS 1 AND 2 +Proposed method +Centralized iLQR +IPOPT +SQP +Single-process +Multi-process +Scenario 1 +356.02 +356.02 +356.04 +347.56 +626.40 +Scenario 2 +2303.7 +2303.7 +2394.4 +2297.7 +—— +(a) β = 1.00 (collision) +(b) β = 1.44 (safe) +(c) β = 1.96 (safe) +(d) β = 2.56 (safe) +Fig. 10. Simulation results for Scenario 2 with different values of β at τ = 72. +To show this, we adapt different values of β for solving +Scenario 2 while keeping all other parameters unchanged, and +we examine the effect of β on both the minimal distance +between vehicles and the overall computation time. The qual- +itative results at τ = 72 are shown in Fig. 10, which is the +time when the closest distance between the corners of two +vehicles occurs. It is clear that when β = 1.00, the yellow +car, and the blue car collide with each other. This collision +is prevented when β is set to be greater than 1.44, and the +spacing between the two vehicles continues to increase with +increasing β. Quantitative results are presented in Table IV. +When β increases from 1.00 to 2.56, the minimal distance +between vehicles also increases, thus the safety is enhanced. +However, a larger β requires more iterations for the algorithm +to converge, which results in a longer computation time. +Based on previous analysis, to ensure safety in real situa- +tions and maintain high computational efficiency, we can first +initialize the algorithm with a reasonable value of β and solve +for a set of candidate trajectories. If collision is detected on +the candidate trajectories, we increase β to a higher value and +solve for the trajectories again. We can keep looping for larger +β until collision-free trajectories are obtained. +TABLE IV +MINIMAL DISTANCE BETWEEN VEHICLE CENTERS, ITERATION NUMBER, +AND COMPUTATION TIME WITH VARYING β FOR SCENARIO 2. +β +Minimal dis. +Iteration num. +Time +1.00 +2.59 m +19 +0.206 s +1.21 +2.81 m +21 +0.228 s +1.44 +2.98 m +24 +0.250 s +1.69 +3.15 m +26 +0.282 s +1.96 +3.30 m +29 +0.314 s +2.25 +3.37 m +32 +0.344 s +2.56 +3.43 m +35 +0.374 s +V. CONCLUSION +This work investigates the cooperative trajectory planning +problem concerning multiple CAVs. The problem is formu- +lated as a strongly non-convex optimization problem with +non-linear vehicle dynamics and other pertinent constraints. +We propose a decentralized optimization framework based +on the dual consensus ADMM algorithm to distribute the +computation load evenly among all CAVs, such that each +CAV is iteratively solving an LQR problem with a fixed +scale. We provide fully parallel implementation to enhance +the efficiency and achieve real-time performance. Simulations +on two traffic scenarios with the proposed method, centralized +iLQR solver, IPOPT, and SQP are performed to validate the +effectiveness and computational efficiency of the proposed +method. Meanwhile, simulations on an increasing number +of CAVs demonstrate the superiority of scalability of the +proposed method compared to the centralized iLQR solver. +A possible future work is to deploy our proposed algorithm +on high-performance parallel processor (HPPP) such as GPU +to scale up our algorithm to excessively large-scale scenarios +containing hundreds of vehicles. Another future work is to per- +form field experiments to further substantiate the effectiveness +of the proposed method. +REFERENCES +[1] X. Sun, F. R. Yu, and P. 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Diehl, +“Casadi: a software framework for nonlinear optimization and optimal +control,” Mathematical Programming Computation, vol. 11, no. 1, pp. +1–36, 2019. + diff --git a/utE3T4oBgHgl3EQfNwlp/content/tmp_files/load_file.txt b/utE3T4oBgHgl3EQfNwlp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b14c703a269170e8a606ef8b6f21f05cdc61a0c0 --- /dev/null +++ b/utE3T4oBgHgl3EQfNwlp/content/tmp_files/load_file.txt @@ -0,0 +1,1000 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf,len=999 +page_content='1 Decentralized iLQR for Cooperative Trajectory Planning of Connected Autonomous Vehicles via Dual Consensus ADMM Zhenmin Huang, Shaojie Shen, and Jun Ma Abstract—Developments in cooperative trajectory planning of connected autonomous vehicles (CAVs) have gathered consider- able momentum and research attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Generally, such problems present strong non-linearity and non-convexity, rendering great difficulties in finding the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Existing methods typ- ically suffer from low computational efficiency, and this hinders the appropriate applications in large-scale scenarios involving an increasing number of vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' To tackle this problem, we propose a novel decentralized iterative linear quadratic regulator (iLQR) algorithm by leveraging the dual consensus alternating direction method of multipliers (ADMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' First, the original non- convex optimization problem is reformulated into a series of convex optimization problems through iterative neighbourhood approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Then, the dual of each convex optimization problem is shown to have a consensus structure, which facilitates the use of consensus ADMM to solve for the dual solution in a fully decentralized and parallel architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Finally, the primal solution corresponding to the trajectory of each vehicle is recovered by solving a linear quadratic regulator (LQR) problem iteratively, and a novel trajectory update strategy is proposed to ensure the dynamic feasibility of vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' With the proposed development, the computation burden is significantly alleviated such that real-time performance is attainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Two traffic scenarios are presented to validate the proposed algorithm, and thorough comparisons between our proposed method and baseline methods (including centralized iLQR, IPOPT, and SQP) are conducted to demonstrate the scalability of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Index Terms—Autonomous driving, multi-agent system, it- erative quadratic regulator (iLQR), differential dynamic pro- gramming (DDP), alternating direction method of multipliers (ADMM), connected autonomous vehicles, cooperative trajectory planning, non-convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' INTRODUCTION The continuous increase in the number of on-road vehicles has imposed a heavy burden on the traffic system, resulting in severe congestion and safety issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Meanwhile, the generally non-cooperative nature of human drivers leads to intense com- petition for limited traffic resources and even worsens the situ- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The existing issues, together with the rapid development of autonomous driving technologies and information technolo- gies, prompt the emergence of connected autonomous vehicles (CAVs), which provide a cooperative driving solution that Zhenmin Huang, Shaojie Shen, and Jun Ma are with the Department of Electronic and Computer Engineering, The Hong Kong University of Science and Technology, Hong Kong SAR, China (email: zhuangdf@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='hk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' eeshaojie@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='hk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='ma@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='hk) This work has been submitted to the IEEE for possible publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Copyright may be transferred without notice, after which this version may no longer be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' can potentially alleviate the problem [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' CAVs are generally equipped with one or more communication devices that enable the information exchange with other on-road vehicles (vehicle- to-vehicle), roadside infrastructure (vehicle-to-infrastructure), and cloud device (vehicle-to-cloud) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Such communication conveys the driving intention between CAVs and forms the physical basis of cooperative driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' However, from the perspective of algorithm development, the problem of cooper- ative trajectory planning between CAVs still remains largely unsolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The main target of cooperative trajectory planning is to make joint decisions and generate cooperative trajectories for all involved CAVs that satisfy vehicle dynamics, traffic rules, safety conditions, and other pertinent constraints while maximizing the overall efficiency [3]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Although various methods are proposed, the complexity of traffic scenarios and the inherent coupling nature of cooperative trajectory planning render this kind of problem hard to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Moreover, with the increasing number of participant CAVs, the scale of the cooperative trajectory planning problem also expands, resulting in deteriorated time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' These issues pose great difficulties in finding the optimal solution in real-time, especially under limited computational resources and for large- scale scenarios, making cooperative trajectory planning a long- standing challenge in the scope of autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Currently, two categories of methods are intensively in- vestigated to tackle the cooperative trajectory planning prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The learning-based methods provide a feasible way to leverage real-world driving data and capture complicated driving behaviours to handle difficult traffic scenarios [6]– [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' However, the requirement for an excessive amount of real-world data and the lack of interpretability of neural networks hinder their wide applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' On the other hand, the optimization-based methods are more mature and well- developed with predictable results and good interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Such methods formulate the cooperative trajectory planning problem in the form of constrained optimization, which in- corporates prescribed constraints concerning safety conditions and control limits, and try to minimize pertinent objective functions such as control costs and deviation from reference track, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Within the family of optimization-based methods, both centralized approaches and decentralized approaches are well researched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The centralized approaches generally adopt a central coordinator that perceives complete information and makes decisions for all CAVs [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' A hierarchical centralized coordination scheme is proposed in [11], which is performed by a central traffic coordinator that handles the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='04386v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='RO] 11 Jan 2023 2 traffics at the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Although centralized methods serve as a natural solution to the cooperative trajectory planning problem, one of the shortcomings is the low computational efficiency caused by the increasing number of CAVs and the limited computing power of the central coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Such poor scalability with respect to the number of vehicles prevents their application to scenarios of large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' On the contrary, decentralized methods require every CAV to make its own decision based on local information [12]–[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Therefore, the computation load is naturally distributed among all vehicles, which results in better scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Nevertheless, incomplete information perceived by each CAV essentially complicates the optimization problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' and apparently, more efforts on algorithm development are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' In general, the optimization problems formulated by optimization-based methods involve strong non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Therefore, nonlinear programming is typically required;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' and in this sense, the efficiency of which relies heavily on the solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Sequential quadratic programming (SQP) and interior point optimizer (IPOPT) are widely used nonlinear programming solvers for solving such optimization problems, but they suffer from low efficiency in most scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Compared with SQP and IPOPT, differential dynamic programming (DDP) presents desirable features such as low memory consumption and significant improvement in computational efficiency, and therefore triggers wide applications to various kinds of optimal control problems [15]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Meanwhile, the iterative linear quadratic regulator (iLQR) provides a simplified version of DDP by eliminating the second-order terms of system dy- namics to further enhance the efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The main obstacle that prevents the usage of the original DDP/iLQR algorithm in trajectory planning for vehicles is that it cannot handle constraints other than system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' To solve this problem, control-limited DDP [18] incorporates the control limits into the DDP framework, while CiLQR [19] combines iLQR with log barrier function to further incorporate general forms of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' These works enable the application of DDP/iLQR to trajectory planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The recent application of iLQR to cooperative trajectory planning is inspired by [20], which formulates the interactive trajectory planning problem as a potential game and obtains the corresponding Nash equilib- rium by solving a single optimal control problem with the iLQR algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Although being one of the most efficient and scalable methods in the scope of optimal control and trajectory planning, the direct application of DDP/iLQR onto high- dimensional dense systems in a centralized fashion still results in suboptimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The development of a distributed version of DDP/iLQR is desirable to further enhance the computational efficiency and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' With its distributed nature, the alternating direction method of multipliers (ADMM) [21] is widely deployed for solving optimization problems in various domains [22]–[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The basic idea of ADMM is to decompose the original problem into smaller, manageable ones such that they can be solved in a parallel and distributed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' In [26], a trajectory planning algorithm based on ADMM is presented to separate the vehicle dynamic constraints from other constraints such that they can be dealt with respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' and superior performance in terms of computational efficiency over SQP and IPOPT is demon- strated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Meanwhile, consensus ADMM is proposed [27] for solving the consensus optimization problem over a connected undirected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Due to the broad existence of systems with a communication network structure, it has been deployed in various domains such as resource allocation [28] and model predictive control [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' This is followed by dual consensus ADMM [30], [31], which extends the consensus ADMM to an optimization problem whose dual problem is a consensus optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The recent applications of ADMM to cooperative trajectory planning are exemplified by [32]–[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Particularly in [32]–[34], ADMM is applied to separate the independent constraints from the coupling constraints of the multi-agent system, such that the former ones can be handled in a parallel manner, yielding a partially decentralized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Meanwhile, a fully decentralized optimization framework is proposed in [35] to solve multi-robot cooperative trajectory planning problems over a large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' However, its perfor- mance is still far away from being real-time, even with a relatively small number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Inheriting both the efficiency of iLQR towards non-linearity and the distributed nature of dual consensus ADMM, this paper presents a novel fully decentralized and parallelizable optimization framework for cooperative trajectory planning of CAVs, based on a convex reformulation methodology to tackle the inherent non-convexity residing in the corresponding optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Through the optimization framework, coupling constraints on safe distances between CAVs are essentially decoupled and handled by all CAVs collaboratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Nonlinear vehicle dynamics and control limits are ensured to be satisfied by a novel dynamically feasible trajectory update strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The main contributions of this paper are listed as follows: A convex reformulation methodology is proposed to address the non-convexity in the original constrained optimization problem for cooperative trajectory planning of CAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' A novel fully decentralized and parallelizable optimiza- tion framework is introduced by inheriting the merits of iLQR and dual consensus ADMM to split the original optimization problem into sub-problems, such that the computation load is distributed evenly among all CAVs and the computational efficiency is enhanced signifi- cantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' An innovative dynamically feasible trajectory update strategy based on iLQR is introduced to guarantee the satisfaction of system dynamics and prescribed control limits of the CAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' A fully parallel implementation of the proposed method based on multi-processing is provided to reach real-time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Since the proposed development is a unified framework for trajectory planning of multi-agent systems, the same implementation can be readily generalized to a wide range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Section II presents the formulation of the cooperative trajectory plan- ning problem for connected autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Section III 3 presents the convex reformulation and the decentralized iLQR algorithm based on dual consensus ADMM for solving the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' In Section IV, two traffic scenarios are provided to demonstrate the effectiveness of the proposed algorithm, and thorough discussions are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' At last, Section V gives the conclusion of this work and several possible future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Notations: Ra×b denotes the space of real matrices con- taining a rows and b columns, and Ra to denote the space of a-dimensional real column vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Similarly, we use Za×b to denote integer matrices containing a rows and b columns and Za to denote a-dimensional integer column vectors, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' 0a×b represents an all-zero matrix with a rows and b columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' In denotes an n by n identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' A⊤ and x⊤ represent the transpose of a matrix and a column vector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The operator || · || denotes the Euclidean norm of a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' We also use blocdiag{A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=', An} to denote the block diagonal matrix with block diagonal entries A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=', An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=', xn) denotes the concatenated vec- tor of the following sets of column vectors {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=', xn}, namely (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=', xn) = [x⊤ 1 , x⊤ 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=', x⊤ n ]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The proximal op- erator is defined as Proxρ f(x) := arg min y {f(y)+ ρ 2||x−y||2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' PROBLEM FORMULATION For a traffic scenario involving N CAVs, we use N = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=', N} to denote the set containing the index of each CAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Considering the discrete-time setting, we use T = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=', T − 1} to denote the time stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' The system dynamics for each vehicle i, i ∈ N, is given by the following discrete-time non-linear equations xi τ+1 = f(xi τ, ui τ), (1) where xi τ ∈ Rn and ui τ ∈ Rm are the state vector and control input vector of vehicle i at time τ for τ ∈ T , and xi 0 is given and fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' We denote xτ = (x1 τ, x2 τ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=', xN τ ) and uτ = (u1 τ, u2 τ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=', uN τ ) as the concatenated vector of all vehicles’ state variables and control inputs at time τ, and furthermore, U = (u0, u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=', uT −1) represents the concatenated vectors of all vehicles’ inputs at all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utE3T4oBgHgl3EQfNwlp/content/2301.04386v1.pdf'} +page_content=' Following [20], the overall cost can be given as J(U) = T −1 � τ=0 Pτ(xτ, uτ) + PT (xT ) (2) where Pτ(xτ, uτ) = N � i=1 Ci τ(xi τ, ui τ) + � 1≤i 0. +(2) The subcategory M generates T in the sense that T itself is the smallest +thick triangulated subcategory containing M. + +A NON-VANISHING RESULT ON THE SINGULARITY CATEGORY +5 +An object X is called presilting (respectively, silting) provided that add X is a +presilting (respectively, silting) subcategory. We mention that the study of silting +objects goes back to [19]. +The following result is due to [2, Proposition 2.4]. +Lemma 3.1. Assume that T has a silting subcategory. Then for any object X, +HomT (X, Σd(X)) = 0 for sufficiently large d. +□ +In what follows, A is an abelian category with enough projective objects. We +have the first consequence of Theorem 2.8. +Proposition 3.2. Assume that A contains a virtually d-periodic object for some +d ≥ 1. Then Dsg(A) has no silting subcategory. +Proof. Take M be a virtually d-periodic object in A. Theorem 2.8 implies that +HomDsg(A)(M, Σnd(M)) ̸= 0 +for any integer n. In particular, n can be sufficiently large. In view of Lemma 3.1, +we have the required non-existence of a silting subcategory. +□ +For a left noetherian ring Λ, we usually write Dsg(Λ) for Dsg(Λ-mod). +The following result strengthens [1, Theorem 1], where the corresponding result +is proved under a finiteness assumption on the selfinjective dimension; compare +[2, Example 2.5(b)] and [16, Corollary 3.12]. +The argument here is completely +different. +Corollary 3.3. Let Λ be a left artinian ring with infinite global dimension. Then +Dsg(Λ) has no silting subcategory. +Proof. By Example 2.3, the semisimple Λ-module Λ0 is virtually 1-periodic. Then +we apply Proposition 3.2. +□ +The above non-existence partially supports the following conjecture [13]; com- +pare [16]. +Singular Presilting Conjecture. +For any artin algebra Λ, there is no nonzero +presilting subcategory in Dsg(Λ). +By [16, Lemma 3.4] or [23, Proposition 1.21], this conjecture implies the following +well-known conjecture, proposed in [6, p.70]; compare [13, Section 1]. +Auslander-Reiten Conjecture. For a non-projective module M over any artin +algebra Λ, we have Extn +Λ(M, M ⊕ Λ) ̸= 0 for some n ≥ 1. +We mention that, by [9, Theorem 4.4], the Singular Presilting Conjecture for +Λ is equivalent to the Auslander-Reiten Conjecture for Λ, provided that Λ is a +Gorenstein artin algebra. +The following result implies that the Singular Presilting Conjecture holds for +ultimately-closed artin algebras; compare [6, Proposition 1.3]. Recall that finite +dimensional periodic algebras and syzygy-finite artin algebras are ultimately-closed; +see Section 2 or [6, p.73]. +Proposition 3.4. Assume that A is ultimately-closed. Then Dsg(A) has no nonzero +presilting subcategory. +In particular, for a ultimately-closed ring Λ, there is no +nonzero presilting subcategory in Dsg(Λ). +Proof. It suffices to prove that Dsg(A) has no nonzero presilting object. Assume +that X is a nonzero presilting object in Dsg(A). Set T to be the smallest thick +subcategory containing X. By [11, Lemma 2.1], there exists an object M in A such +that X is isomorphic to Σn(M) for some integer n. Since M is ultimately-closed, + +6 +CHEN, LI, ZHANG, ZHAO +there exists d ≥ 1 such that E = M ⊕Ω(M)⊕· · ·⊕Ωd−1(M) is virtually 1-periodic; +see Example 2.4. In view of Lemma 2.7, the object E belongs to T . However, by +Theorem 2.8, we have +HomT (E, Σn(E)) ̸= 0 +for any integer n. Since X is a silting object of T , we have a desired contradiction +by Lemma 3.1. +□ +Remark 3.5. It seems that Proposition 3.4 might be strengthened. The abelian +category A is said be virtually ultimately-closed if for each object M, there exists +d ≥ 1 such that Ωd(M) lies in ⟨M ⊕Ω(M)⊕· · ·⊕Ωd−1(M)⟩. In this case, the object +M ⊕ Ω(M) ⊕ · · · ⊕ Ωd−1(M) is still virtually 1-periodic. Then the same argument +above implies that Proposition 3.4 holds for virtually ultimately-closed categories. +However, we do not know any virtually ultimately-closed category, which is not +ultimately-closed. +Remark 3.6. Unlike the ungraded case, the graded singularity category of a graded +artin algebra might have a silting object; for example, see [22, Theorem 3.0.3] and +[20, Theorem 1.4]. +We mention the following known non-existence result. +Proposition 3.7. Let R be a commutative noetherian local ring, which is non- +regular. Then Dsg(R) has no silting subcategory. +Proof. Denote by k the residue field of R. By [3, Theorem 6.5], we infer that +HomDsg(R)(k, Σn(k)) ̸= 0 +(3.1) +for any integer n. +Here, we identify HomDsg(R)(k, Σn(k)) with the stable co- +homology group � +Ext +n +R(k, k); see [3, 1.4.2]. Then the non-existence follows from +Lemma 3.1. +□ +3.2. The dg Leavitt algebra. In this subsection, we assume that Λ is an artin +algebra over a commutative artinian ring k. +Recall that Λ0 = Λ/J with J its Jacobson radical. We will assume that Λ0 is +a subalgebra of Λ with a decomposition Λ = Λ0 ⊕ J of Λ0-Λ0-bimodules. This +assumption holds if Λ is given by a finite quiver with admissible relations, or if Λ +is a finite dimensional algebra over a perfect field. +Consider the left Λ0-dual J∗ = Hom(J, Λ0) of J, which carries a natural Λ0-Λ0- +bimodule structure. We have the Casimir element c = � +i∈S α∗ +i ⊗ αi ∈ J∗ ⊗ J, +where {αi | i ∈ S} and {α∗ +i | i ∈ S} form the dual basis of J. The multiplication +on J induces a map of Λ0-Λ0-bimodules +∂+ : J∗ −→ J∗ ⊗ J∗. +To be more precise, we have ∂+(g) = � g1 ⊗ g2 such that g(ab) = � g2(ag1(b)) for +any a, b ∈ J. +Associated to the artin algebra Λ, the dg Leavitt algebra L = LΛ0(J) is intro- +duced in [12]. As an algebra, it is given by +L = TΛ0(J ⊕ J∗)/(a ⊗ g − g(a), 1 − c | a ∈ J, g ∈ J∗). +Here, TΛ0(J ⊕ J∗) denotes the tensor algebra. It is naturally Z-graded such that +|e| = 0 for any e ∈ Λ0, |a| = −1 for any a ∈ J and |g| = 1 for any g ∈ J∗. +The differential ∂ on L is uniquely determined by the graded Leibniz rule and the +conditions that ∂|Λ0 = 0 and ∂|J∗ = ∂+; see [12, Remark 3.6]. We mention that +the classical Leavitt algebras appear already in [21]. + +A NON-VANISHING RESULT ON THE SINGULARITY CATEGORY +7 +For any dg algebra A, we denote by H∗(A) = ⊕n∈ZHn(A) its total cohomol- +ogy, which inherits a graded algebra structure from A. Recall that A is acyclic if +H∗(A) = 0, which is equivalent to the condition that H0(A) = 0. +We view Λ0 as the corresponding stalk complex concentrated in degree zero, and +as an object in Dsg(Λ). Then the following graded k-module +� +n∈Z +HomDsg(Λ)(Λ0, Σn(Λ0)) +(3.2) +becomes a graded k-algebra, whose multiplication is induced by composition of +morphisms in Dsg(Λ). +The following result is implicitly contained in [12], and indicates the intimate +link between dg Leavitt algebras and singularity categories. +Lemma 3.8. Keep the notation as above. Then there is an isomorphism of graded +algebras +H∗(L)op ≃ +� +n∈Z +HomDsg(Λ)(Λ0, Σn(Λ0)), +where H∗(L)op denotes the opposite algebra of H∗(L). +Proof. By the isomorphism in [12, Theorem 9.5], we infer that H∗(L)op is isomor- +phic to the total cohomology algebra of the dg endomorphism algebra of Λ0 in the +singular Yoneda dg category. By the quasi-equivalence in [12, Corollary 9.3], we +infer that the latter algebra is isomorphic to (3.2). +□ +In general, the structure of the dg Leavitt algebra L seems to be very compli- +cated. The following second consequence of Theorem 2.8 is a dichotomy on its total +cohomology H∗(L). We refer to Proposition 4.3 below for a strengthened version. +Proposition 3.9. Let Λ be an artin algebra with L the associated dg Leavitt algebra. +Then the following statements hold. +(1) The dg Leavitt algebra L is acyclic if and only if Λ has finite global dimen- +sion. +(2) If Λ has infinite global dimension, then Hn(L) ̸= 0 for any integer n. +Proof. In view of Lemma 3.8, the dg Leavitt algebra L is acyclic if and only if +HomDsg(Λ)(Λ0, Λ0) = 0, which is equivalent to the vanishing of Λ0 in Dsg(Λ). Since +Λ0 generates Dsg(Λ), the last condition is equivalent to the vanishing of Dsg(Λ), +which is well known to be further equivalent to the finiteness of the global dimension +of Λ. In summary, we infer (1). +For (2), we assume that Λ has infinite global dimension. By Example 2.3, Λ0 is +virtually 1-periodic. Theorem 2.8 implies that +HomDsg(Λ)(Λ0, Σn(Λ0)) ̸= 0 +(3.3) +for any integer n. Now the required statement follows from the isomorphism in +Lemma 3.8 immediately. +□ +Remark 3.10. The inequality (3.3) is analogous to (3.1). +In the same spirit, +Proposition 3.9 is analogous to the characterization of regular local rings in [3, +Theorem 6.5] via the stable cohomology algebras of the residue fields. +4. The Hom-finiteness +Let k be a commutative ring. +We will assume that the abelian category A +is k-linear. Consequently, the singularity category Dsg(A) is k-linear. We study +the Hom-finiteness of the singularity category, and obtain a trichotomy on the +cohomologies of the dg Leavitt algebras; see Proposition 4.3. + +8 +CHEN, LI, ZHANG, ZHAO +For an object M in A and d ≥ 1, we consider a graded k-algebra +Γ(M; d) = +� +n∈Z +HomDsg(A)(M, Σnd(M)), +(4.1) +whose multiplication is induced by composition of morphisms in Dsg(A). +The proof of the following lemma resembles the one of Theorem 2.8. +Lemma 4.1. Let d ≥ 1 and M be a virtually d-periodic object in A. Assume that +X and Y are objects in Dsg(A). Then the following statements hold. +(1) Assume that the k-module HomDsg(A)(X, M) is of infinite length. Then so +is HomDsg(A)(X, Σnd(M)) for each n ≥ 0. +(2) Assume that the k-module HomDsg(A)(M, Y ) is of infinite length. Then so +is HomDsg(A)(Σnd(M), Y ) for each n ≥ 0. +(3) Each homogeneous component of Γ(M; d) is of infinite length if and only if +so is one of the homogeneous components. +Proof. We will only give the proof of (1), as (2) is proved dually and (3) follows +immediately by combining (1) and (2). +We now prove (1). Thanks to Lemma 2.2, it suffices to claim that the k-module +HomDsg(A)(X, Σd(M)) is of infinite length. By Lemma 2.7, M is isomorphic to +ΣdΩd(M). Therefore, Ωd(M) does not belong to the following full subcategory +S′′ = {Z ∈ A | HomDsg(A)(X, Σd(Z)) is of finite length}. +We observe that S′′ contains P and is closed under direct summands and extensions. +Since Ωd(M) ∈ ⟨M⟩ and Ωd(M) does not belong to S′′, it follows that M does not +belong to S′′ either. This proves the claim and thus (1). +□ +We say that Dsg(A) is Hom-finite over k, if the k-module HomDsg(A)(X, Y ) is +of finite length for any object X and Y . +The following result characterizes the Hom-finiteness of Dsg(A) using virtually +periodic objects. +Proposition 4.2. Let M be a virtually 1-periodic object in A which generates +Dsg(A). Then the following statements are equivalent. +(1) The category Dsg(A) is Hom-finite over k. +(2) One of the homogeneous components of Γ(M; 1) is nonzero and of finite +length. +(3) Each homogeneous component of Γ(M; 1) is nonzero and of finite length. +Proof. By Theorem 2.8, each homogeneous component of Γ(M; 1) is nonzero. Since +M generates Dsg(A), it is a standard fact that Dsg(A) is Hom-finite if and only if +HomDsg(A)(M, Σn(M)) is of finite length for any integer n. This proves “(1) ⇔ (2)”. +The implications “(2) ⇔ (3)” follow from Lemma 4.1(3). +□ +In what follows, we assume that Λ is an artin algebra over a commutative artinian +ring k. We keep the setup in Subsection 3.2. We mention that the Hom-finiteness +of the singularity category of certain artin algebras is studied in [11, Section 5]. +We have the following trichotomy on the Hom-finiteness of the total cohomology +algebra H∗(L) of the associated dg Leavitt algebra. +Proposition 4.3. Let Λ be an artin algebra and L be the associated dg Leavitt +algebra. Then the following statements hold. +(1) The algebra Λ has finite global dimension if and only if H∗(L) = 0. +(2) The algebra Λ has infinite global dimension and Dsg(Λ) is Hom-finite if +and only if each homogeneous component of H∗(L) is nonzero and of finite +length. + +A NON-VANISHING RESULT ON THE SINGULARITY CATEGORY +9 +(3) The category Dsg(Λ) is not Hom-finite if and only if each homogeneous +component of H∗(L) is of infinite length. +Proof. By Example 2.3, Λ0 is virtually 1-periodic and it clearly generates Dsg(Λ). +By Lemma 3.8, we identify H∗(L)op with Γ(Λ0; 1) defined in (4.1). Then the results +follow from Propositions 3.9 and 4.2. +□ +Remark 4.4. We point that Proposition 4.3(2) is somehow analogous to the char- +acterization of Gorenstein local rings in [3, Theorem 6.4]; compare Remark 3.10. +Assume that Λ is a Gorenstein artin algebra. By [9, Theorem 4.4], the singu- +larity category Dsg(Λ) is triangle equivalent to the stable category of Gorenstein- +projective Λ-modules. In particular, Dsg(Λ) is Hom-finite and thus each homoge- +neous component of H∗(L) is of finite length. In general, unlike the commutative +case, the Hom-finiteness of Dsg(Λ) does not imply the Gorensteinness of Λ; see [10, +Example 4.3] for concrete examples. +Acknowledgements. +The authors thank Takuma Aihara, Hongxing Chen, Wei +Hu and Zhengfang Wang for many helpful discussions. This work is supported by +the National Natural Science Foundation of China (No.s 12171207, 12131015 and +12161141001). +References +[1] T. Aihara, T. Honma, K. Miyamoto, and Q. Wang, Report on the finiteness of silting +objects, Proc. Edinb. Math. Soc. 64 (2) (2021), 217–233. +[2] T. Aihara, and O. Iyama, Silting mutation in triangulated categories, J. Lond. Math. +Soc. 85 (3) (2012), 633–668. +[3] L.L. Avramov, and O. Veliche, Stable cohomology of local rings, Adv. 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Usui, Characterization of eventually periodic modules in the singularity categories, J. +Pure Appl. Algebra 226 (12) (2022), 107145. +[25] B. Zimmermann-Huisgen, Predicting syzygies over monomial relation algebras, Manuscr. +Math. 70 (1991), 157–182. +Xiao-Wu Chen +Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, +School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, +Anhui, PR China +Zhi-Wei Li, Xiaojin Zhang +School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, Jiangsu, PR +China +Zhibing Zhao +School of Mathematical Sciences, Anhui University, Hefei 230601, Anhui, PR China + diff --git a/yNAzT4oBgHgl3EQf7_5U/content/tmp_files/load_file.txt b/yNAzT4oBgHgl3EQf7_5U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..38e7b1e82afb92edf5cb761dcb844720ed744c0f --- /dev/null +++ b/yNAzT4oBgHgl3EQf7_5U/content/tmp_files/load_file.txt @@ -0,0 +1,633 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf,len=632 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='01897v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='RT] 5 Jan 2023 A NON-VANISHING RESULT ON THE SINGULARITY CATEGORY XIAO-WU CHEN, ZHI-WEI LI, XIAOJIN ZHANG, ZHIBING ZHAO∗ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We prove that a virtually periodic object in an abelian category gives rise to a non-vanishing result on certain Hom groups in the singularity category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Consequently, for any artin algebra with infinite global dimension, its singularity category has no silting subcategory, and the associated differential graded Leavitt algebra has a non-vanishing cohomology in each degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We verify the Singular Presilting Conjecture for ultimately-closed algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We obtain a trichotomy on the Hom-finiteness of the cohomology of differential graded Leavitt algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Introduction The singularity category is a fundamental homological invariant for a ring with infinite global dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' It is traced back to [9] and is rediscovered in the geometric context in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' It has received increasing attention from people in different subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Recall that a module is periodic if its higher syzygy is isomorphic to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' These modules play a particular role in the singularity category [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We propose a slightly more general notion: a module is called virtually periodic if its higher syzygy lies in the extension closure of the module itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' A prototype is the semisimple quotient module of a left artinian ring modulo its Jacobson radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The central result of this paper is Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8, which states the following non- vanishing property: for a virtually d-periodic module M, the Hom groups between M and its (nd)-th suspension Σnd(M) in the singularity category are always non- vanishing for all integers n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' There are two consequences of the above non-vanishing result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The first one states that the singularity category of a left artinian ring with infinite global di- mension does not have a silting subcategory in the sense of [19, 2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' This strengthens [1, Theorem 1], and partially supports the Singular Presilting Con- jecture [13] and thus the well-known Auslander-Reiten Conjecture [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We prove that the Singular Presilting Conjecture holds for ultimately-closed algebras [17], which include periodic algebras and syzygy-finite algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The second consequence states that the differential graded Leavitt algebra [12] associated to an artin algebra with infinite global dimension has a non-vanishing cohomology in each degree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In Section 4, we use virtually periodic objects to characterize the Hom-finiteness of the singularity category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We obtain a trichotomy on the cohomlogies of the differential graded Leavitt algebras associated to artin algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We mention that the results on Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' 16E05, 18G80, 16S88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' singularity category, periodic module, silting subcategory, Leavitt algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' ∗ The corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' xwchen@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='cn, zhiweili@jsnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='cn, xjzhang@jsnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='cn, zbzhao@ahu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' 1 2 CHEN, LI, ZHANG, ZHAO differential graded Leavitt algebras are analogous to the ones in [3] on the stable cohomology algebras of the residue fields of commutative noetherian local rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We will denote the suspension functor in any triangulated category by Σ, and write dg for ‘differential graded’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' For triangulated categories, we refer to [15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' for artin algebras, we refer to [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Virtually periodic objects Let A be an abelian category with enough projective objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The latter con- dition means that for each object M, there is an epimorphism P → M with P projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We denote by P the full subcategory formed by all projective objects, and by A the stable category of A modulo morphisms factoring through projective objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' For any object M, the first syszygy Ω(M) of M is defined to be the kernel of any epimorphism P → M with P projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We mention that Ω(M) is uniquely defined in the stable category A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Denote by ⟨M⟩ the smallest full subcategory of A, which contains {M} ∪ P and is closed under direct summands and extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The extension-closed condition means that for any short exact sequence 0 → X → Y → Z → 0 with X, Z ∈ ⟨M⟩, we have that Y necessarily lies in ⟨M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Let d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Recall that a non-projective object M is called d-periodic if there is an isomorphism Ωd(M) ≃ M in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' A d-periodic object necessarily has infinite projective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The study of periodic modules is traced back to [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We mention that periodic modules play a role in the singularity category [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Let d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' An object M in A is said to be virtually d-periodic provided that M has infinite projective dimension and that Ωd(M) lies in ⟨M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We observe that a d-periodic object is virtually d-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The following fact is easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Assume that M is virtually d-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then M is virtually (nd)- periodic for any n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We use induction on n and assume that M is virtually (nd)-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By applying Horseshoe Lemma and the fact that Ωnd(M) ∈ ⟨M⟩, we infer that Ω(n+1)d(M) = Ωd(Ωnd(M)) ∈ ⟨Ωd(M)⟩ ⊆ ⟨M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Here, the latter inclusion uses the fact that Ωd(M) ∈ ⟨M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We infer that M is also virtually (n + 1)d-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' □ For a left noetherian ring Λ, we denote by Λ-mod the abelian category of finitely generated left Λ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The following examples motivate Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Let Λ be a left artinian ring with infinite global dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Set Λ0 = Λ/J with J its Jacboson radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then Λ0 is virtually 1-periodic in Λ-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Indeed, the projective dimension of Λ0 is equal to the global dimension of Λ, and thus is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Moreover, Ω(Λ0) is clearly an iterated extension of simple modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then we infer that Ω(Λ0) ∈ ⟨Λ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' For an object X in A, we denote by add X the smallest full additive subcategory which contains X and is closed under direct summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Recall from [17, Section 3] that an object M in A is ultimately- closed, if there exist d ≥ 1 and some object X in add (M ⊕ Ω(M)⊕ · · · ⊕ Ωd−1(M)) such that Ωd(M) and X are isomorphic in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In this case, the object M ⊕ Ω(M) ⊕ · · ⊕ Ωd−1(M) is virtually 1-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The abelian category A is called ultimately-closed, if any object is ultimately- closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Following [17, Section 3], a left noetherian ring Λ is ultimately-closed if Λ-mod is ultimately-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' A NON-VANISHING RESULT ON THE SINGULARITY CATEGORY 3 Recall that a hypersurface ring is of the form S/(x) with S a regular local ring and x a nonzero element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By [14, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1], the higher syzygy of each module over a hypersurface ring is 2-periodic and thus each module is ultimately- closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Consequently, a hypersurface ring is ultimately-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Recall that a finite dimensional algebra over a field is periodic if it has a periodic bimodule resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see [8] for concrete examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By a similar reasoning, we infer that a periodic algebra is ultimately-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In what follows, we give further classes of ultimately-closed rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' For any d ≥ 1, we denote by Ωd(A) the full subcategory of A formed by those objects that are isomorphic to Ωd(M) in A for some object M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The abelian category A is syzygy-finite provided that there exist d ≥ 1 and an object E such that Ωd(A) ⊆ add E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In this case, the object E is virtually d-periodic provided that A has infinite global dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Moreover, if such an object E already belongs to Ωd(A), then E is virtually 1-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We observe that a syzygy-finite Krull-Schmidt abelian category is necessarily ultimately-closed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' compare [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' For example, let Λ be a left artinian ring which is syzygy-finite, that is, Λ-mod is syzygy-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then the module category Λ-mod is ultimately-closed, and thus Λ is ultimately-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We mention that syzygy-finite artinian rings include artinian rings of finite rep- resentation type, artinian rings with square-zero Jacobson radical, and finite di- mensional monomial algebras by [25, Theorem I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We assume that R is a commutative noetherian complete local ring, which is non-regular and Cohen-Macaulay of finite Cohen-Macaulay-type [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We take E to be the direct sum of all indecomposable maximal Cohen-Macaulay R- modules, and let d be the Krull dimension of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then we have Ωd(R-mod) ⊆ add E, because the d-th syzygy of any finitely generated R-module is maximal Cohen- Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In particular, the category R-mod is syzygy-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The completeness of R implies that R-mod is Krull-Schmidt, and thus is ultimately-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Therefore, the ring R is ultimately-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We assume in addition that R is Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then we have Ωd(R-mod) = add E, in which case E is virtually 1-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' For another choice of such a module E, we refer to [18, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Denote by Db(A) the bounded derived category of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Using the canonical func- tor, we view the bounded homotopy category Kb(P) as a thick triangulated sub- category of Db(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Following [9], the singularity category of A is defined to be the following Verdier quotient triangulated category Dsg(A) = Db(A)/Kb(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' As usual, we identify any object M in A with the corresponding stalk complex concentrated in degree zero, which is still denoted by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The latter is also viewed as an object in Dsg(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Consequently, Σn(M) will mean the corresponding stalk complex concentrated in degree −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The following fact is well known;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' compare [11, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2] and [9, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Let M be an object in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then there is an isomorphism M ≃ ΣΩ(M) in Dsg(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Recall that any short exact sequence in A induces canonically an exact triangle in Db(A) and thus an exact triangle in Dsg(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We consider the following exact sequence 0 → Ω(M) → P → M → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' As the projective object P vanishes in Dsg(A), the induced exact triangle in Dsg(A) is the form Ω(M) −→ 0 −→ M −→ ΣΩ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' 4 CHEN, LI, ZHANG, ZHAO By [15, Lemma I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='7], we infer that the morphism M → ΣΩ(M) is an isomorphism, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Let M be a virtually d-periodic object in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then we have HomDsg(A)(M, Σnd(M)) ̸= 0 for any integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In view of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2, it suffices to prove that HomDsg(A)(M, Σd(M)) ̸= 0 ̸= HomDsg(A)(M, Σ−d(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Since M has infinite projective dimension, it does not vanish in Dsg(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Conse- quently, we have HomDsg(A)(M, M) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='7, we have an isomorphism M ≃ ΣdΩd(M) in Dsg(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In particular, we have HomDsg(A)(M, ΣdΩd(M)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1) We claim that HomDsg(A)(M, Σd(M)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Otherwise, the object M belongs to the following full subcategory S = {X ∈ A | HomDsg(A)(M, Σd(X)) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' As any short exact sequence in A induces an exact triangle in Dsg(A), it follows that S is closed under extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Clearly, it contains P and is closed under direct summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' It follows that S contains ⟨M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Since M is virtually d-periodic, we have that Ωd(M) belongs to ⟨M⟩ and thus is contained in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' This contradicts to the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1), and proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Dually, we observe that HomDsg(A)(ΣdΩd(M), M) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Consider the full subcategory S′ = {Y ∈ A | HomDsg(A)(Σd(Y ), M) = 0}, which contains P and is closed under direct summands and extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By a dual argument as above, we prove that HomDsg(A)(Σd(M), M) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' This completes the whole proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Two consequences In this section, we draw two consequences of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We show that the singularity category of a left artinian ring with infinite global dimension does not have a silting subcategory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We verify the Singular Presilting Conjecture for ultimately-closed artin algebras in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' compare Re- mark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We prove that the dg Leavitt algebra [12] associated to any artin algebra with infinite global dimension has a non-vanishing cohomology in each degree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Silting subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Let T be a triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Recall from [2, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1] that a full additive subcategory M is called silting, provided that the following two conditions are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' (1) The subcategory M is presilting, that is, HomT (M, Σn(M)) = 0 for any M ∈ M and n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' (2) The subcategory M generates T in the sense that T itself is the smallest thick triangulated subcategory containing M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' A NON-VANISHING RESULT ON THE SINGULARITY CATEGORY 5 An object X is called presilting (respectively, silting) provided that add X is a presilting (respectively, silting) subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We mention that the study of silting objects goes back to [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The following result is due to [2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Assume that T has a silting subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then for any object X, HomT (X, Σd(X)) = 0 for sufficiently large d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' □ In what follows, A is an abelian category with enough projective objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We have the first consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Assume that A contains a virtually d-periodic object for some d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then Dsg(A) has no silting subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Take M be a virtually d-periodic object in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8 implies that HomDsg(A)(M, Σnd(M)) ̸= 0 for any integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In particular, n can be sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1, we have the required non-existence of a silting subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' □ For a left noetherian ring Λ, we usually write Dsg(Λ) for Dsg(Λ-mod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The following result strengthens [1, Theorem 1], where the corresponding result is proved under a finiteness assumption on the selfinjective dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' compare [2, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='5(b)] and [16, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The argument here is completely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Let Λ be a left artinian ring with infinite global dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then Dsg(Λ) has no silting subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3, the semisimple Λ-module Λ0 is virtually 1-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then we apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' □ The above non-existence partially supports the following conjecture [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' com- pare [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Singular Presilting Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' For any artin algebra Λ, there is no nonzero presilting subcategory in Dsg(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By [16, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4] or [23, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='21], this conjecture implies the following well-known conjecture, proposed in [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='70];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' compare [13, Section 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Auslander-Reiten Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' For a non-projective module M over any artin algebra Λ, we have Extn Λ(M, M ⊕ Λ) ̸= 0 for some n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We mention that, by [9, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4], the Singular Presilting Conjecture for Λ is equivalent to the Auslander-Reiten Conjecture for Λ, provided that Λ is a Gorenstein artin algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The following result implies that the Singular Presilting Conjecture holds for ultimately-closed artin algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' compare [6, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Recall that finite dimensional periodic algebras and syzygy-finite artin algebras are ultimately-closed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see Section 2 or [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Assume that A is ultimately-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then Dsg(A) has no nonzero presilting subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In particular, for a ultimately-closed ring Λ, there is no nonzero presilting subcategory in Dsg(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' It suffices to prove that Dsg(A) has no nonzero presilting object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Assume that X is a nonzero presilting object in Dsg(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Set T to be the smallest thick subcategory containing X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By [11, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1], there exists an object M in A such that X is isomorphic to Σn(M) for some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Since M is ultimately-closed, 6 CHEN, LI, ZHANG, ZHAO there exists d ≥ 1 such that E = M ⊕Ω(M)⊕· · ·⊕Ωd−1(M) is virtually 1-periodic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In view of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='7, the object E belongs to T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' However, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8, we have HomT (E, Σn(E)) ̸= 0 for any integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Since X is a silting object of T , we have a desired contradiction by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' It seems that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4 might be strengthened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The abelian category A is said be virtually ultimately-closed if for each object M, there exists d ≥ 1 such that Ωd(M) lies in ⟨M ⊕Ω(M)⊕· · ·⊕Ωd−1(M)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In this case, the object M ⊕ Ω(M) ⊕ · · · ⊕ Ωd−1(M) is still virtually 1-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then the same argument above implies that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4 holds for virtually ultimately-closed categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' However, we do not know any virtually ultimately-closed category, which is not ultimately-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Unlike the ungraded case, the graded singularity category of a graded artin algebra might have a silting object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' for example, see [22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3] and [20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We mention the following known non-existence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Let R be a commutative noetherian local ring, which is non- regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then Dsg(R) has no silting subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Denote by k the residue field of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By [3, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='5], we infer that HomDsg(R)(k, Σn(k)) ̸= 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1) for any integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Here, we identify HomDsg(R)(k, Σn(k)) with the stable co- homology group � Ext n R(k, k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see [3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then the non-existence follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The dg Leavitt algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In this subsection, we assume that Λ is an artin algebra over a commutative artinian ring k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Recall that Λ0 = Λ/J with J its Jacobson radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We will assume that Λ0 is a subalgebra of Λ with a decomposition Λ = Λ0 ⊕ J of Λ0-Λ0-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' This assumption holds if Λ is given by a finite quiver with admissible relations, or if Λ is a finite dimensional algebra over a perfect field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Consider the left Λ0-dual J∗ = Hom(J, Λ0) of J, which carries a natural Λ0-Λ0- bimodule structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We have the Casimir element c = � i∈S α∗ i ⊗ αi ∈ J∗ ⊗ J, where {αi | i ∈ S} and {α∗ i | i ∈ S} form the dual basis of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The multiplication on J induces a map of Λ0-Λ0-bimodules ∂+ : J∗ −→ J∗ ⊗ J∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' To be more precise, we have ∂+(g) = � g1 ⊗ g2 such that g(ab) = � g2(ag1(b)) for any a, b ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Associated to the artin algebra Λ, the dg Leavitt algebra L = LΛ0(J) is intro- duced in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' As an algebra, it is given by L = TΛ0(J ⊕ J∗)/(a ⊗ g − g(a), 1 − c | a ∈ J, g ∈ J∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Here, TΛ0(J ⊕ J∗) denotes the tensor algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' It is naturally Z-graded such that |e| = 0 for any e ∈ Λ0, |a| = −1 for any a ∈ J and |g| = 1 for any g ∈ J∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The differential ∂ on L is uniquely determined by the graded Leibniz rule and the conditions that ∂|Λ0 = 0 and ∂|J∗ = ∂+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see [12, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We mention that the classical Leavitt algebras appear already in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' A NON-VANISHING RESULT ON THE SINGULARITY CATEGORY 7 For any dg algebra A, we denote by H∗(A) = ⊕n∈ZHn(A) its total cohomol- ogy, which inherits a graded algebra structure from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Recall that A is acyclic if H∗(A) = 0, which is equivalent to the condition that H0(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We view Λ0 as the corresponding stalk complex concentrated in degree zero, and as an object in Dsg(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then the following graded k-module � n∈Z HomDsg(Λ)(Λ0, Σn(Λ0)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2) becomes a graded k-algebra, whose multiplication is induced by composition of morphisms in Dsg(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The following result is implicitly contained in [12], and indicates the intimate link between dg Leavitt algebras and singularity categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Keep the notation as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then there is an isomorphism of graded algebras H∗(L)op ≃ � n∈Z HomDsg(Λ)(Λ0, Σn(Λ0)), where H∗(L)op denotes the opposite algebra of H∗(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By the isomorphism in [12, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='5], we infer that H∗(L)op is isomor- phic to the total cohomology algebra of the dg endomorphism algebra of Λ0 in the singular Yoneda dg category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By the quasi-equivalence in [12, Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3], we infer that the latter algebra is isomorphic to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' □ In general, the structure of the dg Leavitt algebra L seems to be very compli- cated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The following second consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8 is a dichotomy on its total cohomology H∗(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We refer to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3 below for a strengthened version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Let Λ be an artin algebra with L the associated dg Leavitt algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' (1) The dg Leavitt algebra L is acyclic if and only if Λ has finite global dimen- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' (2) If Λ has infinite global dimension, then Hn(L) ̸= 0 for any integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8, the dg Leavitt algebra L is acyclic if and only if HomDsg(Λ)(Λ0, Λ0) = 0, which is equivalent to the vanishing of Λ0 in Dsg(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Since Λ0 generates Dsg(Λ), the last condition is equivalent to the vanishing of Dsg(Λ), which is well known to be further equivalent to the finiteness of the global dimension of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In summary, we infer (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' For (2), we assume that Λ has infinite global dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3, Λ0 is virtually 1-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8 implies that HomDsg(Λ)(Λ0, Σn(Λ0)) ̸= 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3) for any integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Now the required statement follows from the isomorphism in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8 immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3) is analogous to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In the same spirit, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='9 is analogous to the characterization of regular local rings in [3, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='5] via the stable cohomology algebras of the residue fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The Hom-finiteness Let k be a commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We will assume that the abelian category A is k-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Consequently, the singularity category Dsg(A) is k-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We study the Hom-finiteness of the singularity category, and obtain a trichotomy on the cohomologies of the dg Leavitt algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' 8 CHEN, LI, ZHANG, ZHAO For an object M in A and d ≥ 1, we consider a graded k-algebra Γ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' d) = � n∈Z HomDsg(A)(M, Σnd(M)), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1) whose multiplication is induced by composition of morphisms in Dsg(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The proof of the following lemma resembles the one of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Let d ≥ 1 and M be a virtually d-periodic object in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Assume that X and Y are objects in Dsg(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' (1) Assume that the k-module HomDsg(A)(X, M) is of infinite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then so is HomDsg(A)(X, Σnd(M)) for each n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' (2) Assume that the k-module HomDsg(A)(M, Y ) is of infinite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then so is HomDsg(A)(Σnd(M), Y ) for each n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' (3) Each homogeneous component of Γ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' d) is of infinite length if and only if so is one of the homogeneous components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We will only give the proof of (1), as (2) is proved dually and (3) follows immediately by combining (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We now prove (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Thanks to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2, it suffices to claim that the k-module HomDsg(A)(X, Σd(M)) is of infinite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='7, M is isomorphic to ΣdΩd(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Therefore, Ωd(M) does not belong to the following full subcategory S′′ = {Z ∈ A | HomDsg(A)(X, Σd(Z)) is of finite length}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We observe that S′′ contains P and is closed under direct summands and extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Since Ωd(M) ∈ ⟨M⟩ and Ωd(M) does not belong to S′′, it follows that M does not belong to S′′ either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' This proves the claim and thus (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' □ We say that Dsg(A) is Hom-finite over k, if the k-module HomDsg(A)(X, Y ) is of finite length for any object X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The following result characterizes the Hom-finiteness of Dsg(A) using virtually periodic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Let M be a virtually 1-periodic object in A which generates Dsg(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then the following statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' (1) The category Dsg(A) is Hom-finite over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' (2) One of the homogeneous components of Γ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' 1) is nonzero and of finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' (3) Each homogeneous component of Γ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' 1) is nonzero and of finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8, each homogeneous component of Γ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' 1) is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Since M generates Dsg(A), it is a standard fact that Dsg(A) is Hom-finite if and only if HomDsg(A)(M, Σn(M)) is of finite length for any integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' This proves “(1) ⇔ (2)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The implications “(2) ⇔ (3)” follow from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' □ In what follows, we assume that Λ is an artin algebra over a commutative artinian ring k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We keep the setup in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We mention that the Hom-finiteness of the singularity category of certain artin algebras is studied in [11, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We have the following trichotomy on the Hom-finiteness of the total cohomology algebra H∗(L) of the associated dg Leavitt algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Let Λ be an artin algebra and L be the associated dg Leavitt algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' (1) The algebra Λ has finite global dimension if and only if H∗(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' (2) The algebra Λ has infinite global dimension and Dsg(Λ) is Hom-finite if and only if each homogeneous component of H∗(L) is nonzero and of finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' A NON-VANISHING RESULT ON THE SINGULARITY CATEGORY 9 (3) The category Dsg(Λ) is not Hom-finite if and only if each homogeneous component of H∗(L) is of infinite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3, Λ0 is virtually 1-periodic and it clearly generates Dsg(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='8, we identify H∗(L)op with Γ(Λ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' 1) defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Then the results follow from Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' We point that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3(2) is somehow analogous to the char- acterization of Gorenstein local rings in [3, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' compare Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Assume that Λ is a Gorenstein artin algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' By [9, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='4], the singu- larity category Dsg(Λ) is triangle equivalent to the stable category of Gorenstein- projective Λ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In particular, Dsg(Λ) is Hom-finite and thus each homoge- neous component of H∗(L) is of finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' In general, unlike the commutative case, the Hom-finiteness of Dsg(Λ) does not imply the Gorensteinness of Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' see [10, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='3] for concrete examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' The authors thank Takuma Aihara, Hongxing Chen, Wei Hu and Zhengfang Wang for many helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content=' This work is supported by the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQf7_5U/content/2301.01897v1.pdf'} +page_content='s 12171207, 12131015 and 12161141001).' metadata={'source': 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